Add Batch 1828f18b-46df-41dd-bc12-4e239a236b96 data

.gitattributes

@@ -425,3 +425,67 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
 2025/Beyond[[:space:]]Self-Interest_[[:space:]]How[[:space:]]Group[[:space:]]Strategies[[:space:]]Reshape[[:space:]]Content[[:space:]]Creation[[:space:]]in[[:space:]]Recommendation[[:space:]]Platforms_/aa862d38-206b-4c35-a04f-7045f6c8b99b_origin.pdf filter=lfs diff=lfs merge=lfs -text
 2025/Beyond[[:space:]]Self-Repellent[[:space:]]Kernels_[[:space:]]History-Driven[[:space:]]Target[[:space:]]Towards[[:space:]]Efficient[[:space:]]Nonlinear[[:space:]]MCMC[[:space:]]on[[:space:]]General[[:space:]]Graphs/be47e927-4760-4239-a8fa-9e65b93f704b_origin.pdf filter=lfs diff=lfs merge=lfs -text
 2025/Beyond[[:space:]]Sensor[[:space:]]Data_[[:space:]]Foundation[[:space:]]Models[[:space:]]of[[:space:]]Behavioral[[:space:]]Data[[:space:]]from[[:space:]]Wearables[[:space:]]Improve[[:space:]]Health[[:space:]]Predictions/523d721e-7c03-405d-aabf-ba82245c9d27_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Beyond[[:space:]]Task-Specific[[:space:]]Reasoning_[[:space:]]A[[:space:]]Unified[[:space:]]Conditional[[:space:]]Generative[[:space:]]Framework[[:space:]]for[[:space:]]Abstract[[:space:]]Visual[[:space:]]Reasoning/187a29f7-5752-44f7-a8d5-732c4e65a8cc_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Beyond[[:space:]]The[[:space:]]Rainbow_[[:space:]]High[[:space:]]Performance[[:space:]]Deep[[:space:]]Reinforcement[[:space:]]Learning[[:space:]]on[[:space:]]a[[:space:]]Desktop[[:space:]]PC/da17f283-f8ac-4488-b3db-031049104729_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Beyond[[:space:]]Topological[[:space:]]Self-Explainable[[:space:]]GNNs_[[:space:]]A[[:space:]]Formal[[:space:]]Explainability[[:space:]]Perspective/b9cd7a30-96a8-41ab-ad86-4786cf1b9eef_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Beyond[[:space:]]Zero[[:space:]]Initialization_[[:space:]]Investigating[[:space:]]the[[:space:]]Impact[[:space:]]of[[:space:]]Non-Zero[[:space:]]Initialization[[:space:]]on[[:space:]]LoRA[[:space:]]Fine-Tuning[[:space:]]Dynamics/9bd2a800-d91c-4f7b-970e-a45ea7881131_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Beyond[[:space:]]the[[:space:]]Permutation[[:space:]]Symmetry[[:space:]]of[[:space:]]Transformers_[[:space:]]The[[:space:]]Role[[:space:]]of[[:space:]]Rotation[[:space:]]for[[:space:]]Model[[:space:]]Fusion/37b9cc0d-806d-4c07-8dae-3066093d0e30_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bi-perspective[[:space:]]Splitting[[:space:]]Defense_[[:space:]]Achieving[[:space:]]Clean-Seed-Free[[:space:]]Backdoor[[:space:]]Security/1a2572fb-8fce-427d-a09e-f6c5c0d94ca3_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/BiAssemble_[[:space:]]Learning[[:space:]]Collaborative[[:space:]]Affordance[[:space:]]for[[:space:]]Bimanual[[:space:]]Geometric[[:space:]]Assembly/332e0677-babf-4a85-9347-30c2a8d88e3c_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/BiMaCoSR_[[:space:]]Binary[[:space:]]One-Step[[:space:]]Diffusion[[:space:]]Model[[:space:]]Leveraging[[:space:]]Flexible[[:space:]]Matrix[[:space:]]Compression[[:space:]]for[[:space:]]Real[[:space:]]Super-Resolution/015a9533-715f-4e8b-a7b5-d0f1265ce5d2_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/BiMark_[[:space:]]Unbiased[[:space:]]Multilayer[[:space:]]Watermarking[[:space:]]for[[:space:]]Large[[:space:]]Language[[:space:]]Models/873e68de-dae3-493c-9d23-2ce173b24e84_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bifurcate[[:space:]]then[[:space:]]Alienate_[[:space:]]Incomplete[[:space:]]Multi-view[[:space:]]Clustering[[:space:]]via[[:space:]]Coupled[[:space:]]Distribution[[:space:]]Learning[[:space:]]with[[:space:]]Linear[[:space:]]Overhead/1a09335b-24a2-4c4d-8edc-f9d0b1270c6a_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Binary[[:space:]]Hypothesis[[:space:]]Testing[[:space:]]for[[:space:]]Softmax[[:space:]]Models[[:space:]]and[[:space:]]Leverage[[:space:]]Score[[:space:]]Models/91628084-66ed-43fd-99ba-f96918f6eee5_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/BinauralFlow_[[:space:]]A[[:space:]]Causal[[:space:]]and[[:space:]]Streamable[[:space:]]Approach[[:space:]]for[[:space:]]High-Quality[[:space:]]Binaural[[:space:]]Speech[[:space:]]Synthesis[[:space:]]with[[:space:]]Flow[[:space:]]Matching[[:space:]]Models/fb862c9c-e710-433d-a0b5-bd97a172282c_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bipartite[[:space:]]Ranking[[:space:]]From[[:space:]]Multiple[[:space:]]Labels_[[:space:]]On[[:space:]]Loss[[:space:]]Versus[[:space:]]Label[[:space:]]Aggregation/2fb9a425-c1a3-47de-b5f9-dcfab725c5ec_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bivariate[[:space:]]Causal[[:space:]]Discovery[[:space:]]with[[:space:]]Proxy[[:space:]]Variables_[[:space:]]Integral[[:space:]]Solving[[:space:]]and[[:space:]]Beyond/2bd3b2a7-300d-42a2-9891-a5365b73244e_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Black-Box[[:space:]]Adversarial[[:space:]]Attacks[[:space:]]on[[:space:]]LLM-Based[[:space:]]Code[[:space:]]Completion/ef48806d-7e74-4207-9b3a-2bd30865a2dc_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Blink[[:space:]]of[[:space:]]an[[:space:]]eye_[[:space:]]a[[:space:]]simple[[:space:]]theory[[:space:]]for[[:space:]]feature[[:space:]]localization[[:space:]]in[[:space:]]generative[[:space:]]models/f61fe0a6-acb2-4140-8ebf-4ff50e5cc12f_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/BlockDialect_[[:space:]]Block-wise[[:space:]]Fine-grained[[:space:]]Mixed[[:space:]]Format[[:space:]]Quantization[[:space:]]for[[:space:]]Energy-Efficient[[:space:]]LLM[[:space:]]Inference/865b60ba-7de8-4d84-a477-e959ff516328_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/BoA_[[:space:]]Attention-aware[[:space:]]Post-training[[:space:]]Quantization[[:space:]]without[[:space:]]Backpropagation/d1353411-68e4-45a9-afaf-103ad251cf8a_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bongard[[:space:]]in[[:space:]]Wonderland_[[:space:]]Visual[[:space:]]Puzzles[[:space:]]that[[:space:]]Still[[:space:]]Make[[:space:]]AI[[:space:]]Go[[:space:]]Mad_/a68e8431-bce4-438f-a1d1-8aa1e309a3eb_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Boost-and-Skip_[[:space:]]A[[:space:]]Simple[[:space:]]Guidance-Free[[:space:]]Diffusion[[:space:]]for[[:space:]]Minority[[:space:]]Generation/6300f6f2-049a-4f4a-b60b-59cfa35d8460_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Boosting[[:space:]]Adversarial[[:space:]]Robustness[[:space:]]with[[:space:]]CLAT_[[:space:]]Criticality[[:space:]]Leveraged[[:space:]]Adversarial[[:space:]]Training/da4d386b-4d0f-4ac2-8a2f-ccec05c4f23f_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Boosting[[:space:]]Masked[[:space:]]ECG-Text[[:space:]]Auto-Encoders[[:space:]]as[[:space:]]Discriminative[[:space:]]Learners/7acb7957-1ecb-457b-9f74-28ccfcbfb26a_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Boosting[[:space:]]Multi-Domain[[:space:]]Fine-Tuning[[:space:]]of[[:space:]]Large[[:space:]]Language[[:space:]]Models[[:space:]]through[[:space:]]Evolving[[:space:]]Interactions[[:space:]]between[[:space:]]Samples/79d1dff8-2c04-4365-bf64-c99651f42b3e_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Boosting[[:space:]]Protein[[:space:]]Graph[[:space:]]Representations[[:space:]]through[[:space:]]Static-Dynamic[[:space:]]Fusion/18017fd7-14c9-4493-b772-f659933b7127_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Boosting[[:space:]]Virtual[[:space:]]Agent[[:space:]]Learning[[:space:]]and[[:space:]]Reasoning_[[:space:]]A[[:space:]]Step-Wise,[[:space:]]Multi-Dimensional,[[:space:]]and[[:space:]]Generalist[[:space:]]Reward[[:space:]]Model[[:space:]]with[[:space:]]Benchmark/18edb47e-987c-4afb-aec0-4388dc5d59aa_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bootstrapping[[:space:]]Self-Improvement[[:space:]]of[[:space:]]Language[[:space:]]Model[[:space:]]Programs[[:space:]]for[[:space:]]Zero-Shot[[:space:]]Schema[[:space:]]Matching/529f0a7b-e63a-4255-8e1a-c08eb8247304_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/BounDr.E_[[:space:]]Predicting[[:space:]]Drug-likeness[[:space:]]via[[:space:]]Biomedical[[:space:]]Knowledge[[:space:]]Alignment[[:space:]]and[[:space:]]EM-like[[:space:]]One-Class[[:space:]]Boundary[[:space:]]Optimization/2f2fe395-3ccc-4d4c-ae28-69687be128c2_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bounded[[:space:]]Rationality[[:space:]]for[[:space:]]LLMs_[[:space:]]Satisficing[[:space:]]Alignment[[:space:]]at[[:space:]]Inference-Time/fff3349a-df2f-480a-bc1c-30d290a49578_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/BoxLM_[[:space:]]Unifying[[:space:]]Structures[[:space:]]and[[:space:]]Semantics[[:space:]]of[[:space:]]Medical[[:space:]]Concepts[[:space:]]for[[:space:]]Diagnosis[[:space:]]Prediction[[:space:]]in[[:space:]]Healthcare/f2e94902-9fe9-4338-904d-2f3b630b3d05_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Branches_[[:space:]]Efficiently[[:space:]]Seeking[[:space:]]Optimal[[:space:]]Sparse[[:space:]]Decision[[:space:]]Trees[[:space:]]via[[:space:]]AO_/4e7f2975-584e-4bee-b443-882624b0709f_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Breaking[[:space:]]Barriers_[[:space:]]Combinatorial[[:space:]]Algorithms[[:space:]]for[[:space:]]Non-Monotone[[:space:]]Submodular[[:space:]]Maximization[[:space:]]with[[:space:]]Sublinear[[:space:]]Adaptivity[[:space:]]and[[:space:]]$1_e$[[:space:]]Approximation/66ba16d7-1eec-4193-ae4e-fe5c518d7395_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Breaking[[:space:]]Silos_[[:space:]]Adaptive[[:space:]]Model[[:space:]]Fusion[[:space:]]Unlocks[[:space:]]Better[[:space:]]Time[[:space:]]Series[[:space:]]Forecasting/5150ff09-1c78-466e-b8fa-9fabef83a018_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Breaking[[:space:]]the[[:space:]]$n^1.5$[[:space:]]Additive[[:space:]]Error[[:space:]]Barrier[[:space:]]for[[:space:]]Private[[:space:]]and[[:space:]]Efficient[[:space:]]Graph[[:space:]]Sparsification[[:space:]]via[[:space:]]Private[[:space:]]Expander[[:space:]]Decomposition/1d258e37-9ce9-450b-be82-dabf0d7c387a_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Breaking[[:space:]]the[[:space:]]Barrier[[:space:]]of[[:space:]]Hard[[:space:]]Samples_[[:space:]]A[[:space:]]Data-Centric[[:space:]]Approach[[:space:]]to[[:space:]]Synthetic[[:space:]]Data[[:space:]]for[[:space:]]Medical[[:space:]]Tasks/1ddeae5d-54be-46cb-89aa-becb33f8ee44_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Breaking[[:space:]]the[[:space:]]Curse[[:space:]]of[[:space:]]Multiagency[[:space:]]in[[:space:]]Robust[[:space:]]Multi-Agent[[:space:]]Reinforcement[[:space:]]Learning/8805932f-46cb-456e-9863-1be53738a0e9_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Breaking[[:space:]]the[[:space:]]Quadratic[[:space:]]Barrier_[[:space:]]Robust[[:space:]]Cardinality[[:space:]]Sketches[[:space:]]for[[:space:]]Adaptive[[:space:]]Queries/57a6c149-665b-4ccc-a483-c61a5f5c2200_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bridging[[:space:]]Fairness[[:space:]]and[[:space:]]Efficiency[[:space:]]in[[:space:]]Conformal[[:space:]]Inference_[[:space:]]A[[:space:]]Surrogate-Assisted[[:space:]]Group-Clustered[[:space:]]Approach/af420e35-5043-4527-9ef9-59430b68cef8_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bridging[[:space:]]Layout[[:space:]]and[[:space:]]RTL_[[:space:]]Knowledge[[:space:]]Distillation[[:space:]]based[[:space:]]Timing[[:space:]]Prediction/2c436b80-250d-4c7c-a424-502c72cff453_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bridging[[:space:]]Protein[[:space:]]Sequences[[:space:]]and[[:space:]]Microscopy[[:space:]]Images[[:space:]]with[[:space:]]Unified[[:space:]]Diffusion[[:space:]]Models/aa80d97d-ee5b-4091-a91c-ec96d1f16b14_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Bring[[:space:]]Reason[[:space:]]to[[:space:]]Vision_[[:space:]]Understanding[[:space:]]Perception[[:space:]]and[[:space:]]Reasoning[[:space:]]through[[:space:]]Model[[:space:]]Merging/0458ce9d-4b76-4e61-96ff-a09b32b1410f_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Broadband[[:space:]]Ground[[:space:]]Motion[[:space:]]Synthesis[[:space:]]by[[:space:]]Diffusion[[:space:]]Model[[:space:]]with[[:space:]]Minimal[[:space:]]Condition/cf0daa6b-16e7-4e31-a540-e6042791e2ea_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Byzantine-Resilient[[:space:]]Federated[[:space:]]Alternating[[:space:]]Gradient[[:space:]]Descent[[:space:]]and[[:space:]]Minimization[[:space:]]for[[:space:]]Partly-Decoupled[[:space:]]Low[[:space:]]Rank[[:space:]]Matrix[[:space:]]Learning/3d96d90c-766b-42b7-b118-de9e85019a12_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/C-3PO_[[:space:]]Compact[[:space:]]Plug-and-Play[[:space:]]Proxy[[:space:]]Optimization[[:space:]]to[[:space:]]Achieve[[:space:]]Human-like[[:space:]]Retrieval-Augmented[[:space:]]Generation/c22ecf23-77b4-4844-a3cf-9909e05de3cb_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/C2IQL_[[:space:]]Constraint-Conditioned[[:space:]]Implicit[[:space:]]Q-learning[[:space:]]for[[:space:]]Safe[[:space:]]Offline[[:space:]]Reinforcement[[:space:]]Learning/7cb6d6fc-8203-4837-89da-e088921c566a_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CABS_[[:space:]]Conflict-Aware[[:space:]]and[[:space:]]Balanced[[:space:]]Sparsification[[:space:]]for[[:space:]]Enhancing[[:space:]]Model[[:space:]]Merging/c5b5761d-711b-42af-b71e-11c36de675c1_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CACTI_[[:space:]]Leveraging[[:space:]]Copy[[:space:]]Masking[[:space:]]and[[:space:]]Contextual[[:space:]]Information[[:space:]]to[[:space:]]Improve[[:space:]]Tabular[[:space:]]Data[[:space:]]Imputation/2ea1931e-2603-4792-afd0-2159864145e9_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CAD-Editor_[[:space:]]A[[:space:]]Locate-then-Infill[[:space:]]Framework[[:space:]]with[[:space:]]Automated[[:space:]]Training[[:space:]]Data[[:space:]]Synthesis[[:space:]]for[[:space:]]Text-Based[[:space:]]CAD[[:space:]]Editing/c3a1e3f6-48f5-4c8d-9118-7dfa72794663_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CALM_[[:space:]]Consensus-Aware[[:space:]]Localized[[:space:]]Merging[[:space:]]for[[:space:]]Multi-Task[[:space:]]Learning/3c6e192b-48ab-428e-b48e-eead98a9c732_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CAN_[[:space:]]Leveraging[[:space:]]Clients[[:space:]]As[[:space:]]Navigators[[:space:]]for[[:space:]]Generative[[:space:]]Replay[[:space:]]in[[:space:]]Federated[[:space:]]Continual[[:space:]]Learning/c2f93a9e-7f31-4e52-898f-24f5437bfe1e_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CASE-Bench_[[:space:]]Context-Aware[[:space:]]SafEty[[:space:]]Benchmark[[:space:]]for[[:space:]]Large[[:space:]]Language[[:space:]]Models/10d50aa8-e709-4a20-9d47-5baa852f9b35_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CAT[[:space:]]Merging_[[:space:]]A[[:space:]]Training-Free[[:space:]]Approach[[:space:]]for[[:space:]]Resolving[[:space:]]Conflicts[[:space:]]in[[:space:]]Model[[:space:]]Merging/58152b9d-db49-4b3e-8598-3669c97dc53e_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CAT_[[:space:]]Contrastive[[:space:]]Adversarial[[:space:]]Training[[:space:]]for[[:space:]]Evaluating[[:space:]]the[[:space:]]Robustness[[:space:]]of[[:space:]]Protective[[:space:]]Perturbations[[:space:]]in[[:space:]]Latent[[:space:]]Diffusion[[:space:]]Models/c3aa0b2c-8690-470d-83c5-5160dc94613e_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CEGA_[[:space:]]A[[:space:]]Cost-Effective[[:space:]]Approach[[:space:]]for[[:space:]]Graph-Based[[:space:]]Model[[:space:]]Extraction[[:space:]]and[[:space:]]Acquisition/de736e28-a604-46e8-ad87-6c090f8c6423_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CERTAIN_[[:space:]]Context[[:space:]]Uncertainty-aware[[:space:]]One-Shot[[:space:]]Adaptation[[:space:]]for[[:space:]]Context-based[[:space:]]Offline[[:space:]]Meta[[:space:]]Reinforcement[[:space:]]Learning/114dc187-134f-4faa-b334-ad0916ac1e0d_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CFP-Gen_[[:space:]]Combinatorial[[:space:]]Functional[[:space:]]Protein[[:space:]]Generation[[:space:]]via[[:space:]]Diffusion[[:space:]]Language[[:space:]]Models/2effccd1-4349-47e1-b985-c74d042f2377_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CFPT_[[:space:]]Empowering[[:space:]]Time[[:space:]]Series[[:space:]]Forecasting[[:space:]]through[[:space:]]Cross-Frequency[[:space:]]Interaction[[:space:]]and[[:space:]]Periodic-Aware[[:space:]]Timestamp[[:space:]]Modeling/29bf894a-773f-4c75-82de-cc80566183db_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CHATS_[[:space:]]Combining[[:space:]]Human-Aligned[[:space:]]Optimization[[:space:]]and[[:space:]]Test-Time[[:space:]]Sampling[[:space:]]for[[:space:]]Text-to-Image[[:space:]]Generation/f84ef880-f4c0-47f8-9cec-5d40a11115c2_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CLARIFY_[[:space:]]Contrastive[[:space:]]Preference[[:space:]]Reinforcement[[:space:]]Learning[[:space:]]for[[:space:]]Untangling[[:space:]]Ambiguous[[:space:]]Queries/922c0c1e-a54e-4a76-a018-c9223ea7bdff_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CLIMB_[[:space:]]Data[[:space:]]Foundations[[:space:]]for[[:space:]]Large[[:space:]]Scale[[:space:]]Multimodal[[:space:]]Clinical[[:space:]]Foundation[[:space:]]Models/9c5e4978-9f5a-495d-9332-3f56b87649b1_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CLOVER_[[:space:]]Cross-Layer[[:space:]]Orthogonal[[:space:]]Vectors[[:space:]]Pruning/45da6a31-48b5-4ec4-b933-1f44195705c8_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/CMoS_[[:space:]]Rethinking[[:space:]]Time[[:space:]]Series[[:space:]]Prediction[[:space:]]Through[[:space:]]the[[:space:]]Lens[[:space:]]of[[:space:]]Chunk-wise[[:space:]]Spatial[[:space:]]Correlations/6c5dac77-4a97-4bb0-97cf-984801f4bde7_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/COExpander_[[:space:]]Adaptive[[:space:]]Solution[[:space:]]Expansion[[:space:]]for[[:space:]]Combinatorial[[:space:]]Optimization/c810731e-8f3b-4175-85f7-771a83011964_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/COGNATE_[[:space:]]Acceleration[[:space:]]of[[:space:]]Sparse[[:space:]]Tensor[[:space:]]Programs[[:space:]]on[[:space:]]Emerging[[:space:]]Hardware[[:space:]]using[[:space:]]Transfer[[:space:]]Learning/aa4f3385-be13-4d70-bd5c-cf7818417798_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/COKE_[[:space:]]Core[[:space:]]Kernel[[:space:]]for[[:space:]]More[[:space:]]Efficient[[:space:]]Approximation[[:space:]]of[[:space:]]Kernel[[:space:]]Weights[[:space:]]in[[:space:]]Multiple[[:space:]]Kernel[[:space:]]Clustering/3f5efe43-2d0f-4e9f-ac5f-bb392ca3cd4a_origin.pdf filter=lfs diff=lfs merge=lfs -text

2025/Beyond Task-Specific Reasoning_ A Unified Conditional Generative Framework for Abstract Visual Reasoning/187a29f7-5752-44f7-a8d5-732c4e65a8cc_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2eda0adbd52ddd9f089c7daa038ec45c952ffc05cef0fbad6a96ffb11cf86a41
+size 2685087

2025/Beyond Task-Specific Reasoning_ A Unified Conditional Generative Framework for Abstract Visual Reasoning/full.md

@@ -0,0 +1,576 @@
+# Beyond Task-Specific Reasoning: A Unified Conditional Generative Framework for Abstract Visual Reasoning
+
+Fan Shi $^{1}$  Bin Li $^{*1}$  Xiangyang Xue $^{1}$
+
+# Abstract
+
+Abstract visual reasoning (AVR) enables humans to quickly discover and generalize abstract rules to new scenarios. Designing intelligent systems with human-like AVR abilities has been a longstanding topic in the artificial intelligence community. Deep AVR solvers have recently achieved remarkable success in various AVR tasks. However, they usually use task-specific designs or parameters in different tasks. In such a paradigm, solving new tasks often means retraining the model, and sometimes retuning the model architectures, which increases the cost of solving AVR problems. In contrast to task-specific approaches, this paper proposes a novel Unified Conditional Generative Solver (UCGS), aiming to address multiple AVR tasks in a unified framework. First, we prove that some well-known AVR tasks can be reformulated as the problem of estimating the predictability of target images in problem panels. Then, we illustrate that, under the proposed framework, training one conditional generative model can solve various AVR tasks. The experiments show that with a single round of multi-task training, UCGS demonstrates abstract reasoning ability across various AVR tasks. Especially, UCGS exhibits the ability of zero-shot reasoning, enabling it to perform abstract reasoning on problems from unseen AVR tasks in the testing phase.
+
+# 1. Introduction
+
+Abstract visual reasoning (AVR) is a fundamental cognitive ability that enables humans to identify visual concepts, infer underlying abstract rules, and generalize the rules to new scenarios effectively (Cattell, 1963; Zhuo & Kankanhalli,
+
+$^{1}$ Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University. Correspondence to: Bin Li <libin@fudan.edu.cn>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+2021; Małkinski & Mandziewski, 2022a). This ability is pivotal in cognitive activities like decision-making, visual analogy and spatial reasoning (Gray & Thompson, 2004; McCloskey, 1983). Many psychological tests are designed to assess the AVR ability of humans, which often requires the subject to discover specific rules that connect different visual concepts (Raven & Court, 1938; McMillen, 2007; Nie et al., 2020). For example, Raven's Progressive Matrix (RPM) (Raven & Court, 1938) focuses on assessing generalization ability, as the tests include unseen combinations of visual concepts and rules, which is a classical intelligence quotient test.
+
+Developing models that reveal human-like AVR ability remains a significant challenge in artificial intelligence research (Zheng et al., 2019; Chollet, 2019; Małkinski & Mandziuk, 2022b). Many AVR benchmarks have been proposed to evaluate intelligent systems in a way that mirrors human-like AVR ability (Barrett et al., 2018; Hill et al., 2019; Zhang et al., 2019; Nie et al., 2020). These benchmarks usually adopt the problem structure of traditional psychological tests and provide programs to generate a large number of problems automatically, e.g., the RAVEN dataset generates RPM problems by combining predefined visual concepts and abstract rules (Zhang et al., 2019). In recent years, deep AVR solvers have achieved remarkable progress on the proposed AVR benchmarks (Barrett et al., 2018; Zhuo & Kankanhalli, 2021; Mondal et al., 2023). However, they often rely on task-specific inductive biases of model architecture and hyperparameters, and solving different tasks typically means training different instances of models. This task-specific paradigm arouses interest in exploring a more unified framework for AVR solvers (Webb et al., 2024).
+
+AVR tasks take different structures of problem panels to assess the subject's ability to understand abstract rules. RPM-style tasks (Barrett et al., 2018; Zhang et al., 2019) require discovering abstract rules of a  $3 \times 3$  image matrix and selecting answers from a candidate panel to complete the matrix via analogy. Odd-one-out problems (McMillen, 2007) test the ability to identify the image that violates the abstract rules of a sequence. In addition, AVR tasks can be distinguished as generative tasks and selective tasks. Generative tasks require the synthesizing or reconstruction of images (Bar et al., 2022; Dedhia et al., 2023; Bai et al., 2023), while
+
+selective tasks provide predefined options for the selection of answers (Barrett et al., 2018; Hill et al., 2019; Nie et al., 2020). These AVR tasks have different structures and goals, posing challenges in proposing a unified problem-solving framework.
+
+Conditional generative models can discover the latent structure of observations, capture complex conditional dependencies between latent factors, and generate outputs conditioned on specific input variables or context (Mirza, 2014; Sohn et al., 2015; Chen et al., 2016). This ability is beneficial for AVR tasks that require understanding the abstract rules underlying the problem panels, where the latent structures can represent the abstract rules like shape transformations and spatial arrangements. Conditional generative models offer a powerful tool for solving AVR problems. Previous works have shown that both generative and selective AVR tasks can be tackled via conditional generation (Pekar et al., 2020; Shi et al., 2021; 2024). The solvers can generate the missing part of the panel and then compare it with the candidates to select the correct answer of an RPM test, which sheds light on unifying selective and generative AVR tasks.
+
+We propose a novel framework, Unified Conditional Generative Solver (UCGS), to address multiple AVR tasks within a unified framework. UCGS frames various AVR tasks as a conditional generation process, which does not need repeated model training or task-specific hyperparameters when solving different AVR tasks. We prove that some classical AVR tasks, including RPM (Zhang et al., 2019; Barrett et al., 2018), Visual Analogy Problem (VAP) (Hill et al., 2019), Odd-One-Out (O3) (Mandziuk & Zychowski, 2019) and Synthetic Visual Reasoning Task (SVRT) (Fleuret et al., 2011), can be solved by estimating the probability of generating the target images conditioned on the remaining images within a problem panel. We design a conditional generative network to instantiate the proposed framework, which can infer visual concepts from image patches and conduct abstract visual reasoning in terms of the concepts. The experiments demonstrate that UCGS exhibits not only the abstract reasoning ability on in-distribution tasks, but also the zero-shot reasoning ability to handle unseen tasks during the test phase.
+
+# 2. Related Work
+
+Abstract Visual Reasoning. Abstract visual reasoning (AVR) tasks are taken to measure the ability of abstract rule learning and problem-solving through analogical reasoning. Some early models (Lovett et al., 2010; Little et al., 2012) relied on artificially designed features to perform abstract visual reasoning. In recent years, a large number of models based on deep features have emerged (Barrett et al., 2018; Wang et al., 2020; Yun et al., 2020; Zhuo & Kankanhalli, 2021; Depeweg et al., 2024). Different abstract features,
+
+such as hierarchical features (Zheng et al., 2019; Benny et al., 2021; Hu et al., 2021), disentangled representations (Van Steenkiste et al., 2019; Wu et al., 2020), and object-centric representations (Mondal et al., 2023; Webb et al., 2024), have been introduced as cues for abstract reasoning. Some approaches solve RPMs based on neuro-symbolic architectures (Hersche et al., 2023). On the other hand, some approaches focus on solving generative AVR tasks (Pekar et al., 2020; Shi et al., 2021; Zhang et al., 2021a;b; Shi et al., 2023; 2024). They directly complete problem panels from the given contexts and select answers by comparing the generated results with candidates. Inspired by language-prompted LLMs (Brown et al., 2020), visual in-context learners have been proposed to solve analogical reasoning problems in visual data (Bar et al., 2022; Dedhia et al., 2023; Bai et al., 2023; Zhao et al., 2023). Compared with the previous solvers, UCGS focuses on solving AVR tasks in a unified conditional generation framework.
+
+Deep Conditional Generative Models. Deep generative models learn distributions of entire datasets, then they can generate novel samples that do not appear in the training stage (Kingma & Welling, 2013; Goodfellow et al., 2014; Du & Mordatch, 2019; Rombach et al., 2022). Deep conditional generative models (DCGMs) estimate the conditional probability distribution of the output given a certain input (Mirza, 2014; Sohn et al., 2015; Chen et al., 2016). Conditional Generative Adversarial Network (CGAN) (Mirza, 2014) is a classical type of DCGM, where a generator produces data conditioned on the input, and a discriminator distinguishes between the real and generated data. Conditional Variational Autoencoder (CVAE) (Sohn et al., 2015) introduces conditional inputs to conventional VAE (Kingma & Welling, 2013), where conditional inputs are encoded with input data into a latent space, and the latent representation is decoded to generate new samples in terms of the conditional inputs. Conditional generative models reveal a deep understanding of complex structures behind conditional inputs and data. Some DCGMs discover the underlying structure behind complex data by predicting the missing part of the data from the given context. Masked autoencoder (MAE) (He et al., 2022) is given an input where some portions are masked, and the task is to generate the missing pieces. The NP family (Garnelo et al., 2018; Kim et al., 2019; Dutordoir et al., 2023) can simulate a continuous function based on given points. In this paper, UCGS solves different AVR tasks with a unified conditional generative model, which extends the application of DCGMs in learning underlying structures from data.
+
+# 3. Methods
+
+In this section, we illustrate how to solve commonly used AVR tasks through conditional generation and leverage a
+
+![](images/271f2f15cf438efb07bc8fa83fc838662c5cb2e2fba9b5e0ff5ea896308686a1.jpg)  
+(a) Raven's Progressive Matrix (RPM)  
+Problem Panel  $I^p$
+
+![](images/645d5d19d263a75bbe98bea2a7bad0f4cefa1b6879fb95a4832b1466c89f4c49.jpg)  
+(b) Visual Analogy Problem (VAP)  
+Problem Panel  $I^p$
+
+![](images/c4e0e8090bb2d20cf1a76a9685e7d30c0a0cefe1edc02dcd4e6abc0734db3b2b.jpg)  
+Candidates  $I^s$
+
+![](images/c907c30ac411a83d2206dd72abb95f2fa4443fd5744618d27c0312c34f0af6f4.jpg)  
+(c) Odd-One-Out (O3) Problem
+
+![](images/8617f44e76d595f370cc834db0b72afb840309274bbb3b439de2da00c3c00edb.jpg)  
+Problem Panel  $I^p$
+
+![](images/9080f0602b0f93b0c7c9ea05613b712e969e14e51533a80c69247ec8cd203dbe.jpg)  
+(d) Synthetic Visual Reasoning Task (SVRT)  
+Left Problem Panel  $I^l$
+
+![](images/85d05a1bdbb864245d48d0c78ba6897e6fa1feca51d408029dcbbcdea9523d99.jpg)  
+Right Problem Panel  $I^r$
+
+![](images/6c53d6d605c448a3d52f9697fbdaa04765056f560dffdac9522def59f158453a.jpg)  
+Query Panel  $I^q$  
+Figure 1: An illustration of abstract visual reasoning (AVR) tasks. This paper involves four AVR tasks, despite their differences in the predefined visual concepts (e.g., object shape and color) and abstract rules (e.g., progressive change and logical rule), all assess the ability to infer abstract rules from visual stimuli. (a) Raven's Progressive Matrices (RPMs) are visual puzzles where participants are given a problem panel with one missing piece. The task is to select the correct image (e.g., the 6th candidate image in the example) from the candidate panel that completes the problem panel. (b) Visual Analogy Problems (VAPs) are similar to RPMs. The participant must choose candidate images that complete the analogy and fit the missing piece of problem panels. (c) In an Odd-One-Out (O3) problem, solvers are given a problem panel to find the odd image that breaks the abstract rule (e.g., the 2nd image in the example panel). (d) A problem of Synthetic Visual Reasoning Task (SVRT) consists of two problem panels following different abstract rules. The task is to deduce the rule that differentiates the problem panels, and solvers are required to categorize the query image into the left or right panel.
+
+unified conditional generative network for abstract visual reasoning. We then introduce an instance of UCGS, where the conditional generative network can generate missing panels from context across various tasks.
+
+# 3.1. Unified Conditional Generative Solver
+
+As shown in Figure 1, the AVR tasks define different visual concepts and the abstract rules on the concepts, which test the understanding of abstract rules in different ways of obtaining answers. We will introduce how to solve different AVR tasks in a unified way as follows.
+
+Definition 3.1.  $\pmb{I}^p = \{\pmb{I}_i^p | i = 1, \dots, N\}$  is a problem panel where  $\pmb{I}_i^p$  is the  $i$ -th panel image of  $\pmb{I}^p$ . The correctness of  $\pmb{I}^p$  is the joint probability  $p(\pmb{I}^p)$ .  $\pmb{I}^p$  is rule-compliant if  $p(\pmb{I}^p)$  is large, and rule-violating if  $p(\pmb{I}^p) \to 0$ .
+
+By modeling the correctness of a problem panel as a joint probability over panel images, we can define the conditional probability of predicting a specific image in the panel.
+
+Definition 3.2. For panel  $\pmb{I}^p$ , the predictability  $p(\pmb{x}|\pmb{I}_{\neg i}^{p})$  indicates the probability that the  $i$ -th panel image is  $\pmb{x}$  given the context  $\pmb{I}_{\neg i}^{p}$ , where  $\pmb{I}_{\neg i}^{p} = \{\pmb{I}_{j}^{p}\}_{j \neq i}$ .
+
+Estimating and sampling results from the predictability can solve generative RPM problems, since the problems require solvers to generate answers from the context (Pekar et al., 2020). But the predictability cannot directly solve selective RPM problems where the solvers must pick answers from
+
+candidate panels (Zhang et al., 2019). Let  $I_{i \to x}^{p}$  be the panel created by replacing the  $i$ -th image of the panel  $I^{p}$  to  $x$ . Using Definitions 3.1 and 3.2, UCGS can solve the following selective tasks by estimating the predictability.
+
+Proposition 3.3. Given the problem panel  $\pmb{I}^p$  and candidate panel  $\pmb{I}^s$  of an RPM test or VAP, where  $N = |\pmb{I}^p|$ , the correct answer  $\pmb{x}^\star = \operatorname*{argmax}_{\pmb{x} \in \pmb{I}^s} p(\pmb{x}|\pmb{I}_{\neg N}^p)$ .
+
+Proof. The context panel  $\pmb{I}_{\neg N}^{p}$  is created by removing the last image from  $\pmb{I}^p$ . The modified panels  $\{\pmb{I}_{N\rightarrow \pmb{x}}^p |\pmb {x}\in \pmb{I}^s\}$  are created by placing the candidate images to the removed position. The correctness of  $\pmb{I}_{N\rightarrow \pmb{x}}^{p}$  is
+
+$$
+p \left(\boldsymbol {I} _ {N \rightarrow \boldsymbol {x}} ^ {p}\right) = p (\boldsymbol {x} \mid \boldsymbol {I} _ {\neg N} ^ {p}) p \left(\boldsymbol {I} _ {\neg N} ^ {p}\right) \propto p (\boldsymbol {x} \mid \boldsymbol {I} _ {\neg N} ^ {p}). \tag {1}
+$$
+
+The correct answer  $\pmb{x}^*$  make up the modified panel with the largest correctness, given by  $\operatorname{argmax}_{\pmb{x} \in I^s} p(\pmb{x} | I_{\neg N}^p)$ .
+
+Proposition 3.4. Given the panel  $\mathbf{I}^p$  of an O3 problem, the index of the odd image  $i^\star = \operatorname*{argmin}_{i=1,\dots,N} p(\mathbf{I}_i^p | \mathbf{I}_{\neg i}^p)$ , where  $N = |\mathbf{I}^p|$ .
+
+Proof. We create context panels  $\{I_{\neg i}^p |i = 1,\dots ,N\}$  by removing each image from  $I^{p}$ . The correctness of  $I_{\neg i}^{p}$  is
+
+$$
+p \left(\boldsymbol {I} _ {\neg i} ^ {p}\right) = \frac {p \left(\boldsymbol {I} ^ {p}\right)}{p \left(\boldsymbol {I} _ {i} ^ {p} \mid \boldsymbol {I} _ {\neg i} ^ {p}\right)} \propto \frac {1}{p \left(\boldsymbol {I} _ {i} ^ {p} \mid \boldsymbol {I} _ {\neg i} ^ {p}\right)}. \tag {2}
+$$
+
+If  $I_{i}^{p}$  is the odd image,  $p(I_{\neg i}^{p})$  will be large since the rule-breaking image is removed, otherwise  $I_{\neg i}^{p}$  is still rule-
+
+violating and  $p(\mathbf{I}_{\neg i}^p) \to 0$ . Hence the index of the odd image is  $i^\star = \operatorname*{argmin}_{i=1,\dots,N} p(\mathbf{I}_i^p|\mathbf{I}_{\neg i}^p)$ .
+
+Proposition 3.5. Given a left problem panel  $\pmb{I}^l$ , a right problem panel  $\pmb{I}^r$  and a query panel  $\pmb{I}^q$  of SVRT, where  $N = |\pmb{I}^l| = |\pmb{I}^r|$ , the query image  $\pmb{x} \in \pmb{I}^q$  belongs to the panel  $\pmb{I}^\star = \operatorname{argmax}_{\pmb{I}^p \in \{\pmb{I}^l, \pmb{I}^r\}} p(\pmb{x} | \pmb{I}^p)$ .
+
+Proof. By appending the query image  $\pmb{x} \in I^q$  to the end of the problem panels  $\pmb{I}^l$  and  $\pmb{I}^r$ , we get the extended panels  $\pmb{I}_{+x}^l$  and  $\pmb{I}_{+x}^r$ . The correctness of the extended panels is
+
+$$
+p \left(\boldsymbol {I} _ {+ \boldsymbol {x}} ^ {l}\right) = p (\boldsymbol {x} \mid \boldsymbol {I} ^ {l}) p \left(\boldsymbol {I} ^ {l}\right), \tag {3}
+$$
+
+$$
+p (\boldsymbol {I} _ {+ \boldsymbol {x}} ^ {r}) = p (\boldsymbol {x} | \boldsymbol {I} ^ {r}) p (\boldsymbol {I} ^ {r}).
+$$
+
+If  $\pmb{x}$  belongs to  $\pmb{I}^p$ , the extended panel  $\pmb{I}_{+\pmb{x}}^p$  remains rulecompliant because the panel rule is not changed. Therefore,  $\pmb{I}_{+\pmb{x}}^p$  has the largest correctness. Consider that  $\pmb{I}^l$  and  $\pmb{I}^r$  are sampled from the dataset  $\mathcal{D}$  uniformly without overlap, in Equation 3 we have  $p(\pmb{I}^l) = p(\pmb{I}^r) = 1 / |\mathcal{D}|$ . Therefore,  $\pmb{x}$  belongs to the panel  $\pmb{I}^\star = \operatorname{argmax}_{\pmb{I}^p \in \{\pmb{I}^l, \pmb{I}^r\}} p(\pmb{x} | \pmb{I}^p)$ .
+
+Propositions 3.3, 3.4, and 3.5 demonstrate that although the answers to AVR problems have different forms, they can be converted into the estimation of predictability on panel images. In this way, answer selection and generation are unified. If the  $i$ -th image of an RPM problem panel is missing, we can generate answers by sampling from  $p(\boldsymbol{x}|\boldsymbol{I}_{\neg i}^{p})$ , as well as select answers by filling in the candidate  $\boldsymbol{x}$  and computing the panel correctness  $p(\boldsymbol{I}_{i\rightarrow \boldsymbol{x}}^{p}) = p(\boldsymbol{x}|\boldsymbol{I}_{\neg i}^{p})p(\boldsymbol{I}_{\neg i}^{p})$ . With a unified predictability estimator, UCGS can solve different AVR problems after a round of multi-task training, preventing repeated training or tuning of AVR solvers. The core of UCGS is a shared predictability estimator and training-free judgment functions. The predictability estimator is a conditional generative network trained on different AVR tasks to estimate the predictability of panel images. As shown in Propositions 3.3, 3.4, and 3.5, the judgment functions transform outputs of the predictability estimator to obtain the final results, which are task-specific. In the next sections, we provide an instance of UCGS and introduce the architecture of the conditional generative network in detail.
+
+# 3.2. Instantiation of UCGS
+
+We instantiate UCGS with a Transformer-based conditional generative network, called UCGS-T. Figure 2a illustrates the architecture of UCGS-T via an RPM problem. The problem panel of an RPM contains  $N = 9$  images, where we leave one as the prediction target and the others as the context. We denote the index of the target image as  $t$ , and the indices of the context images as a set  $C$ . The objective of UCGS-T is to estimate the predictability  $p(\mathbf{I}_t^p | \mathbf{I}_C^p)$ . UCGS-T consists of five modules, which will be introduced as follows.
+
+# 3.2.1. IMAGE ENCODER AND DECODER
+
+Figure 2b depicts the architecture of the image encoder and decoder. UCGS-T learns patch representations of the input image by mapping continuous representations into a finite set of discrete codes, which is then decoded back into the original data space (Van Den Oord et al., 2017; Razavi et al., 2019). The image encoder extracts a feature map with  $M$  visual features from  $I_{i}^{p}$  using a CNN-based neural network, which are projected to the nearest vectors in a fixed-size codebook  $e = (e_1, e_2, \dots, e_L)$  to obtain  $M$  quantized patch representations  $Z_{i} = \{Z_{i,1}, Z_{i,2}, \dots, Z_{i,M}\}$ . The image decoder takes the quantized patch representations as input and reconstructs the image from the discrete representation space. The input images are compressed into quantized patch representations to learn more abstract and general representations for reasoning. We can estimate the predictability  $p(I_t^p | I_C^p)$  over the high-dimensional images through  $p(Z_t|Z_C)$  on the patch representations to reduce the complexity of reasoning.
+
+# 3.2.2.PATCH ENCODER
+
+The quantized patch representations focus on local regions of the image, potentially corresponding to a specific geometric shape or pattern learned by the codebook. As shown in Figure 2c, the patch encoder computes the relation between the local patch representations to understand the role of each patch in the overall image and extract the image-level visual concepts (e.g., the number of entities in the image). We add  $M$  learnable positional embeddings to the patches of each panel image, which is projected to the high-dimensional representations  $\tilde{Z}$  by a linear network to encode spatial position information, where  $\tilde{Z}_{i,m}$  is the position-augmented representations of  $Z_{i,m}$  for  $i = 1,\dots ,N$  and  $m = 1,\ldots ,M$ .  $\tilde{Z}_i$  are passed through a Transformer decoder to capture global dependencies and relationships between the patches. In addition to the patch representations, we introduce class tokens  $\{\mathrm{CLS}_1,\dots ,\mathrm{CLS}_K\}$  as learnable slots to aggregate global information from the patches and obtain the visual concepts  $S_{i} = \{S_{i,k}|k = 1,\dots ,K\}$  where
+
+$$
+\boldsymbol {S} _ {i, k} = \operatorname {T r a n s f o r m e r D e c o d e r} \left(\mathrm {C L S} _ {k}, \tilde {\boldsymbol {Z}} _ {i}\right). \tag {4}
+$$
+
+The independent slots capture  $K$  visual concepts from the patches. The Transformer decoder updates the slots based on attention mechanisms, where the slots act as queries and the patch representations  $\tilde{Z}_i$  are keys and values. The final output of the patch encoder is the visual concepts  $S_i$  that consider relationships across the patches of the entire image.
+
+# 3.2.3. CONCEPT ENCODER
+
+The concept encoder combines the visual concepts of the context images (i.e., context concepts)  $S_{C} = \{S_{i}|i\in C\}$ , understanding the overall layout and abstract rule within the
+
+![](images/edecd33bc47b6754a36c521a08ce9cf4381d3eee9ca9c2ce5e1ee905d714bb2e.jpg)  
+(a) Overview
+
+![](images/caa348279ca4311fe8b4d277b57431409ecfb970570f9dea015009bdde6fb640.jpg)  
+Problem Panel
+
+![](images/338a3857d23a6293754e9763749e8ce0ee03974f5084858b1956adceb3c2447b.jpg)  
+Legend  
+Concept Slots  
+Context Patches
+
+![](images/027c515a5179861e7fa68c08ec08f0e8cba14bc6f646fc901132c75256151e1c.jpg)
+
+![](images/51ead5271f0ef6d207f7073adcbb162ccaba843b4820578d7d397840e0db03db.jpg)  
+(b) Image Encoder and Decoder
+
+![](images/207e3343e568c2001ac80f3273b77fc69b71cab2b4e8e7ac6e1f68eef0bc1e60.jpg)  
+(c) Patch Encoder
+
+![](images/d2146a79f8266bf2fcc6081b84f6e42019b7d55f917c983ddd273552e95619a8.jpg)  
+(d) Concept Encoder  
+Figure 2: An Overview of UCGS-T. The image encoder extracts high-level features from each context panel image, which are mapped into discrete context patch representations via vector quantization. The patch encoder captures image-level visual concepts from context patches that encode local information. The panel encoder integrates visual concepts of the context images, understanding abstract rules on the panel, and predicting visual concepts for the target image. The patch decoder generates patch representations of the target image from the target visual concepts predicted by the panel encoder. The image decoder reconstructs the target image from the target patches.
+
+![](images/f6001aa431193ad5bfd8807598292986f7b3f0218fa48eb820d5574d97dfd537.jpg)  
+(e) Patch Decoder
+
+panel, and predicting the visual concepts of the target image (i.e., target concepts)  $S_{t}$ . Figure 2d displays the architecture of the concept encoder. First, the input context concepts  $S_{C}$  are reorganized into  $K$  groups  $\{G_1,\dots ,G_K\}$  in terms of the concepts, where the  $k$ -th group  $G_{k} = \{S_{i,k}|i\in C\}$ . We group the context concepts since the abstract rules are typically global and shared among different visual concepts (Wu et al., 2020; Shi et al., 2024), e.g., the tuples (Triangle, Square, Pentagon) and (Small, Middle, Large) point to the same abstract rule Increase. To encode the position of concepts within the group  $G_{k}$ , learnable positional embeddings are added to the visual concepts in  $G_{k}$ , followed by a linear projection to produce the position-augmented representations  $\tilde{G}_k$ . Each group is processed independently by a shared Transformer decoder to capture concept-specific abstract rules and predict the target visual concepts  $S_{t} = \{S_{t,k}|k = 1,\ldots ,K\}$  where
+
+$$
+\boldsymbol {S} _ {t, k} = \text {T r a n s f o r m e r D e c o d e r} \left(\mathrm {P E} _ {t}, \tilde {\boldsymbol {G}} _ {k}\right). \tag {5}
+$$
+
+$\mathrm{PE}_t$  is a learnable target positional embedding. The output  $S_{t}$  is used to reconstruct the patches of the target image.
+
+# 3.2.4.PATCH DECODER
+
+The patch decoder in Figure 2e generates target patches from the target visual concepts  $S_{t}$  in an autoregressive manner (Yan et al., 2021). The predictability is factorized by
+
+$$
+p \left(\boldsymbol {Z} _ {t} \mid \boldsymbol {Z} _ {C}\right) = \prod_ {m = 1} ^ {M} p \left(\boldsymbol {Z} _ {t, m} \mid \boldsymbol {Z} _ {t, <   m}, \boldsymbol {Z} _ {C}\right) \tag {6}
+$$
+
+where a learnable token  $Z_{t,0}$  is used to indicate the beginning of the autoregressive decoding process. The process generates the patch sequence one step at a time, where the  $m$ -th step depends on the previously generated patches  $Z_{t, < m}$  and the target concepts  $S_{t}$  computed from  $Z_{C}$ . The  $m$ -th step  $p(Z_{t,m}|Z_{t, < m},Z_C)$  is parameterized by a Transformer decoder with a prediction head:
+
+$$
+\tilde {\boldsymbol {Z}} _ {t, <   m} = \text {P o s t i o n a l E m b e d d i n g} \left(\boldsymbol {Z} _ {t, <   m}\right),
+$$
+
+$$
+\boldsymbol {H} = \text {T r a n s f o r m e r D e c o d e r} \left(\tilde {\boldsymbol {Z}} _ {t, <   m}, \boldsymbol {S} _ {t}\right), \tag {7}
+$$
+
+$$
+\boldsymbol {\pi} _ {m} = \operatorname {S o f t m a x} \left(\operatorname {L i n e a r} \left(\boldsymbol {H} _ {m}\right)\right),
+$$
+
+$$
+\boldsymbol {Z} _ {t, m} = \boldsymbol {e} [ c _ {m} ], \text {w h e r e} c _ {m} \sim \text {C a t e g o r i c a l} (\boldsymbol {\pi} _ {m}).
+$$
+
+The output of the Transformer decoder is passed through a prediction head, which maps the hidden representations into the probabilities over the discrete codes in the codebook. The final prediction of the target patch  $Z_{t,m}$  is given by querying the  $c_m$ -th vector  $e[c_m]$  in the codebook.
+
+# 3.2.5. MODEL TRAINING
+
+The total loss of the model consists of an image reconstruction loss  $\mathcal{L}_{\mathrm{recon}}$  and a patch prediction loss  $\mathcal{L}_{\mathrm{pred}}$ . We adopt the training objective of VQVAE (Van Den Oord et al., 2017) as the image reconstruction loss to ensure that the input image can be correctly reconstructed from its quantized discrete representations.  $\mathcal{L}_{\mathrm{recon}}$  is defined as the distance between the input images and the reconstructions with a commitment loss to ensure the continuous representations are not too far from the codebook vectors. The patch prediction loss evaluates the difference between the predicted target patches (discrete codes) and the target patches encoded from the image encoder.  $\mathcal{L}_{\mathrm{pred}}$  is defined as
+
+$$
+\mathcal {L} _ {\text {p r e d}} = - \sum_ {m = 1} ^ {M} \log p \left(\boldsymbol {Z} _ {t, m} \mid \boldsymbol {Z} _ {t, <   m}, \boldsymbol {Z} _ {C}\right). \tag {8}
+$$
+
+By combining the image reconstruction loss and patch prediction loss, the total loss is given as
+
+$$
+\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {p r e d}} + \lambda \cdot \mathcal {L} _ {\text {r e c o n}} \tag {9}
+$$
+
+where  $\lambda$  is a hyperparameter that balances the losses. The image encoder and decoder are pretrained before training the remaining modules. We can set  $\lambda > 0$  to finetune the image encoder and decoder in the following training stage. But we find that freezing their parameters is the best choice.
+
+# 4. Experiments
+
+In this section, we compare UCGS-T with ablation baselines, task-specific solvers and Multimodal LLMs (MLLMs) on four classical AVR tasks shown in Figure 1. We also conduct qualitative experiments to illustrate the performance of UCGS-T in answer generation tasks by visualizing the generation results of RAVEN and PGM.
+
+Datasets. The models are evaluated on RAVEN (Zhang et al., 2019) and PGM (Barrett et al., 2018) to assess their reasoning abilities on RPM tasks. The ability to reason over O3 problems is tested using the G1-set dataset (Mańdziuk & Žychowski, 2019). Performance on the remaining AVR tasks is evaluated using the VAP (Hill et al., 2019) and SVRT (Fleuret et al., 2011) datasets. RAVEN and PGM are used for both training and testing, while G1-set, VAP, and SVRT are used only during testing to assess the zero-shot reasoning capabilities of the models. In addition to the public datasets, we construct three new datasets, i.e., O3-ID, VAP-ID, and SVRT-ID, based on RAVEN. These
+
+datasets are designed to evaluate the ability to reason on unseen AVR tasks that share in-distribution visual concepts and abstract rules with the training data. See Appendix A for more details on the dataset construction.
+
+Metric. We compute the selection accuracy, a standard metric that evaluates model performance on AVR tasks (Barrett et al., 2018). This metric is applicable to both selective and generative solvers. For selective solvers, which predict the index of answers, selection accuracy can be calculated straightforwardly. Although candidate panels guarantee a unique correct answer, many AVR problems admit multiple valid solutions, making it challenging to enumerate all possible correct answers when evaluating generative solvers (Shi et al., 2024). We consider the output of a generative solver to be correct if it is the closest match to the ground truth answer among the candidate panels. Accordingly, both selective and generative solvers are evaluated using selection accuracy in our experiments.
+
+Ablation Baselines. We employ three backbone architectures and two conditional generative solvers to construct a set of ablation baselines. One of the backbones is the patch-based backbone (Patch) used in UCGS-T. Given that recent task-specific solvers (Mondal et al., 2023; Webb et al., 2024) have demonstrated strong performance using object-centric representations, we introduce an object-centric backbone (OCL) (Locatello et al., 2020; Yuan et al., 2023), which extracts slot representations to capture individual objects in panel images. We also incorporate a monolithic backbone (Mono) that encodes each panel image as a single representation. We introduce two baseline conditional generative solvers for predictability estimation. The Transformer-based solver (TF) maps context images to target images using a Transformer encoder-decoder architecture. The ANP-based solver (ANP) (Kim et al., 2019) models the distribution over image panels using stochastic functions where context images are used to infer the functions, and target images are sampled from fixed locations of the functions. By combining the backbones and conditional generative solvers, we construct six ablation baselines. Further details of UCGS-T and these baselines are provided in Appendix B. UCGS-T and the ablation baselines are trained under a multi-task setting using all seven configurations of RAVEN and the neutral configuration of PGM. The models are evaluated on VAP, G1-set, and SVRT without retraining or fine-tuning to assess zero-shot reasoning ability. We conduct qualitative experiments by visualizing the target images generated by the models to illustrate and compare the answer-generation capabilities of UCGS-T and the ablation baselines.
+
+Task-specific Solvers and MLLMs. Since UCGS-T is designed to handle both selective and generative tasks, we compare it against several task-specific solvers that support both answer selection and generation: PrAE (Zhang
+
+Table 1: Selection accuracy (%) on AVR tasks. ID tasks have only in-distribution visual concepts and abstract rules, while OOD tasks contain out-of-distribution visual concepts and abstract rules. The problems of zero-shot tasks only appear in the testing stage. The models marked with * are evaluated on subsets of the datasets.  
+
+<table><tr><td rowspan="2">MODEL</td><td colspan="2">ID TASKS</td><td colspan="3">ID AND ZERO-SHOT TASKS</td><td colspan="3">OOD AND ZERO-SHOT TASKS</td></tr><tr><td>RAVEN</td><td>PGM</td><td>O3-ID</td><td>VAP-ID</td><td>SVRT-ID</td><td>G1-SET</td><td>VAP</td><td>SVRT</td></tr><tr><td>ANP + MONO</td><td>6.0</td><td>14.1</td><td>11.9</td><td>5.8</td><td>51.9</td><td>32.9</td><td>30.7</td><td>48.4</td></tr><tr><td>ANP + OCL</td><td>8.3</td><td>12.3</td><td>13.3</td><td>8.5</td><td>49.4</td><td>29.3</td><td>30.4</td><td>49.8</td></tr><tr><td>ANP + PATCH</td><td>10.5</td><td>12.3</td><td>11.7</td><td>10.9</td><td>49.8</td><td>30.6</td><td>30.9</td><td>51.0</td></tr><tr><td>TF + MONO</td><td>11.2</td><td>14.1</td><td>13.6</td><td>6.5</td><td>56.4</td><td>32.7</td><td>30.2</td><td>50.8</td></tr><tr><td>TF + OCL</td><td>7.4</td><td>12.3</td><td>12.7</td><td>7.1</td><td>49.6</td><td>33.7</td><td>30.6</td><td>50.0</td></tr><tr><td>TF + PATCH</td><td>21.0</td><td>15.6</td><td>35.8</td><td>15.5</td><td>71.8</td><td>30.5</td><td>27.1</td><td>51.4</td></tr><tr><td>UCGS-T</td><td>64.6</td><td>38.1</td><td>33.6</td><td>35.8</td><td>84.6</td><td>30.4</td><td>28.8</td><td>52.8</td></tr><tr><td>GPT-40*</td><td>12.6</td><td>21.4</td><td>30.4</td><td>12.6</td><td>40.8</td><td>52.9</td><td>29.3</td><td>64.2</td></tr><tr><td>UCGS-T*</td><td>60.9</td><td>37.8</td><td>32.7</td><td>34.0</td><td>85.9</td><td>30.4</td><td>30.1</td><td>52.8</td></tr><tr><td>RANDOM GUESS</td><td>12.5</td><td>12.5</td><td>11.1</td><td>12.5</td><td>50.0</td><td>22.5</td><td>25.0</td><td>50.0</td></tr></table>
+
+et al., 2021a), NVSA (Hersche et al., 2023), GCA (Pekar et al., 2020), ALANS (Zhang et al., 2021b), and RAISE (Shi et al., 2024). To ensure consistency with the experimental setup used for UCGS-T and the ablation baselines, all task-specific solvers are trained and tested on each dataset separately, without supervision from rule or attribute annotations. In addition, we include GPT-4o (Achiam et al., 2023), a powerful general-purpose multimodal language model (MLLM), as a reference point for comparison. Task-specific prompts for GPT-4o are designed based on the prior benchmark study (Cao et al., 2024).
+
+# 4.1. Performance on AVR Tasks
+
+The experiments evaluate model performance across four AVR tasks. We categorize the evaluation settings into the following three groups. In-Distribution (ID) Tasks include problems that are seen during training, whose abstract rules and visual concepts are also encountered during training. In-Distribution Zero-Shot (ID-ZS) Tasks are not used during training, but their abstract rules and visual concepts are present in the training data. Out-of-Distribution Zero-Shot (OOD-ZS) Tasks are the most challenging tasks, which are entirely unseen during training and involve novel abstract rules or visual concepts. We compare UCGS-T with both ANP-based ablation baselines (ANP + Mono, ANP + OCL, ANP + Patch) and Transformer-based baselines (TF + Mono, TF + OCL, TF + Patch), as well as GPT-4o. For reference, we also include a random guess baseline.
+
+# 4.1.1. IN-DISTRIBUTION TASKS
+
+UCGS-T and the baselines are trained on RAVEN and PGM, whose test splits are treated as in-distribution (ID) tasks. While the training and testing samples of RAVEN and PGM
+
+are generated using the same sets of abstract rules and visual concepts, the rules and concepts are composed differently between the splits. The purpose of the ID tasks is to evaluate the ability to reason over the abstract rules and visual concepts that have been learned during training. As shown in Table 1, Transformer-based baselines outperform their ANP-based counterparts, with TF + Patch achieving the strongest results among them. UCGS-T further achieves  $64.6\%$  accuracy on RAVEN and  $38.1\%$  on PGM, demonstrating the abstract reasoning capabilities on ID tasks. In contrast, ANP-based baselines perform poorly, with accuracies close to random guessing. This suggests that ANP-based models struggle with compositional reasoning in AVR tasks such as RAVEN and PGM.
+
+# 4.1.2. IN-DISTRIBUTION ZERO-SHOT TASKS
+
+We introduce ID-ZS tasks to evaluate the ability of models to generalize their reasoning capabilities to novel tasks. In this setting, all models are trained on RPM tasks and evaluated on other types of AVR tasks. They are required to solve problems from zero-shot tasks that contain in-distribution abstract rules and visual concepts. We construct three ID-ZS datasets, i.e., O3-ID, VAP-ID, and SVRT-ID, which follow the forms of O3, VAP, and SVRT, respectively. These datasets are built on the abstract rules and visual concepts of RAVEN. Across all ID-ZS tasks, models exhibit a noticeable drop in performance compared to ID tasks, highlighting the difficulty of transferring learned reasoning abilities to new task formats. Transformer-based baselines continue to outperform their ANP-based counterparts. UCGS-T achieves the best results overall, with accuracies of  $33.6\%$  on O3-ID,  $35.8\%$  on VAP-ID, and  $84.6\%$  on SVRT-ID. These results suggest that UCGS-T is more effective at generalizing rea
+
+soning abilities from the training data to novel tasks. For example, UCGS-T can perform abstract reasoning on VAPs with six-image panels, despite having been trained solely on RPMs with nine-image panels.
+
+# 4.1.3. OUT-OF-DISTRIBUTION ZERO-SHOT TASKS
+
+ODD-ZS tasks evaluate models on novel abstract rules and visual concepts that are entirely unseen during training, with test problems presented in forms that differ from those used in the training phase. We use the G1-set, VAP, and SVRT datasets to assess the out-of-distribution and zero-shot reasoning capabilities of models. As shown in Table 1, both UCGS-T and the baselines experience a substantial drop in performance compared to the ID tasks. On G1-set and VAP, UCGS-T and the baselines achieve accuracies around  $30\%$ , which is  $5 - 8\%$  higher than random guessing. This indicates that the UCGS framework enables a certain degree of generalization in abstract reasoning. UCGS-T slightly outperforms the other baselines on SVRT, a task that requires holistic pattern recognition based on spatial relationships among randomly generated abstract shapes. This requirement is similar to image-level visual concept recognition in datasets such as PGM and RAVEN (e.g., recognizing the number of objects in a scene). The results on SVRT suggest that UCGS-T may benefit from its panel encoder, which is specifically designed to extract high-level image abstractions. Furthermore, UCGS-T achieves a large accuracy improvement over the baselines in ID tasks, while the performance gains are relatively smaller in OOD tasks. We suppose that the unseen visual concepts and abstract rules in OOD tasks lead to a significant drop in accuracy for both UCGS-T and the baselines, thereby narrowing the performance gap that arises from different model architectures.
+
+# 4.1.4. COMPARISON WITH GPT-40
+
+Table 1 also presents the results of the general-purpose system GPT-4o. We randomly sampled subsets from the test sets of RAVEN, PGM, and VAP to evaluate both GPT-4o and UCGS-T. The results show that UCGS-T outperforms GPT-4o significantly on ID and ID-ZS tasks, while GPT-4o achieves better performance on OOD-ZS tasks. This may be attributed to GPT-4o's ability to recognize unseen visual concepts or abstract rules. However, this does not necessarily imply that GPT-4o possesses stronger zero-shot reasoning ability, as it is difficult to determine whether similar visual concepts or rules were encountered during its training. In contrast, UCGS-T is trained on a clearly defined and controlled set of samples, making its evaluation setting more transparent. Notably, GPT-4o shows weaker generalization on ID-ZS tasks, suggesting that UCGS-T is more effective at transferring its reasoning capabilities to novel task formats. In this experiment, GPT-4o is provided with task-specific prompts, following the setup in (Cao et al., 2024). We also
+
+Table 2: Selection accuracy (\%) of UCGS-T and the task-specific solvers on RAVEN and PGM. The task-specific solvers are trained and tested separately on each dataset. They are trained without additional annotations to keep consistent with the experimental setup of UCGS-T and baselines. PrAE and ALANS only define the architecture to solve RAVEN. The codebase of NVSA is not applicable to PGM, and the reported accuracy on PGM is obtained by using additional annotation information. We only report their performance on RAVEN here.
+
+<table><tr><td>MODEL</td><td>RAVEN</td><td>PGM</td></tr><tr><td>PRAE (ZHANG ET AL., 2021A)</td><td>13.6</td><td>-</td></tr><tr><td>NVSA (HERSCHE ET AL., 2023)</td><td>11.5</td><td>-</td></tr><tr><td>GCA (PEKAR ET AL., 2020)</td><td>37.3</td><td>31.7</td></tr><tr><td>ALANS (ZHANG ET AL., 2021B)</td><td>50.1</td><td>-</td></tr><tr><td>RAISE (SHI ET AL., 2024)</td><td>54.5</td><td>14.0</td></tr><tr><td>UCGS-T</td><td>64.6</td><td>38.1</td></tr><tr><td>RANDOM GUESS</td><td>12.5</td><td>12.5</td></tr></table>
+
+incorporate prior knowledge in the prompts, such as stating that an RPM problem is a  $3 \times 3$  matrix, which is not provided to UCGS-T during either training or testing. Prior work (Cao et al., 2024) has shown that GPT-4o achieves a notable performance boost (approximately  $20\%$ ) when provided with language descriptions of candidate images. Although GPT-4o is a powerful general-purpose model, it is not specifically designed for AVR tasks. Its performance on RPM tasks may benefit from more advanced prompt engineering and chain-of-thoughts design. Appendix B.4 provides the prompts used in this experiment.
+
+# 4.1.5. COMPARISON WITH TASK-SPECIFIC SOLVERS
+
+This experiment compares UCGS-T with classic generative task-specific RPM solvers. Table 2 shows the experimental results where UCGS-T outperforms the task-specific solvers under the same experimental setup. The experimental results reveal that, without additional annotation information, the performance of generative task-specific solvers drops significantly. For example, PrAE achieves an accuracy of  $65.0\%$  with rule supervision during training (Zhang et al., 2021a), but its accuracy drops to  $13.6\%$  in our experiment. The reasoning process of GCA is specifically designed for predicting the bottom-right image in an RPM matrix, which limits its ability to generalize to other AVR tasks. Additionally, models like PrAE, ALANS, and NVSA rely on predefined rules and explicit representations tailored to specific datasets, making it difficult for them to handle unseen or undefined visual concepts and abstract rules. UCGS-T does not rely on manually designed concepts or rules. It performs an independent reasoning process over each visual concept, inferring both the underlying visual concepts and abstract rules without the need for additional annotations.
+
+![](images/46a27e4ee0f21451fbc97b6d3d026a378b7efde281a5b2cb69dba17282f69563.jpg)
+
+![](images/d7d31a516ad005b6334083d9d05ddd042eaa944563935031cce2ce09fe447f10.jpg)
+
+![](images/94f661cf40eaccec670266e70071a238fd23878924aa367543782e64100d91a6.jpg)
+
+![](images/99e88881fb3d5bb088501477815491b9501026e6bb5ba27bf3556ea5736100d6.jpg)
+
+![](images/5cb175649da92183e7711daf90eb0c8f43b689e8a6c97fc389f1ea115aa8b865.jpg)
+
+![](images/4d77a35d1be8acf2e58c5ec2bd356db3e6cbb228aa2558da678d4ab0986fc5f3.jpg)
+
+![](images/169e90205754b7853342bdd2f5af363ee5922a0c136f2cb5227d18023f42aa83.jpg)  
+(a) RAVEN
+
+![](images/69dee89b5efcc2836d6bb4f9a41c9c8503d2b713795d832c261d6767fff7eb7f.jpg)  
+Figure 3: Comparison of generation results on RAVEN (left) and PGM (right). In the visualization result of each model, the bottom-right position is the prediction and the remaining context images are ground truths. The figure illustrates qualitative differences between models, with errors and artifacts appearing in some results predicted by baselines. Note that there is noise in the problem panel, therefore the generated result of UCGS-T on PGM is correct though it is a little different from the ground truth.
+
+![](images/b21a7f93d7eed05b2276a0a63ae200eaad2117770d7a08f0f5584b53240a0bd3.jpg)
+
+![](images/be891c96b2434a584d55124b80e45096cda738e10f739b0900847dc575d9f923.jpg)
+
+![](images/4e56c565859acde00daecc95e5e2d6ad2074fcb9f46a286fcf3f2676662ad43f.jpg)
+
+![](images/bcc4d86dd3a0ecf2c213f2f90e02a16c8ec7528ef2eb1c927ce7d7d2b777385f.jpg)
+
+![](images/f591b12ef324b2ec8a908a0df7a41729be422a9ebf971d02b0df8af4e30f2c5b.jpg)
+
+![](images/7d698a094d9579db39eb7c4d9bdce3b9b1f7f3eb8697ee284ef81d0fb3d11036.jpg)
+
+![](images/8d480c66b90ca4f1292054d3e29b3f2b4ec678e50901d7b68417bdc655729573.jpg)  
+(b) PGM
+
+![](images/afb8ed333164afd3c4b345b101c9b1b15cc01cf1e315c09b06d339e6d96d6739.jpg)
+
+# 4.2. Answer Generation on RAVEN and PGM
+
+Figure 3 visualizes the generated answers produced by UCGS-T and the baselines. UCGS-T can generate rulecompliant answers from the context on RAVEN and PGM. While TF-Patch is capable of generating clear and reasonable outputs on the PGM example, it fails to produce the correct answer for the RAVEN problem. Other baselines often generate images containing incorrect visual concepts. For instance, the solutions predicted by TF-Mono and ANP-Mono on RAVEN display scenes with four objects, whereas the correct answer should contain three objects. Models using the object-centric backbone generally struggle to generate high-quality outputs where the mismatch between predicted target slots and the true scene structure significantly degrades the pixel reconstruction quality. We observe that UCGS-T and TF-Patch are better at capturing diversity in answer generation. Since RAVEN and PGM introduce a degree of randomness or noise in data generation, UCGS-T's outputs may differ slightly from the ground truth but still conform to the task rules. For example, in Figure 3a, the answer should contain three objects, but their positions within the panel may vary without affecting correctness.
+
+# 5. Conclusion and Limitations
+
+We propose UCGS, a unified AVR framework to solve multiple AVR tasks using a single conditional generative network. We instantiate UCGS with a model that infers visual concepts from image patches to perform reasoning on the problem panel. Experimental results demonstrate that UCGS not only exhibits abstract reasoning ability on ID tasks but also zero-shot generalization, enabling it to solve unseen tasks during testing, even when those tasks involve OOD visual concepts and abstract rules.
+
+Limitations. While UCGS-T outperforms the generative task-specific RPM solvers, its accuracy is still lower than state-of-the-art selective solvers. Reducing the performance gap remains an important direction for future work. We believe this paper provides a unified perspective for solving AVR tasks. This work does not address AVR tasks involving real-world scenes. Real-world problems introduce more complex visual concepts and abstract rules, which may require more powerful image tokenizers and decoders. Effectively discovering and utilizing such rules in realistic settings remains a challenge for future investigation.
+
+# Impact Statement
+
+This paper presents work whose goal is to construct unified abstract visual reasoning solvers. We feel that there are no potential societal impacts that must be specifically highlighted and discussed here.
+
+# Acknowledgements
+
+This work was supported by the National Natural Science Foundation of China (No.62176060) and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.
+
+# References
+
+Achiam, J., Adler, S., Agarwal, S., Ahmad, L., Akkaya, I., Aleman, F. L., Almeida, D., Altenschmidt, J., Altman, S., Anadkat, S., et al. Gpt-4 technical report. arXiv preprint arXiv:2303.08774, 2023.  
+Bai, Y., Geng, X., Mangalam, K., Bar, A., Yuille, A., Darrell, T., Malik, J., and Efros, A. A. Sequential modeling enables scalable learning for large vision models. arXiv preprint arXiv:2312.00785, 2023.  
+Bar, A., Gandelsman, Y., Darrell, T., Globerson, A., and Efros, A. Visual prompting via image inpainting. Advances in Neural Information Processing Systems, 35: 25005-25017, 2022.  
+Barrett, D., Hill, F., Santoro, A., Morcos, A., and Lillicrap, T. Measuring abstract reasoning in neural networks. In International conference on machine learning, pp. 511-520. PMLR, 2018.  
+Benny, Y., Pekar, N., and Wolf, L. Scale-localized abstract reasoning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12557-12565, 2021.  
+Brown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J. D., Dhariwal, P., Neelakantan, A., Shyam, P., Sastry, G., Askell, A., et al. Language models are few-shot learners. Advances in neural information processing systems, 33: 1877-1901, 2020.  
+Cao, X., Lai, B., Ye, W., Ma, Y., Heintz, J., Chen, J., Cao, J., and Rehg, J. M. What is the visual cognition gap between humans and multimodal llms? arXiv preprint arXiv:2406.10424, 2024.  
+Cattell, R. B. Theory of fluid and crystallized intelligence: A critical experiment. Journal of educational psychology, 54(1):1, 1963.  
+Chen, X., Duan, Y., Houthooft, R., Schulman, J., Sutskever, I., and Abbeel, P. Infogan: Interpretable representation
+
+learning by information maximizing generative adversarial nets. Advances in neural information processing systems, 29, 2016.  
+Chollet, F. On the measure of intelligence. arXiv preprint arXiv:1911.01547, 2019.  
+Dedhia, B., Chang, M., Snell, J. C., Griffiths, T. L., and Jha, N. K. Im-promptu: In-context composition from image prompts. arXiv preprint arXiv:2305.17262, 2023.  
+Depeweg, S., Rothkopf, C. A., and Jäkel, F. Solving bongard problems with a visual language and pragmatic constraints. Cognitive Science, 48(5):e13432, 2024.  
+Du, Y. and Mordatch, I. Implicit generation and modeling with energy based models. Advances in Neural Information Processing Systems, 32, 2019.  
+Dutordoir, V., Saul, A., Ghahramani, Z., and Simpson, F. Neural diffusion processes. In International Conference on Machine Learning, pp. 8990-9012. PMLR, 2023.  
+Garnelo, M., Schwarz, J., Rosenbaum, D., Viola, F., Rezende, D. J., Eslami, S., and Teh, Y. W. Neural processes. arXiv preprint arXiv:1807.01622, 2018.  
+Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y. Generative adversarial nets. Advances in neural information processing systems, 27, 2014.  
+Gray, J. R. and Thompson, P. M. Neurobiology of intelligence: science and ethics. Nature Reviews Neuroscience, 5(6):471-482, 2004.  
+He, K., Chen, X., Xie, S., Li, Y., Dollar, P., and Girshick, R. Masked autoencoders are scalable vision learners. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 16000-16009, 2022.  
+Hersche, M., Zeqiri, M., Benini, L., Sebastian, A., and Rahimi, A. A neuro-vector-symbolic architecture for solving raven's progressive matrices. Nature Machine Intelligence, 5(4):363-375, 2023.  
+Hill, F., Santoro, A., Barrett, D. G., Morcos, A. S., and Lillicrap, T. Learning to make analogies by contrasting abstract relational structure. arXiv preprint arXiv:1902.00120, 2019.  
+Hu, S., Ma, Y., Liu, X., Wei, Y., and Bai, S. Stratified rule-aware network for abstract visual reasoning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 1567-1574, 2021.  
+Kim, H., Mnih, A., Schwarz, J., Garnelo, M., Eslami, A., Rosenbaum, D., Vinyals, O., and Teh, Y. W. Attentive neural processes. arXiv preprint arXiv:1901.05761, 2019.
+
+Kingma, D. P. and Welling, M. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.  
+Little, D. R., Lewandowsky, S., and Griffiths, T. L. A bayesian model of rule induction in raven's progressive matrices. In Proceedings of the Annual Meeting of the Cognitive Science Society, volume 34, 2012.  
+Locatello, F., Weissenborn, D., Unterthiner, T., Mahendran, A., Heigold, G., Uszkoreit, J., Dosovitskiy, A., and Kipf, T. Object-centric learning with slot attention. Advances in Neural Information Processing Systems, 33:11525-11538, 2020.  
+Lovett, A., Forbus, K., and Usher, J. A structure-mapping model of raven's progressive matrices. In Proceedings of the Annual Meeting of the Cognitive Science Society, 2010.  
+Małkinski, M. and Mandziuk, J. Deep learning methods for abstract visual reasoning: A survey on raven's progressive matrices. arXiv preprint arXiv:2201.12382, 2022a.  
+Malkinski, M. and Mandziuk, J. A review of emerging research directions in abstract visual reasoning. arXiv preprint arXiv:2202.10284, 2022b.  
+McCloskey, M. Intuitive physics. Scientific american, 248 (4):122-131, 1983.  
+McMillen, J. The colossal book of short puzzles and problems, 2007.  
+Mirza, M. Conditional generative adversarial nets. arXiv preprint arXiv:1411.1784, 2014.  
+Mondal, S. S., Webb, T., and Cohen, J. D. Learning to reason over visual objects. arXiv preprint arXiv:2303.02260, 2023.  
+Nie, W., Yu, Z., Mao, L., Patel, A. B., Zhu, Y., and Anandkumar, A. Bongard-logo: A new benchmark for human-level concept learning and reasoning. Advances in Neural Information Processing Systems, 33:16468-16480, 2020.  
+Pekar, N., Benny, Y., and Wolf, L. Generating correct answers for progressive matrices intelligence tests. arXiv preprint arXiv:2011.00496, 2020.  
+Raven, J. C. and Court, J. Raven's progressive matrices. Western Psychological Services Los Angeles, CA, 1938.  
+Razavi, A., Van den Oord, A., and Vinyals, O. Generating diverse high-fidelity images with vq-vae-2. Advances in neural information processing systems, 32, 2019.  
+Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. High-resolution image synthesis with latent
+
+diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10684-10695, 2022.  
+Shi, F., Li, B., and Xue, X. Raven's progressive matrices completion with latent gaussian process priors. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 9612-9620, 2021.  
+Shi, F., Li, B., and Xue, X. Compositional law parsing with latent random functions. In The Eleventh International Conference on Learning Representations, 2023.  
+Shi, F., Li, B., and Xue, X. Towards generative abstract reasoning: Completing raven's progressive matrix via rule abstraction and selection. In The Twelfth International Conference on Learning Representations, 2024.  
+Sohn, K., Lee, H., and Yan, X. Learning structured output representation using deep conditional generative models. Advances in neural information processing systems, 28, 2015.  
+Van Den Oord, A., Vinyals, O., et al. Neural discrete representation learning. Advances in neural information processing systems, 30, 2017.  
+Van Steenkiste, S., Locatello, F., Schmidhuber, J., and Bachem, O. Are disentangled representations helpful for abstract visual reasoning? Advances in Neural Information Processing Systems, 32, 2019.  
+Wang, D., Jamnik, M., and Lio, P. Abstract diagrammatic reasoning with multiplex graph networks. arXiv preprint arXiv:2006.11197, 2020.  
+Webb, T., Mondal, S. S., and Cohen, J. D. Systematic visual reasoning through object-centric relational abstraction. Advances in Neural Information Processing Systems, 36, 2024.  
+Wu, Y., Dong, H., Grosse, R., and Ba, J. The scattering compositional learner: Discovering objects, attributes, relationships in analogical reasoning. arXiv preprint arXiv:2007.04212, 2020.  
+Yan, W., Zhang, Y., Abbeel, P., and Srinivas, A. Videogpt: Video generation using vq-vae and transformers. arXiv preprint arXiv:2104.10157, 2021.  
+Yuan, J., Chen, T., Li, B., and Xue, X. Compositional scene representation learning via reconstruction: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(10):11540-11560, 2023.  
+Yun, X., Bohn, T., and Ling, C. A deeper look at bongard problems. In Advances in Artificial Intelligence: 33rd Canadian Conference on Artificial Intelligence, Canadian AI 2020, Ottawa, ON, Canada, May 13-15, 2020, Proceedings 33, pp. 528-539. Springer, 2020.
+
+Zhang, C., Gao, F., Jia, B., Zhu, Y., and Zhu, S.-C. Raven: A dataset for relational and analogical visual reasoning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5317-5327, 2019.  
+Zhang, C., Jia, B., Zhu, S.-C., and Zhu, Y. Abstract spatial-temporal reasoning via probabilistic abduction and execution. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9736-9746, 2021a.  
+Zhang, C., Xie, S., Jia, B., Wu, Y. N., Zhu, S.-C., and Zhu, Y. Learning algebraic representation for systematic generalization in abstract reasoning. arXiv preprint arXiv:2111.12990, 2021b.  
+Zhao, H., Cai, Z., Si, S., Ma, X., An, K., Chen, L., Liu, Z., Wang, S., Han, W., and Chang, B. Mmicl: Empowering vision-language model with multi-modal in-context learning. arXiv preprint arXiv:2309.07915, 2023.  
+Zheng, K., Zha, Z.-J., and Wei, W. Abstract reasoning with distracting features. Advances in Neural Information Processing Systems, 32, 2019.  
+Zhuo, T. and Kankanhalli, M. Effective abstract reasoning with dual-contrast network. In International Conference on Learning Representations, 2021.
+
+# A. Datasets
+
+In this section, we will introduce the datasets used for model training and testing. Tables 3 and 4 introduce the training, validation, and test splits of the datasets.
+
+Table 3: The ID tasks used to train and evaluate the baselines, task-specific solvers and UCGS-T. #Samples = number of samples per dataset, #Images = number of images per problem, #Candidates = number of candidate per problem.  
+
+<table><tr><td>Dataset</td><td colspan="3">RAVEN</td><td colspan="3">PGM</td></tr><tr><td>Split</td><td>Train</td><td>Valid</td><td>Test</td><td>Train</td><td>Valid</td><td>Test</td></tr><tr><td>#Samples</td><td>42K</td><td>14K</td><td>14K</td><td>1.4M</td><td>5K</td><td>200K</td></tr><tr><td>#Images</td><td colspan="3">9</td><td colspan="3">9</td></tr><tr><td>#Candidates</td><td colspan="3">8</td><td colspan="3">8</td></tr><tr><td>Image Size</td><td colspan="3">128×128</td><td colspan="3">128×128</td></tr></table>
+
+Table 4: The ID-ZS and OOD-ZS tasks used to evaluate the baselines and UCGS-T. #Samples = number of samples per dataset, #Images = number of images per problem, #Candidates = number of candidate per problem.  
+
+<table><tr><td>Dataset</td><td>G1-set</td><td>VAP</td><td>SVRT</td><td>O3-ID</td><td>VAP-ID</td><td>SVRT-ID</td></tr><tr><td>Split</td><td>Test</td><td>Test</td><td>Test</td><td>Test</td><td>Test</td><td>Test</td></tr><tr><td>#Samples</td><td>1K</td><td>200K</td><td>253</td><td>14K</td><td>14K</td><td>14K</td></tr><tr><td>#Images</td><td>4~5</td><td>6</td><td>8</td><td>5</td><td>6</td><td>8</td></tr><tr><td>#Candidates</td><td>NA</td><td>4</td><td>2</td><td>NA</td><td>8</td><td>2</td></tr><tr><td>Image Size</td><td>128×128</td><td>128×128</td><td>128×128</td><td>128×128</td><td>128×128</td><td>128×128</td></tr></table>
+
+Table 5: The subsets used to evaluate GPT-4o and UCGS-T. The datasets noted with * are subsets sampled from the corresponding complete datasets. #Samples = number of samples per dataset, #Images = number of images per problem, #Candidates = number of candidate per problem.  
+
+<table><tr><td>Dataset</td><td>RAVEN*</td><td>PGM*</td><td>G1-set</td><td>VAP*</td><td>SVRT</td><td>O3-ID</td><td>VAP-ID</td><td>SVRT-ID</td></tr><tr><td>Split</td><td>Test</td><td>Test</td><td>Test</td><td>Test</td><td>Test</td><td>Test</td><td>Test</td><td>Test</td></tr><tr><td>#Samples</td><td>700</td><td>1K</td><td>1K</td><td>1K</td><td>253</td><td>700</td><td>700</td><td>700</td></tr><tr><td>#Images</td><td>9</td><td>9</td><td>4~5</td><td>6</td><td>8</td><td>5</td><td>6</td><td>8</td></tr><tr><td>#Candidates</td><td>8</td><td>8</td><td>NA</td><td>4</td><td>2</td><td>NA</td><td>8</td><td>2</td></tr><tr><td>Image Size</td><td>128×128</td><td>128×128</td><td>128×128</td><td>128×128</td><td>128×128</td><td>128×128</td><td>128×128</td><td>128×128</td></tr></table>
+
+# A.1. ID Tasks
+
+RAVEN (Zhang et al., 2019). The RAVEN dataset is inspired by the widely used RPM tests. RAVEN provides a structured and systematically generated set of problems with clearly defined rules and visual concepts. Each RAVEN problem consists of a nine-image matrix  $(3 \times 3$  grid) where the bottom-right image is removed. The task is to infer the missing image from a set of candidate choices based on the underlying abstract rules. RAVEN contains seven configurations, each containing 10K problems with a specific scene structure. The configurations incorporate five concepts that define problem variations: Shape (e.g., triangle, square, pentagon, etc.), Size (relative scale of objects), Color (e.g., white, gray, black, etc.) and Position/Number (spatial arrangement and number of objects within each image). Each problem follows one or more concept-specific rules that govern changes in these visual concepts: Constant (the concept remains unchanged), Progression
+
+(the concept changes sequentially), Arithmetic (the concept follows a mathematical relationship) and Distribution-of-Three (the concept has three permutable values). We use problems from all seven configurations of RAVEN to train and test the models.
+
+PGM (Barrett et al., 2018). The PGM dataset is a large-scale RPM-like dataset. The problem structure of PGM problems is similar to RAVEN. A PGM problem also consists of a nine-image matrix, with the bottom-right image removed. The models infer the missing image based on the abstract rules of the given context and choose the correct answer from eight candidate images. The dataset contains 1.4M unique problems. The visual concepts are shape and line of the color, number, position, size and type. The visual concepts follow systematically defined rules, which have five categories: progression, XOR, OR, AND, and consistent union. The PGM dataset provides eight generalization regimes, and we use the neutral regime for model training and evaluation.
+
+# A.2. ID-ZS Tasks
+
+![](images/112d96069ad4b04f2e4e99d026ee327bf1db08a73dd3dc1a50ec6a0d70a99b75.jpg)
+
+![](images/4f9959299d5c70cea132857a454ad0a6c220b382f9c085125b306cb10acc90ef.jpg)
+
+![](images/c78566458bb61d226f404a3d725fd313ac430574846cf83dfa14e544357ace99.jpg)  
+Figure 4: Construction of the ID tasks.
+
+O3-ID. As shown in Figure 4a, we replace the bottom-right image of each  $3 \times 3$  image panel with a randomly selected distractor from the candidate panel to construct odd-one-out tests from RAVEN. This process transforms the bottom-right image into a rule-breaking image by modifying a specific visual concept of the correct image. The constructed O3-ID dataset shares the same visual concepts and abstract rules as RAVEN but follows the form of odd-one-out, where the models must identify the image breaking the rule of the problem panel.
+
+VAP-ID. The construction of the VAP-ID task is relatively straightforward in Figure 4b. Unlike RAVEN, a VAP presents a  $2 \times 3$  image matrix as its problem panel, and the abstract rules can change different visual concepts in different rows. Here, we construct VAP problems using RAVEN's design of analogy. Specifically, we remove the final row and retain the first two rows (i.e., the first six images) of each RAVEN problem panel to generate a problem of VAP-ID. In this way, VAP-ID problems preserve the same in-distribution visual concepts and abstract rules as RAVEN, while adopting a different structure of problem panels. These problems are designed to assess whether models can still parse the learned abstract rules and complete the task when the structure of problem panels is changed.
+
+SVRT-ID. In Figure 4c, we split each RAVEN problem panel into two parts to construct SVRT-style problems. The first two rows form the left panel, and the last two rows form the right panel. We then extract the bottom-right image from both panels to form the query panel, while the remaining images are kept as the corresponding problem panels. In this way of construction, the left and right problem panels exhibit different abstract rules of the same type. Therefore, the association between the query images and the problem panels is typically unambiguous.
+
+# A.3. OOD-ZS Tasks
+
+G1-set (Mandyckiak & Zychowski, 2019). The G1-set dataset evaluates the AVR ability of models through odd-one-out problems. Unlike RPMs, where the task is to infer missing images, an odd-one-out problem requires the models to recognize the oddest figure in a panel based on relational differences in visual concepts. In G1-set, the odd figure can differ on the visual concepts size, shape, shading, rotation, etc. To ensure the problems are well-defined, the G1-set problems are constructed using a strict rule. In each problem with  $N$  figures, there is one subset of  $N - 1$  figures that share a common set of  $1 \sim 3$  features. The remaining figure is the odd image that lacks this commonality. The difficulty of problems is controlled by the number of figures and the number of shared features.
+
+VAP (Hill et al., 2019). The VAP dataset evaluates the analogy-making ability of models. Inspired by previous human and machine reasoning tasks like RPMs, VAP requires the models to identify abstract rules and apply them across different visual concepts. Each problem panel consists of a source sequence and a target sequence. The source sequence has three images where an abstract rule is instantiated. The target sequence has two images where the removed position must be completed according to the rule of the source sequence. The candidate panel provides four candidate answer images, including one correct answer and three distractors. The goal is to select the correct answer that best completes the target sequence by analogy with the source sequence. The core challenge in VAP is to apply learned rules across different visual domains. VAP defines four logical or mathematical abstract rules: XOR, OR, AND and progression. The abstract rules can be instantiated in seven different visual concepts: line type, line color, shape type, shape color, shape size, shape quantity and shape position. Each visual concept contains 10 possible values to increase the complexity of problems.
+
+SVRT (Fleuret et al., 2011). The SVRT dataset evaluates the relational visual reasoning ability of models. Unlike traditional image classification tasks that rely on texture, SVRT problems require holistic pattern recognition based on spatial relationships among randomly generated abstract shapes. Each of the 23 problems in the dataset requires assigning a query image to one of two panels, where the distinction is defined by an abstract rule governing the global arrangement of parts. Importantly, these rules are not based on the appearance, location, or topology of individual parts but on how multiple parts interact as a whole. The problems in SVRT involve relational concepts rather than low-level features. Some examples of reasoning principles used in the dataset include: proximity (are two or more shapes close to each other), similarity (do multiple shapes share the same size/form), symmetry (are the parts arranged symmetrically), topology (are certain shapes enclosed within others), counting (does the image contain an even or odd number of objects) and identity relations (do two objects have the same shape or orientation). For example, one problem might require distinguishing two identical shapes vs. two different images.
+
+# B. Details of Models
+
+# B.1.UCGS-T
+
+This section describes the architectures of learnable networks in UCGS-T and the choice of hyperparameters. We introduce the networks in the order of image encoder, image decoder, patch encoder, panel encoder and patch decoder.
+
+- Image Encoder:
+
+-  $4 \times 4$  Conv, stride 2, padding 1, 64, ReLU
+
+-  $4 \times 4$  Conv, stride 2, padding 1, 64, ReLU  
+-  $4 \times 4$  Conv, stride 2, padding 1, 128, ReLU  
+-  $3\times 3$  Conv, stride 1, padding 1, 128  
+-ResBlock,hidden32,128  
+- ResBlock, hidden 32, 128, ReLU  
+-  $4 \times 4$  Conv, stride 2, padding 1, 64, ReLU  
+-  $4 \times 4$  Conv, stride 2, padding 1, 128, ReLU  
+-  $3\times 3$  Conv, stride 1, padding 1, 128  
+-ResBlock,hidden32,128  
+- ResBlock, hidden 32, 128, ReLU  
+-  $1\times 1$  Conv, stride 1, 64
+
+# - Image Decoder:
+
+-  $3 \times 3$  Conv, stride 1, padding 1, 128, ReLU  
+-ResBlock,hidden32,128  
+- ResBlock, hidden 32, 128, ReLU  
+-  $4 \times 4$  Deconv, stride 2, padding 1, 64, ReLU  
+-  $4\times 4$  Deconv, stride 2, padding 1, 128  
+-ResBlock,hidden32,128  
+- ResBlock, hidden 32, 128, ReLU  
+-  $4 \times 4$  Deconv, stride 2, padding 1, 64, ReLU  
+-  $4 \times 4$  Deconv, stride 2, padding 1, 64, ReLU  
+-  $4 \times 4$  Deconv, stride 2, padding 1, 3
+
+- Patch Encoder: The linear network to encode position information is
+
+LayerNorm, 128  
+- Fully Connected, 128, ReLU  
+- Fully Connected, 128
+
+The patch encoder is a 12-layer Transformer decoder where the hidden size is 128 and the number of attention heads is 8.
+
+- Panel Encoder: The panel encoder is a 12-layer Transformer decoder with the hidden size 128 and the number of attention heads 8. We use a linear layer to convert the input keys and values to hidden vectors. Another linear layer is introduced to convert the input queries to hidden vectors.  
+- Patch Decoder: The panel encoder is a 12-layer Transformer decoder with the hidden size 128 and the number of attention heads 8. We use a linear layer to convert the input keys and values to hidden vectors. Another linear layer is introduced to convert the input queries to hidden vectors. The output of the Transformer decoder is projected to the index of the discrete vector in the codebook  $e$  by a linear layer without the bias parameter.
+
+The image encoder and image decoder are first trained by setting the learning rate as  $4 \times 10^{-4}$  and batch size as 64, referring to the original configuration in VQVAE (Van Den Oord et al., 2017). Then we freeze the parameters of the image encoder and image decoder, training the remaining part by setting the learning rate as  $3 \times 10^{-4}$ , batch size as 128. We monitor the performance of UCGS-T on the validation set after each training epoch and save the checkpoint with the best validation accuracy. The parameters are updated by Adam optimizer (Kingma & Ba, 2014).
+
+# B.2. Ablation Baselines
+
+Besides the patch-based backbone of UCGS-T, we introduce another two backbones. The object-centric backbone can distinguish objects in scenes and extract representations for individual objects. We adopt the encoder of STSN (Mondal et al., 2023) as the object-centric backbone, and train the backbone using the official code<sup>1</sup>. The monolithic backbone encodes a
+
+scene into one representation. The encoder and decoder of the monolithic backbone consist of continuous convolutional layers that map the image into a single representation of size 512. We call the slots and representations extracted from the object-centric and monolithic backbones as patches for convenience. We introduce two types of baseline conditional generative solvers: the Transformer-based solver (Vaswani et al., 2017) and the ANP-based solver (Kim et al., 2019). All the parameters are updated by Adam optimizer (Kingma & Ba, 2014).
+
+Transformer-based Solver (Vaswani et al., 2017). The Transformer-based conditional generative network inputs all context patches  $Z^{c} = \{Z_{i}^{c}|i = 1,\dots ,N_{c}\}$  to a Transformer encoder. The output of the Transformer decoder is the key and query of a Transformer decoder, and the embeddings of the target positions  $\{\mathrm{PE}_i|i = 1,\dots ,N_t\}$  are regarded as the query of the Transformer decoder. The generative process is
+
+$$
+\pmb {h} _ {i} ^ {k v} = f _ {\mathrm {K V}} \left(\pmb {Z} _ {i} ^ {c}\right), \qquad \qquad i = 1, \dots , N _ {c},
+$$
+
+$$
+\hat {\boldsymbol {h}} ^ {k v} = \operatorname {T F E n c o d e r} \left(\boldsymbol {h} ^ {k v}\right),
+$$
+
+$$
+\boldsymbol {h} _ {i} ^ {q} = f _ {\mathrm {Q}} \left(\mathrm {P E} _ {i}\right), \quad i = 1, \dots , N _ {t}, \tag {10}
+$$
+
+$$
+\hat {\boldsymbol {h}} ^ {q} = \operatorname {T F D e c o d e r} \left(\boldsymbol {h} ^ {q}, \hat {\boldsymbol {h}} ^ {k v}\right),
+$$
+
+$$
+\boldsymbol {Z} _ {i} ^ {t} = f _ {\mathrm {O}} \left(\hat {\boldsymbol {h}} _ {i} ^ {q}\right), \quad i = 1, \ldots , N _ {t}.
+$$
+
+In the generative process,  $f_{\mathrm{KV}}$ ,  $f_{\mathrm{Q}}$ , and  $f_{\mathrm{O}}$  are single-layer feedforward networks. We set the hidden size of the Transformer encoder and decoder as 512, the number of Transformer layers as 12, and the number of attention heads as 8.
+
+ANP-based Solver (Kim et al., 2019). The ANP conditional generative baseline regards the mapping from context patches to target patches as stochastic functions. The context slots are used to construct the context of stochastic functions, and the model will sample a function from the posterior to map the embeddings of target positions into target patches. We set the size of the global latent as 512, the number of attention layers as 12, and the number of attention heads as 8, and other hyperparameters and the model architecture follow the 2D regression configuration in (Kim et al., 2019).
+
+# B.3. Task-specific Solvers
+
+We use the official codes of  $\mathrm{PrAE}^2$ ,  $\mathrm{NVSA}^3$ ,  $\mathrm{GCA}^4$ ,  $\mathrm{ALANS}^5$  and  $\mathrm{RAISE}^6$  in the experiments. All task-specific solvers are trained without additional annotation information. To stabilize the training process, ALANS is trained based on the pretrained checkpoint provided by the authors.
+
+# B.4. GPT-40
+
+We design task-specific prompts for GPT-4o, which are illustrated in Figure 5. In the prompts, we will provide the information about the problem structures, e.g., the problem panel of PGM is a  $3 \times 3$  matrix and the problem panel of VAP is a  $2 \times 3$  matrix. Each panel is regarded as an input image of GPT-4o.
+
+# B.5. Computational Resource
+
+All the models are trained on a single 24GB NVIDIA GeForce RTX 4090 GPU and implemented using PyTorch (Paszke et al., 2019).
+
+# C. Additional Experimental Results
+
+# C.1. Answer Generation
+
+We provide additional examples of answer generation. Figures 6 and 7 show the generated results for various AVR problems on the RAVEN and PGM datasets, respectively. Overall, the generated results on RAVEN are more accurate, corresponding
+
+to the selection accuracy, primarily because the problems of RAVEN contain less noise, which has been discussed in previous works. UCGS-T achieves the best generation performance on both RAVEN and PGM. UCGS-T can handle the uncertainty caused by noise effectively. For example, in the 4th sample of PGM, the shape and position of the generated object differ from the ground truth. TF-Patch generates relatively accurate results on RAVEN, but it tends to produce rule-violating answers on PGM. Overall, TF-Patch generates fewer artifacts in the images. Baselines using the object-centric backbone (TF-OCL and ANP-OCL) struggle to generate clear images, often producing large black regions. Baselines based on the monolithic backbone tend to produce samples that deviate significantly from the ground truths.
+
+# References
+
+Barrett, D., Hill, F., Santoro, A., Morcos, A., and Lillicrap, T. Measuring abstract reasoning in neural networks. In International conference on machine learning, pp. 511-520. PMLR, 2018.  
+Fleuret, F., Li, T., Dubout, C., Wampler, E. K., Yantis, S., and Geman, D. Comparing machines and humans on a visual categorization test. Proceedings of the National Academy of Sciences, 108(43):17621-17625, 2011.  
+Hill, F., Santoro, A., Barrett, D., Morcos, A., and Lillicrap, T. Learning to make analogies by contrasting abstract relational structure. In International Conference on Learning Representations, 2019.  
+Kim, H., Mnih, A., Schwarz, J., Garnelo, M., Eslami, A., Rosenbaum, D., Vinyals, O., and Teh, Y. W. Attentive neural processes. arXiv preprint arXiv:1901.05761, 2019.  
+Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.  
+Mańdzik, J. and Žychowski, A. Deepiq: A human-inspired ai system for solving iq test problems. In 2019 International Joint Conference on Neural Networks (IJCNN), pp. 1-8. IEEE, 2019.  
+Mondal, S. S., Webb, T., and Cohen, J. D. Learning to reason over visual objects. arXiv preprint arXiv:2303.02260, 2023.  
+Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al. Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems, 32, 2019.  
+Van Den Oord, A., Vinyals, O., et al. Neural discrete representation learning. Advances in neural information processing systems, 30, 2017.  
+Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., and Polosukhin, I. Attention is all you need. Advances in neural information processing systems, 30, 2017.  
+Zhang, C., Gao, F., Jia, B., Zhu, Y., and Zhu, S.-C. Raven: A dataset for relational and analogical visual reasoning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5317-5327, 2019.
+
+(a) RPM  
+![](images/af1a6fa70c84868e866adb6bd051618df561b988d3bdc756d7979f42d7ee58cf.jpg)  
+Each task consists of a question image with a  $3 \times 3$  matrix. Eight of the nine resulting cells contain an abstract shape, while one cell on the bottom right-hand side of the matrix is empty. Your task is to complete the matrix by finding the missing shape among multiple possible alternatives. One of the option images is the correct answer. The first image is the question image. The second image is the option images, where the options are given from left to right, and the index of the first image is 1. Please output the index of the correct answer in the following JSON format: {"Result": "Index"}
+
+(b) VAP  
+![](images/fad406b421fdabea9ba8bb19a83be566565cc0afbf8a1c194bb10d9389a19b34.jpg)  
+Each task consists of a question image with a  $2 \times 3$  matrix. Five of the six resulting cells contain an abstract shape, while one cell on the bottom right-hand side of the matrix is empty. Your task is to complete the matrix by finding the missing shape among multiple possible alternatives. One of the option images is the correct answer. The first image is the question image. The second image is the option images, where the options are given from left to right, and the index of the first image is 1. Please output the index of the correct answer in the following JSON format: {"Result": "Index"}
+
+(c) Odd-one-out  
+![](images/24f0ab94776421da91587d9807d2bad96714a00db9f1c95c6977cf507dc8a7d1.jpg)  
+The input is a question panel containing N images, where each image contains an abstract shape. Your task is to identify the single image that does not share the common attribute, pattern, category, or relationship exhibited by the other N-1 images. The images are given from left to right, and the index of the first image is 1. Please output the index of the answer in the following JSON format: {"Result": "Index"}
+
+(d) SVRT  
+Figure 5: Task-specific prompts of GPT-4o.  
+![](images/d9f85854f745cc7523e6628edcf9e3b2bfe4c9dd7fda0e6da1643ba82600935d.jpg)  
+Each task consists of two question panels and a query panel. Each question panel contains 8 images. Each image contains simple geometric shapes. The two question panels show sets of geometric shapes organized according to different rules. The query panel has two query images. Your task is to determine which question panel's rule matches the rule displayed in the query image. The input images are question panel 1, question panel 2 and query panel in order. The images in panels are given from left to right, and the index of the first image is 1. Please output which panel the query images belong to. Using the following JSON format: {"Query1": "Index", "Query2": "Index"}
+
+![](images/5afe597cdcb4275591b30d88b8e486427ff32f9deed44ef868a59e60875055e5.jpg)  
+Figure 6: Comparison of generation results on RAVEN.
+
+![](images/565781e8be6a2917f0dc8b4b943fc971c90613a417f6f652dddc2186be42ec1e.jpg)  
+Figure 7: Comparison of generation results on PGM.

2025/Beyond Task-Specific Reasoning_ A Unified Conditional Generative Framework for Abstract Visual Reasoning/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8d150fc06a990b412c95ca5c22ad15673f7c571d522f03018aa23291f529f11e
+size 1364427

2025/Beyond The Rainbow_ High Performance Deep Reinforcement Learning on a Desktop PC/da17f283-f8ac-4488-b3db-031049104729_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:271aecf4a0ef2ef18184b7928eacd87cfb85e984b3db501a3f6ad8371c236af2
+size 4807473

2025/Beyond The Rainbow_ High Performance Deep Reinforcement Learning on a Desktop PC/full.md

@@ -0,0 +1,526 @@
+# Beyond The Rainbow: High Performance Deep Reinforcement Learning on a Desktop PC
+
+Tyler Clark<sup>1</sup> Mark Towers<sup>1</sup> Christine Evers<sup>1</sup> Jonathon Hare<sup>1</sup>
+
+# Abstract
+
+Rainbow Deep Q-Network (DQN) demonstrated combining multiple independent enhancements could significantly boost a reinforcement learning (RL) agent's performance. In this paper, we present "Beyond The Rainbow" (BTR), a novel algorithm that integrates six improvements from across the RL literature to Rainbow DQN, establishing a new state-of-the-art for RL using a desktop PC, with a human-normalized interquartile mean (IQM) of 7.4 on Atari-60. Beyond Atari, we demonstrate BTR's capability to handle complex 3D games, successfully training agents to play Super Mario Galaxy, Mario Kart, and Mortal Kombat with minimal algorithmic changes. Designing BTR with computational efficiency in mind, agents can be trained using a high-end desktop PC on 200 million Atari frames within 12 hours. Additionally, we conduct detailed ablation studies of each component, analyzing the performance and impact using numerous measures. Code is available at https://github.com/VIPTankz/BTR.
+
+# 1. Introduction
+
+Deep Reinforcement Learning (RL) has achieved numerous successes in complex sequential decision-making tasks, most rapidly since Mnih et al. (2015) proposed Deep Q-Learning (DQN). With this success, RL has become increasingly popular among smaller research labs, the hobbyist community, and even the general public. However, recent state-of-the-art approaches (Schrittwieser et al., 2020; Badia et al., 2020a; Hessel et al., 2021; Kapturowski et al., 2023) are increasingly out of reach for those with more lim
+
+$^{1}$ School of Electronics and Computer Science, University Of Southampton, Southampton, UK. Correspondence to: Tyler Clark <tjc2g19@soton.ac.uk>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+![](images/82ee7208ab96ca6dd8a6e9d869f034ed73bb26174b8bb308d37e91da3d7900ae.jpg)  
+Figure 1: Interquartile mean human-normalized performance for BTR against other RL algorithms on the Atari benchmark in terms of walltime performance (all results use 200M frames). The results for DQN and Rainbow DQN are those reported in RLiable (Agarwal et al., 2021), and Dreamer-v3 refers to Hafner et al. (2023). Shaded areas show  $95\%$  bootstrapped confidence intervals, with BTR using 4 seeds.
+
+ited compute resources, either in terms of the required hardware or the walltime necessary to train a single agent. This is a unique issue in RL compared to natural language processing or image recognition which have foundation models that can be efficiently fine-tuned for a new task or problem (Lv et al., 2024). Meanwhile, RL agents must be trained afresh for each environment. Therefore, the development of powerful RL algorithms that can be trained quickly on inexpensive hardware is crucial for smaller research labs and the hobbyist community.
+
+These concerns are not new. Ceron & Castro (2021) highlighted that Rainbow DQN (Hessel et al., 2018) required 34,200 GPU hours (equivalent to 1435 days) of training, making the research impossible for anyone except a few research labs, with more recent algorithms exacerbating this problem. Recurrent network architectures (Horgan et al., 2018), high update to sample ratio (D'Oro et al., 2022), and the use of world-models and search-based techniques (Schrittwieser et al., 2020), all increase the computational resources necessary to train agents. Many of these use distributed approaches requiring multiple CPUs and GPUs (or TPU's), or requiring numerous days and weeks to train a single agent, dramatically decreasing RL's accessibility.
+
+For this purpose, we develop "Beyond the Rainbow" (BTR), taking the same principle as Rainbow DQN (Hessel et al., 2018), selecting 6 previously independently evaluated improvements and combining them into a singular algorithm (Section 3). These components were chosen for their performance qualities or to reduce the computational requirements for training an agent. As a result, BTR sets a new state-of-the-art score for Atari-60 (Bellemare et al., 2013) (excluding recurrent approaches) with an Interquartile Mean (IQM) of  $7.4^{1}$  using a single desktop machine in less than 12 hours, and outperforms Rainbow DQN on Procgen (Cobbe et al., 2020) in less than a fifth of the wait-time (Section 4.1). Further, we demonstrate BTR's potential by training agents to solve three modern 3D games for the first time, Mario Kart Wii, Super Mario Galaxy and Mortal Kombat, that each contain complex mechanics and graphics (Section 4.2). To verify the effectiveness and effect of the six improvements to BTR, in Section 5.1, we conduct a thorough ablation of each component, plotting their impact on the Atari-5 environments and in Section 5.2, we utilize seven different measures to analyse the component's impact on the agent's policy and network weights. This allows us to more precisely understand how the components impact BTR beyond performance or waittime.
+
+In summary, we make the following contributions to state-of-the-art RL.
+
+- High Performance (Section 4.1) - BTR outperforms the state-of-the-art for non-recurrent RL on the Atari-60 benchmark, with an IQM of 7.4 (compared to Rainbow DQN's 1.9), outperforming humans on 52/60 games. Furthermore, BTR outperforms Rainbow DQN with Impala on the Progen benchmark despite using a smaller model and  $80\%$  less walltime.  
+- Modern Environments (Section 4.2) - Testing beyond Atari, we demonstrate BTR can train agents for three modern games: Super Mario Galaxy (final stage), Mario Kart Wii (Rainbow Road), and Mortal Kombat (Endurance mode). These environments contain 3D graphics and complex physics and have never been solved using RL.  
+- Computationally Accessible (Figure 6) - Using a high-end desktop PC, BTR trains Atari agents for 200 million frames in under 12 hours, significantly faster than Rainbow DQN's 35 hours. This increases RL research's accessibility for smaller research labs and hobbyists without the need for GPU clusters or excessive walltime.
+
+- Component Impact Analysis (Section 5) - We conduct thorough ablations of BTR without each component, investigating performance and other measures. We discover that BTR widens action gaps (reducing the effects of approximation errors), is robust to observation noise, and reduces neuron dormancy and weight matrix norm (shown to improve plasticity throughout training).
+
+# 2. Background
+
+Before describing BTR's extensions, we outline standard RL mathematics, how DQN is implemented, and Rainbow DQN's extensions.
+
+# 2.1. RL Problem Formulation
+
+We adopt the standard formulation of RL (Sutton & Barto, 2018), described as a Markov Decision Process (MDP) defined by the tuple  $(\mathcal{S},\mathcal{A},\mathcal{P},\mathcal{R})$ , where  $\mathcal{S}$  is the set of states,  $\mathcal{A}$  is the set of actions,  $\mathcal{P}:\mathcal{S}\times \mathcal{A}\to \Delta (\mathcal{S})$  is the stochastic transition function, and  $\mathcal{R}:\mathcal{S}\times \mathcal{A}\to \mathbb{R}$  is the reward function. The agent's objective is to learn a policy  $\pi :S\rightarrow \Delta (\mathcal{A})$  that maximizes the expected sum of discounted rewards  $\mathbb{E}_{\pi}[\sum_{t = 0}^{\infty}\gamma^{t}r(s_{t},a_{t})]$ , where  $\gamma \in [0,1)$  is the discount rate.
+
+# 2.2. Deep Q-Learning (DQN)
+
+One popular method for solving MDPs is Q-Learning (Watkins & Dayan, 1992) where an agent learns to predict the expected sum of discounted future rewards for a given state-action pair. To allow agents to generalize over states and thus be applied to problems with larger state spaces, Mnih et al. (2013) successfully combined Q-Learning with neural networks. To do this, training minimizes the error between the predictions from a parameterized network  $\mathrm{Q}_{\theta}$  and a target defined by
+
+$$
+r _ {t} + \gamma \max  _ {a \in A} Q _ {\theta^ {\prime}} \left(s _ {t + 1}, a\right), \tag {1}
+$$
+
+where  $Q_{\theta'}$  is an earlier version of the network referred to as the target network, which is periodically updated from the online network  $Q_{\theta}$ . The data used to perform updates is gathered by sampling from an Experience Replay Buffer (Lin, 1992), which stores states, actions, rewards, and next states experienced by the agent while interacting with the environment. To effectively explore the environment, greedy exploration is used, where each observation has a  $\frac{\epsilon}{1}$  probability of choosing a random action.
+
+# 2.3. Rainbow DQN and Improvements to DQN
+
+In collecting 6 different improvements to DQN, Rainbow DQN (Hessel et al., 2018) proved cumulatively that these improvements could achieve a greater performance
+
+than any individually. We briefly explain the individual improvements, ordered by performance impact, most of which are preserved within BTR (see Table 1). For more detail, we refer readers to the extension's respective papers:
+
+1. Prioritized Experience Replay - To select training examples, DQN sampled uniformly from an Experience Replay Buffer, assuming that all examples are equally important to train with. Schaul et al. (2015) proposed sampling training examples proportionally to their last seen absolute temporal difference error, increasing training on samples for which the network most inaccurately predicts their future rewards.  
+2. N-Step - Q-learning utilizes bootstrapping to minimize the difference between the predicted value and the resultant reward plus the maximum value of the next state (Eq. 1). N-step (Sutton et al., 1998) reduces the reliance on this bootstrapped next value by considering the next  $n$  rewards and the observation in  $n$  timesteps (Rainbow DQN used  $n = 3$ ).  
+3. Distributional RL - Due to the stochastic nature of RL environments and agent policies, Bellemare et al. (2017) proposed learning the return distribution rather than scalar expectation. This was done through modeling the return distributions using probability masses and the Kullbeck-Leibler divergence loss function.  
+4. Noisy Networks - Agents can often insufficiently explore their environment resulting in sub-optimal policies. Fortunato et al. (2018) added parametric noise to the network weights, causing the model's outputs to be randomly perturbed, increasing exploration during training, particularly for states where the agent has less confidence.  
+5. Dueling DQN - The agent's Q-value can be rewritten as the sum of state-value and advantage  $(Q(s,a) = V(s) + A(s,a))$ . Looking to improve action generalization, Wang et al. (2016) split the hidden layers into two separate streams for the value and advantage, recombining them with  $Q(s,a) = V(s) + (A(s,a) - \frac{1}{|A|}\sum_{a'}A(s,a'))$ .  
+6. **Double DQN - Selecting a target Q-value with the maximum Q-value from the next observation (Eq. 1) can frequently cause overestimation, negatively affecting agent performance.** To reduce this overestimation, Van Hasselt et al. (2016) propose utilizing the online network rather than the target network to select the next action when forming targets, defined as:
+
+$$
+r _ {t} + \gamma \mathrm {Q} _ {\theta^ {\prime}} \left(s _ {t + 1}, \underset {a \in A} {\arg \max } \mathrm {Q} _ {\theta} \left(s _ {t + 1}, a\right)\right). \tag {2}
+$$
+
+# 3. Beyond the Rainbow - Extensions and Improvements
+
+Building on Rainbow DQN (Hessel et al., 2018), BTR includes 6 more improvements undiscovered in 2018.2 Additionally, as hyperparameters are critical to agent performance, Section 3.2 discusses key hyperparameters and our choices. In the appendices, we include a table of hyperparameters, a figure of the network architecture and the agent's loss function (Appendices D.2, E and E.2). Finally, the source code using Gymnasium (Towers et al., 2024) is included within the supplementary material to help future work build upon or utilize BTR.
+
+# 3.1. Extensions
+
+Impala Architecture + Adaptive Maxpooling - Espeholt et al. (2018) proposed a convolutional residual neural network architecture based on He et al. (2016), featuring three residual blocks<sup>3</sup>, substantially increasing performance over DQN's three-layer convolutional network. Following Cobbe et al. (2020), we scale the width of the convolutional layers by 2 to improve performance. We include an additional 6x6 adaptive max pooling layer after the convolutional layers (Schmidt & Schmied, 2021), which was found to speed up learning and support different input resolutions. The adaptive maxpooling is identical to a standard 2D maxpooling layer, but can be used with any input resolution as it automatically adjusts the stride and kernel size to fit a specified output size.
+
+Spectral Normalization (SN) - To help stabilize the training of discriminators in Generative Adversarial Networks (GANs), Miyato et al. (2018) proposed Spectral Normalization to help control the Lipschitz constant of convolutional layers. SN works to normalize the weight matrices of each layer in the network by their largest singular value, ensuring that the transformation applied by the weights does not distort the input data excessively, which can lead to instability during training. Bjorck et al. (2021) and Gogianu et al. (2021) found that SN could improve performance in RL, especially for larger networks and Schmidt & Schmied (2021) found SN reduced the number of updates required before initial progress is made.
+
+Implicit Quantile Networks (IQN) - Dabney et al. (2018) improved upon Bellemare et al. (2017), used in Rainbow DQN, learning the return distribution over the probability space rather than probability distribution over return val
+
+Table 1: A comparison of components between Rainbow DQN (Hessel et al., 2018) and BTR.  
+
+<table><tr><td>Added To Rainbow DQN</td><td>Same As Rainbow DQN</td><td>Removed From Rainbow DQN</td></tr><tr><td>Impala (Scale=2)</td><td>N-Step TD Learning</td><td>Double (N/A with Munchausen)</td></tr><tr><td>Adaptive Maxpooling (6x6)</td><td>Prioritized Experience Replay</td><td>C51 (Upgraded to IQN)</td></tr><tr><td>Spectral Normalization</td><td>Dueling</td><td></td></tr><tr><td>Implicit Quantile Networks</td><td>Noisy Networks</td><td></td></tr><tr><td>Munchausen</td><td></td><td></td></tr><tr><td>Vectorized Environments</td><td></td><td></td></tr></table>
+
+ues. This removes the limit on the range of Q-values that can be expressed, and enables learning the expected return at every probability.
+
+Munchausen RL - Bootstrapping is a core aspect of RL; used to calculate target values (Eq. 1) with most algorithms using the reward,  $r_t$ , and the optimal Q-value of the next state,  $Q^*$ . However, since in practice the optimal policy is not known, the current policy  $\pi$  is used. Munchausen RL (Vieillard et al., 2020) leverages an additional estimate in the bootstrapping process by adding the scaled-log policy to the loss function (Eq. 3 where  $\alpha \in [0,1]$  is a scaling factor,  $\sigma$  is the softmax function, and  $\tau$  is the softmax temperature). This assumes a stochastic policy, therefore DQN is converted to Soft-DQN with  $\pi_{\theta'} = \sigma(\frac{Q_{\theta'}}{\tau})$ . As Munchausen does not use argmax over the next state, Double DQN is obsolete. Munchausen RL's update rule is
+
+$$
+\begin{array}{l} Q _ {\theta} \left(s _ {t}, a _ {t}\right) = r _ {t} + \alpha \tau \ln \pi_ {\theta^ {\prime}} \left(a _ {t} \mid s _ {t}\right) + \\ \gamma \sum_ {a ^ {\prime} \in A} \pi_ {\theta^ {\prime}} \left(a ^ {\prime} \mid s _ {t + 1}\right) \left(Q _ {\theta^ {\prime}} \left(s _ {t + 1}, a ^ {\prime}\right) - \tau \ln \left(\pi_ {\theta^ {\prime}} \left(a ^ {\prime} \mid s _ {t + 1}\right)\right)\right). \tag {3} \\ \end{array}
+$$
+
+Vectorization - RL agents typically take multiple steps in a single environment, followed by a gradient update with a small batch size (Rainbow DQN took 4 environment steps, followed by a batch of 32). However, taking multiple steps in parallel and performing updates on larger batches can significantly reduce walltime. We follow Schmidt & Schmied (2021), taking 1 step in 64 parallel environments with one gradient update with batch size 256 (Schmidt & Schmied (2021) took two gradient updates). This results in a replay ratio (ratio of gradient updates to environment steps) of  $\frac{1}{64}$ . Higher replay ratios have been shown to improve performance (D'Oro et al., 2022), however we opt to keep this value low to reduce walltime.
+
+# 3.2. Hyperparameters
+
+Hyperparameters have repeatedly shown to have a very large impact on performance in RL (Ceron et al., 2024), thus we perform a small amount of tuning to improve per
+
+formance. Firstly, how frequently the target network is updated is closely intertwined with batch size and replay ratio. We found that updating the target network every 500 gradient steps<sup>4</sup> performed best. Given our high batch size, we additionally performed minor hyperparameter tests using different learning rates finding that a slightly higher learning rate of  $1 \times 10^{-4}$  performed best, compared to  $6.25 \times 10^{-5}$  in Rainbow DQN. In Appendix D.2, we clarify the meaning of the terms frames, steps and transitions.
+
+For many years, RL algorithms have used a discount rate of 0.99, however, when reaching high performance, lower discount rates alter the optimal policy, causing even optimally performing agents to not collect the maximum cumulative rewards. To prevent this, we follow MuZero Reanalyse (Schrittwieser et al., 2021) using  $\gamma = 0.997$ . For our Prioritized Experience Replay, we use the lower value of  $\alpha = 0.2$ , the parameter used to determine sample priority, recommended by Toromanoff et al. (2019) when using IQN. Lastly, many previous experiments used only noisy networks or  $\epsilon$ -greedy exploration, however, we opt to use both until 100M frames, then set  $\epsilon$  to zero, effectively disabling it. We elaborate on this decision in Appendix F.
+
+# 4. Evaluation
+
+To assess BTR, we test on two standard RL benchmarks, Atari (Bellemare et al., 2013) and Progen (Cobbe et al., 2020) in Section 4.1. Secondly, we train agents for three modern games (Super Mario Galaxy, Mario Kart Wii, and Mortal Kombat) with complex 3D graphics and physics in Section 4.2, never before shown to be trainable with RL.
+
+# 4.1. Atari and Progen Performance
+
+We evaluate BTR on the Atari-60 benchmark following (Machado et al., 2018) and without life information (see Appendix I for the impact), evaluating every million frames on 100 episodes. Figure 1 plots BTR against DQN, Rainbow DQN and Dreamer-v3, showing BTR's competitive performance despite using significantly less walltime. Fig
+
+![](images/dc33f5fc2c87a8cdb9aa91e60292e33e9c4f480d04bb84dcf6392b2c98710356.jpg)  
+Figure 2: Box plot performance of BTR (4 seeds) against other popular algorithms such as MEME (Kapturowski et al., 2023), Dreamer v3 (Hafner et al., 2023), Bigger, Better, Faster (BBF) (Schwarzer et al., 2023) and EfficientZero-v2 (EZV2) (Wang et al., 2024). Brackets show the number of frames the algorithms use, the number of walltime hours and the hardware used respectively. Shaded areas show  $95\%$  confidence intervals. Top: Atari 55 game benchmark - we used the overlapping games 55 between the popular Atari-57 benchmark, and the 60 games used in RLiable (Agarwal et al., 2021). Bottom: Atari-26 benchmark, commonly used for testing sample-efficient algorithms.
+
+![](images/f34a22271f965801167daad3b2cbcef50713070559a34b3aa7b3e581e9eb5bc4.jpg)  
+Figure 3: BTR compared to Rainbow DQN + Impala (width x4) (Cobbe et al., 2020) after 200M frames on the Procgen benchmark. Shaded areas show  $95\%$  CIs, with results averaged over 5 seeds.
+
+ure 2 shows a box plot comparison of final performance. In comparison to human expert performance, BTR equals or exceeds them in 52 of 60. Importantly, we find that BTR appears to continue increasing performance beyond 200 million frames, indicating that higher performance is still possible with more time and data. Results tables and graphs can be found in Appendices A and B, respectively.
+
+To further confirm BTR's performance, we benchmark on Progen (Cobbe et al., 2020), a procedurally generated set of environments aiming to prevent overfitting to specific tasks, a prevalent problem in RL (Justesen et al., 2018; Juliani et al., 2019). The results are shown in Figure 3 with individual games in Appendix B. BTR is able to exceed Rainbow DQN + Impala's performance, despite using sig-
+
+nificantly fewer convolutional filters (which Cobbe et al. (2020) found to significantly improve performance) and using 8 hours of waittime compared to 41. While BTR provides an improvement over Rainbow DQN in Procgen, we did not target procedurally generated environments thus it does not currently compete with the state-of-the-art (Cobbe et al., 2021; Hafner et al., 2023). There are numerous ways performance can be improved (Jesson & Jiang, 2024; Cobbe et al., 2020) which we leave to future work.
+
+# 4.2. Applying BTR to Modern Games
+
+To demonstrate BTR's capabilities beyond standard RL benchmarks, we utilized Dolphin (Dolphin-Emulator, 2024), a Nintendo Wii emulator, to train agents for a range of modern 3D games: Super Mario Galaxy, Mario Kart Wii and Mortal Kombat. Using a desktop PC, we were able to train the agent to complete some of the most difficult tasks within each game. Namely, the final level in Super Mario Galaxy, Rainbow Road (a notoriously difficult track in Mario Kart Wii), and defeating all opponents in Mortal Kombat Endurance mode. For details about the environments and setup, see Appendix J. To achieve this, BTR required minimal adjustments: changing the input image resolution to  $140 \times 114$  (from Atari's  $84 \times 84$ ) due to the game's higher resolution and aspect ratio, and to reduce the number of vectorized environments to 4 due to the games' memory and CPU requirements.
+
+BTR was able to solve all three games, including consistently finishing first place in Mario Kart. In contrast, Rainbow DQN's performance plateaued before completing any of the games. We provide videos of our agent playing all
+
+![](images/525a0565ef41921248ffa939a5960cf5e5dc61c0fb67ee0bd3d6aef07a0fb85d.jpg)  
+Figure 4: BTR being used to play Super Mario Galaxy (final level), Mario Kart Wii (Rainbow Road) and Mortal Kombat: Armageddon (Endurance Mode) respectively. Consistent completion is defined as over  $90\%$ .
+
+![](images/906bb292c4b58250d87bcf1042531f78df45701a5bb48e16a7bb0e7a9937e7c5.jpg)
+
+![](images/2cc3558a4556511e14012721c5cb01bfffbfc6527fd868dc5a780389cf2ce554.jpg)
+
+three Wii games and all games in the Atari-5 benchmark  $^{5}$ .
+
+# 5. Analysis
+
+Given BTR's performance demonstrated in Section 4, in this section, we ablate each component to evaluate their performance impact (Section 5.1). Using the ablated agents, we measure numerous attributes during and after training to assess each component's impact (Section 5.2).
+
+$^5$ https://www.youtube.com/playlist?list=PL4geUsKi0NN-sjbuZP_fU28AmAPQunLoI
+
+# 5.1. Ablations Studies
+
+BTR amalgamates independently evaluated components into a single algorithm. To understand and verify each component's contribution, Figure 5 plots BTR's performance without each component on the Atari-5 benchmark. $^6$
+
+We find that Impala had the largest effect on performance  $(+142\%$  IQM), with the other components generally causing a less significant effect. Despite this, simply using Rainbow with Impala does not produce similar results (6.3 IQM compared to 7.7 on Atari-5). Munchausen and IQN have a strong impact on environments requiring fine
+
+Due to the resources required to evaluate on all environments, Aitchison et al. (2023) proposes a subset of 5 games that closely correlate with the performance across all of them.
+
+Figure 5: BTR's human-normalized scores without different components, with shaded areas showing  $95\%$  bootstrapped confidence intervals averaged over 4 seeds. Left: Predicted Atari-57 median score using the regression procedure defined in Aitchison et al. (2023). However, we find the prediction does not match the true median (see Appendix K). Right: Interquartile mean across the 5 games. For individual game graphs and additional ablations, see Appendices B and C.  
+![](images/24903756d56235a05ead1fa9b19738876905b803ec0dde44483b107dd2f89a82.jpg)  
+Beyond The Rainbow (BTR) no IQN no Munchausen Rainbow + Impala no Impala no Maxpool no Spectral
+
+![](images/55135f5567756c8fc8618efd0ac9b4288a1410117f8451e50a7ef805aab4f624.jpg)
+
+grained control such as Phoenix, as explored in Section 5.2.
+
+For vectorization and maxpooling, while their inclusion reduces performance, we find their secondary effects crucial to keep BTR computationally accessible. Omitting vectorization increases walltime by  $328\%$  (Figure 6) by processing environment steps in parallel and taking fewer gradient steps (781,000 compared to Rainbow DQN's 12.5 million).<sup>7</sup> We find that maxpooling decreases the model's parameters by  $77\%$ , and makes using wider convolutional layers possible without causing the total number of parameters to increase drastically.
+
+![](images/279a2c88530565e15ad9ccd42929d26eacad860bf9c5a303c16fc2dac74dfcc6.jpg)  
+Figure 6: Walltime of BTR on a desktop PC with components removed, compared with Hessel et al. (2018) and Schmidt & Schmied (2021). For hardware details, see Appendix G.
+
+# 5.2. What are the effects of BTR's components?
+
+To help interpret the results in Section 5.1, Table 2 measures seven different attributes of the agent either during or after training: action gaps and action swaps (linked to causing approximation errors (Bellemare et al., 2016)); policy churn (which can cause excessive off-policyness and instability (Schaul et al., 2022)) and score with additional noise (indicating robustness of the policies).
+
+While it is clear that Impala strongly contributes to performance, we find that without BTR's other components the learned policy is highly noisy and unstable. Table 2, demonstrates that without IQN and Munchausen the agent experiences very low action gaps (absolute Q-value difference between the highest two valued actions), causing the agent to swap its argmax action almost every other step. This is likely to result in approximation errors altering the policy and causing a high degree of off-policyness in the replay buffer. This is particularly detrimental in games requiring fine-grained control, such as Phoenix where the agent needs to narrowly dodge many projectiles, reflected in BTR's performance without these components.
+
+![](images/c89b93f31c18105b8a1d9c6d25bb03f1758d22e8924b464ccf8ed4948316f456.jpg)
+
+![](images/42152fde9bccb710b7b210223c96db65cb6de578913b6134a912b52747675d0a.jpg)
+
+![](images/ffa7864512c6c1f322e56ce4fa84e486909fe708212564dce42569c45ad7ff9f.jpg)  
+Figure 7: Plot showing  $\%$  of dormant neurons (Sokar et al., 2023), SRank with  $\delta = 0.01$  (Kumar et al., 2021) and L2 norm of network weights, for details see Appendix E.3. Results are averaged over 3 seeds and 5 tasks (Atari-5). Shading shows  $95\%$  confidence intervals.
+
+Furthermore, we find that maxpooling produces a more robust policy. To test this, we evaluate the performance of BTR's ablations when taking different quantities of  $\epsilon$ -actions and with altered observations and find maxpooling alleviates some of the performance loss (Table 2). Lastly, we find Munchausen and IQN to have a significant impact on Policy Churn (Schaul et al., 2022), with Munchausen reducing it by  $6.4\%$  and IQN increasing it by  $3.3\%$ . As a result, when these components are used together, they appear to reach a level of churn which does not harm learning and potentially provides some exploratory benefits.
+
+Lastly, Figure 7 shows an analysis of the trained model weights across the Atari-5 benchmark. We find little difference between trained models other than when removing Impala, which decreases dormant neurons and increases the L2 Norm of different layers, which have been linked with plasticity loss (Lyle et al., 2024).
+
+# 6. Related Work
+
+The most similar work to BTR, developing a computationally-limited non-distributed RL algorithm, is "Fast and Efficient Rainbow" (Schmidt & Schmied, 2021). They optimized Rainbow DQN to maximize performance for 10 million frames through parallelizing the environments and dropping C51 along with hyperparameter optimizations. This differs from our goals of producing
+
+Table 2: Comparison of policy churn, action gaps, actions swaps and evaluation performance with different quantities of  $\epsilon$ -actions and color jitter (both only applied for evaluation). All measurements use the final agent, trained on 200 million frames, for Atari Phoenix, averaged over 3 seeds. Action Gap is the average absolute Q-value difference between the highest two valued actions. % Actions Swap is the percentage of times the agent's argmax action has changed from the last timestep. Policy churn is the percentage of states in which the agent's argmax action has changed after a single gradient step. Color jitter applies a random  $10\%$  change to the brightness, saturation and hue of each frame. For associated errors with these values, please see Appendix E.3.
+
+<table><tr><td>Category</td><td>BTR</td><td>w/o Munchausen</td><td>w/o IQN</td><td>w/o SN</td><td>w/o Impala</td><td>w/o Maxpool</td></tr><tr><td>Action Gap</td><td>0.282</td><td>0.055</td><td>0.180</td><td>0.274</td><td>0.215</td><td>0.264</td></tr><tr><td>% Action Swaps</td><td>36.6%</td><td>47.7%</td><td>42.2%</td><td>40.3%</td><td>41.1%</td><td>39.3%</td></tr><tr><td>Policy Churn</td><td>3.8%</td><td>11.0%</td><td>0.5%</td><td>3.3%</td><td>4.5%</td><td>4.2%</td></tr><tr><td>Score ColorJitter</td><td>212k</td><td>85k</td><td>110k</td><td>187k</td><td>19k</td><td>187k</td></tr><tr><td>Score ε = 0.03</td><td>94k</td><td>42k</td><td>62k</td><td>75k</td><td>10k</td><td>86k</td></tr><tr><td>Score ε = 0.01</td><td>194k</td><td>70k</td><td>110k</td><td>132k</td><td>13k</td><td>171k</td></tr><tr><td>Score ε = 0</td><td>330k</td><td>184k</td><td>187k</td><td>296k</td><td>21k</td><td>406k</td></tr></table>
+
+an algorithm that scales across training regimes (up to 200 million frames) and domains (Atari, Progen, Super Mario Galaxy, Mario Kart and Mortal Kombat), resulting in different design decisions.
+
+For less computation-limited approaches, Ape-X (Horgan et al., 2018) was the first to explore highly distributed training, allowing agents to be trained on a billion frames in 120 hours through using  $>100$  CPUs. Following this, Kapturowski et al. (2018) proposed R2D2 using a recurrent neural network, increasing sample efficiency but slowing down gradient updates by  $38\%$ . Agent57 (Badia et al., 2020a) was the first RL agent to achieve superhuman performance across 57 Atari games, though required 90 billion frames. MEME (Kapturowski et al., 2023), Agent57's successor, focused on achieving superhuman performance within the standard 200 million frames limit, achieved by using a significantly higher replay ratio and larger network architecture. Most recently, Dreamer-v3 (Hafner et al., 2023) used a 200 million parameter model requiring over a week of training, achieving similar results as MEME. We detail some key differences between BTR, MEME and Dreamer-v3 in Table 3. While these approaches perform equally or
+
+better than BTR, all are inaccessible to smaller research labs or hobbyists due to their required computational resources and walltime. Therefore, while these algorithms have important research value demonstrating the possible performance of RL agents, performative algorithms with a lower cost of entry, like BTR, are necessary for RL to become widely applicable and accessible.
+
+# 7. Conclusions
+
+We have demonstrated that, once again, independent improvements from across Deep Reinforcement Learning can be combined into a single algorithm capable of pushing the state-of-the-art far beyond what any single improvement is capable of. Importantly, we find that this can be accomplished on desktop PCs, increasing the accessibility of RL for smaller research labs and hobbyists.
+
+We acknowledge there exists many more promising improvements we could not include in BTR, leaving room for more future work to create stronger integrated agents in a few years. For example, BTR does not add an explicit exploration component, resulting in it struggling in hard-
+
+Table 3: Comparison of performance, walltime, observations and complexity of different algorithms.  
+
+<table><tr><td>Category</td><td>BTR</td><td>MEME</td><td>Dreamer-v3</td></tr><tr><td>A100 GPU Days</td><td>0.9</td><td>Not Reported</td><td>7.7</td></tr><tr><td>Recurrent?</td><td>No (4 stacked frames)</td><td>Yes</td><td>Yes</td></tr><tr><td>Learns from?</td><td>Single Transitions</td><td>Trajectories (length 160)</td><td>Trajectories (length 64)</td></tr><tr><td>World Model</td><td>No</td><td>No</td><td>Yes</td></tr><tr><td>Parameters</td><td>2.9M</td><td>Not Reported (≈&gt;20M)</td><td>200M</td></tr><tr><td>Observation Shape</td><td>84x84</td><td>210x160</td><td>64x64</td></tr><tr><td>Gradient Steps</td><td>781K</td><td>3.75M</td><td>1.5M</td></tr><tr><td>Atari-60 IQM</td><td>7.4</td><td>9.6</td><td>9.6</td></tr></table>
+
+exploration tasks such as Montezuma's Revenge; therefore, mechanisms used in Never Give Up (Badia et al., 2020b), etc may prove useful. Section 5.1 found that the neural network's core architecture, Impala, had the largest impact on performance, an area we believe is generally underappreciated in RL. Previous work (Kaptuowski et al., 2018) has incorporated recurrent models enhancing performance, though we are uncertain how this can be incorporated into BTR without affecting its computational accessibility, a question which warrants future research.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Reinforcement Learning, particularly to improve accessibility to those with limited computational resources. As with any work increasing accessibility, this has potential for misuse by bad actors. However, we believe these concerns are offset by the field's potential to tackle key societal problems.
+
+# Acknowledgments
+
+This work was supported by the UK Research and Innovation (UKRI) Centre for Doctoral Training in Machine Intelligence for Nano-electronic Devices and Systems [EP/S024298/1] and the Engineering and Physical Sciences Research Council (EPSRC) ActivATOR project [EP/W017466/1]. The authors acknowledge the use of the IRIDIS X High Performance Computing Facility, and the Southampton-Wolfson AI Research Machine (SWARM) GPU cluster generously funded by the Wolfson Foundation, together with the associated support services at the University of Southampton in the completion of this work. The authors dedicate this work to the memory of George Morton-Fallows, whose passion for computer science inspired this research.
+
+# References
+
+Agarwal, R., Schwarzer, M., Castro, P. S., Courville, A. C., and Bellemare, M. Deep reinforcement learning at the edge of the statistical precipice. Advances in neural information processing systems, 34:29304-29320, 2021.  
+Aitchison, M., Sweetser, P., and Hutter, M. Atari-5: Distilling the arcade learning environment down to five games. In International Conference on Machine Learning, pp. 421-438. PMLR, 2023.  
+Badia, A. P., Piot, B., Kapturowski, S., Sprechmann, P., Vitvitskyi, A., Guo, Z. D., and Blundell, C. Agent57: Outperforming the atari human benchmark. In International conference on machine learning, pp. 507-517. PMLR, 2020a.  
+Badia, A. P., Sprechmann, P., Vitvitskyi, A., Guo, D., Piot, B., Kapturowski, S., Tieleman, O., Arjovsky, M., Pritzel, A., Bolt, A., et al. Never give up: Learning directed exploration strategies. In International Conference on Learning Representations, 2020b.  
+Ball, P. J., Smith, L., Kostrikov, I., and Levine, S. Efficient online reinforcement learning with offline data. In International Conference on Machine Learning, pp. 1577-1594. PMLR, 2023.  
+Bellemare, M. G., Naddaf, Y., Veness, J., and Bowling, M. The arcade learning environment: An evaluation platform for general agents. Journal of Artificial Intelligence Research, 47:253-279, 2013.  
+Bellemare, M. G., Ostrovski, G., Guez, A., Thomas, P., and Munos, R. Increasing the action gap: New operators for reinforcement learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 30, 2016.  
+Bellemare, M. G., Dabney, W., and Munos, R. A distributional perspective on reinforcement learning. In International conference on machine learning, pp. 449-458. PMLR, 2017.  
+BJORCK, N., Gomes, C. P., and Weinberger, K. Q. Towards deeper deep reinforcement learning with spectral normalization. Advances in neural information processing systems, 34:8242-8255, 2021.  
+Ceron, J. S. O. and Castro, P. S. Revisiting rainbow: Promoting more insightful and inclusive deep reinforcement learning research. In International Conference on Machine Learning, pp. 1373-1383. PMLR, 2021.  
+Ceron, J. S. O., Araújo, J. G. M., Courville, A., and Castro, P. S. On the consistency of hyper-parameter selection in value-based deep reinforcement learning. In Reinforcement Learning Conference, 2024.
+
+Cobbe, K., Hesse, C., Hilton, J., and Schulman, J. Leveraging procedural generation to benchmark reinforcement learning. In International conference on machine learning, pp. 2048-2056. PMLR, 2020.  
+Cobbe, K. W., Hilton, J., Klimov, O., and Schulman, J. Phasic policy gradient. In International Conference on Machine Learning, pp. 2020-2027. PMLR, 2021.  
+Dabney, W., Ostrovski, G., Silver, D., and Munos, R. Implicit quantile networks for distributional reinforcement learning. In International conference on machine learning, pp. 1096-1105. PMLR, 2018.  
+Dolphin-Emulator. Dolphin emulator. https://github.com/dolphin-emu/dolphin, 2024. Accessed: 2024-09-30.  
+D'Oro, P., Schwarzer, M., Nikishin, E., Bacon, P.-L., Bellemare, M. G., and Courville, A. Sample-efficient reinforcement learning by breaking the replay ratio barrier. In Deep Reinforcement Learning Workshop NeurIPS 2022, 2022.  
+Espeholt, L., Soyer, H., Munos, R., Simonyan, K., Mnih, V., Ward, T., Doron, Y., Firoiu, V., Harley, T., Dunning, I., et al. Impala: Scalable distributed deep-rl with importance weighted actor-learner architectures. In International conference on machine learning, pp. 1407-1416. PMLR, 2018.  
+Fortunato, M., Azar, M. G., Piot, B., Menick, J., Hessel, M., Osband, I., Graves, A., Mnih, V., Munos, R., Hassabis, D., et al. Noisy networks for exploration. In International Conference on Learning Representations, 2018.  
+Gallici, M., Fellows, M., Ellis, B., Pou, B., Masmitja, I., Foerster, J. N., and Martin, M. Simplifying deep temporal difference learning. arXiv preprint arXiv:2407.04811, 2024.  
+Gogianu, F., Berariu, T., Rosca, M. C., Clopath, C., Busoniu, L., and Pascanu, R. Spectral normalisation for deep reinforcement learning: an optimisation perspective. In International Conference on Machine Learning, pp. 3734-3744. PMLR, 2021.  
+Hafner, D., Pasukonis, J., Ba, J., and Lillicrap, T. Mastering diverse domains through world models, 2023. URL https://arxiv.org/abs/2301.04104, 2023.  
+He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016.  
+Hessel, M., Modayil, J., Van Hasselt, H., Schaul, T., Ostrovski, G., Dabney, W., Horgan, D., Piot, B., Azar, M., and
+
+Silver, D. Rainbow: Combining improvements in deep reinforcement learning. In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018.  
+Hessel, M., Danihelka, I., Viola, F., Guez, A., Schmitt, S., Sifre, L., Weber, T., Silver, D., and Van Hasselt, H. Muesli: Combining improvements in policy optimization. In International conference on machine learning, pp. 4214-4226. PMLR, 2021.  
+Horgan, D., Quan, J., Budden, D., Barth-Maron, G., Hessel, M., van Hasselt, H., and Silver, D. Distributed prioritized experience replay. In International Conference on Learning Representations, 2018.  
+Jesson, A. and Jiang, Y. Improving generalization on the procgen benchmark with simple architectural changes and scale. arXiv preprint arXiv:2410.10905, 2024.  
+Juliani, A., Khalifa, A., Berges, V.-P., Harper, J., Teng, E., Henry, H., Crespi, A., Togelius, J., and Lange, D. Obstacle tower: A generalization challenge in vision, control, and planning. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, 2019.  
+Justesen, N., Rodriguez Torrado, R., Bontrager, P., Khalifa, A., Togelius, J., and Risi, S. Illuminating generalization in deep reinforcement learning through procedural level generation. In NeurIPS Workshop on Deep Reinforcement Learning, 2018.  
+Kaptuowski, S., Ostrovski, G., Quan, J., Munos, R., and Dabney, W. Recurrent experience replay in distributed reinforcement learning. In International conference on learning representations, 2018.  
+Kapturowski, S., Campos, V., Jiang, R., Rakicevic, N., van Hasselt, H., Blundell, C., and Badia, A. P. Human-level atari 200x faster. In The Eleventh International Conference on Learning Representations, 2023.  
+Kumar, A., Agarwal, R., Ghosh, D., and Levine, S. Implicit under-parameterization inhibits data-efficient deep reinforcement learning. In International Conference on Learning Representations, 2021.  
+Lin, L.-J. Self-improving reactive agents based on reinforcement learning, planning and teaching. Machine learning, 8:293-321, 1992.  
+Loshchilov, I. and Hutter, F. Decoupled weight decay regularization. In International Conference on Learning Representations, 2019.  
+Lv, K., Yang, Y., Liu, T., Guo, Q., and Qiu, X. Full parameter fine-tuning for large language models with limited resources. In ACL (1), 2024.
+
+Lyle, C., Zheng, Z., Khetarpal, K., van Hasselt, H., Pascanu, R., Martens, J., and Dabney, W. Disentangling the causes of plasticity loss in neural networks. arXiv preprint arXiv:2402.18762, 2024.  
+Machado, M. C., Bellemare, M. G., Talvitie, E., Veness, J., Hausknecht, M., and Bowling, M. Revisiting the arcade learning environment: Evaluation protocols and open problems for general agents. Journal of Artificial Intelligence Research, 61:523-562, 2018.  
+Miyato, T., Kataoka, T., Koyama, M., and Yoshida, Y. Spectral normalization for generative adversarial networks. In International Conference on Learning Representations, 2018.  
+Mnih, V., Kavukcuoglu, K., Silver, D., Graves, A., Antonoglou, I., Wierstra, D., and Riedmiller, M. Playing atari with deep reinforcement learning. arXiv preprint arXiv:1312.5602, 2013.  
+Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., Graves, A., Riedmiller, M., Fidjeland, A. K., Ostrovski, G., et al. Human-level control through deep reinforcement learning. nature, 518 (7540):529-533, 2015.  
+Obando Ceron, J., Bellemare, M., and Castro, P. S. Small batch deep reinforcement learning. Advances in Neural Information Processing Systems, 36, 2024.  
+Schaul, T., Quan, J., Antonoglou, I., and Silver, D. Prioritized experience replay. arXiv preprint arXiv:1511.05952, 2015.  
+Schaul, T., Barreto, A., Quan, J., and Ostrovski, G. The phenomenon of policy churn. Advances in Neural Information Processing Systems, 35:2537-2549, 2022.  
+Schmidt, D. and Schmied, T. Fast and data-efficient training of rainbow: an experimental study on atari. In Deep RL Workshop NeurIPS 2021, 2021.  
+Schrittwieser, J., Antonoglou, I., Hubert, T., Simonyan, K., Sifre, L., Schmitt, S., Guez, A., Lockhart, E., Hassabis, D., Graepel, T., et al. Mastering atari, go, chess and shogi by planning with a learned model. Nature, 588 (7839):604-609, 2020.  
+Schrittwieser, J., Hubert, T., Mandhane, A., Barekatain, M., Antonoglou, I., and Silver, D. Online and offline reinforcement learning by planning with a learned model. Advances in Neural Information Processing Systems, 34: 27580-27591, 2021.  
+Schwarzer, M., Ceron, J. S. O., Courville, A., Bellemare, M. G., Agarwal, R., and Castro, P. S. Bigger, better, faster: Human-level atari with human-level efficiency.
+
+In International Conference on Machine Learning, pp. 30365-30380. PMLR, 2023.  
+Sokar, G., Agarwal, R., Castro, P. S., and Evci, U. The dormant neuron phenomenon in deep reinforcement learning. In International Conference on Machine Learning, pp. 32145-32168. PMLR, 2023.  
+Sutton, R. S. and Barto, A. G. Reinforcement learning: An introduction. MIT press, 2018.  
+Sutton, R. S., Barto, A. G., et al. Introduction to reinforcement learning, volume 135. MIT press Cambridge, 1998.  
+Toromanoff, M., Wirbel, E., and Moutarde, F. Is deep reinforcement learning really superhuman on atari? In Deep Reinforcement Learning Workshop of 39th Conference on Neural Information Processing Systems (Neurips' 2019), 2019.  
+Towers, M., Kwiatkowski, A., Terry, J., Balis, J. U., De Cola, G., Deleu, T., Goulão, M., Kallinteris, A., Krimmel, M., KG, A., et al. Gymnasium: A standard interface for reinforcement learning environments. arXiv preprint arXiv:2407.17032, 2024.  
+Van Hasselt, H., Guez, A., and Silver, D. Deep reinforcement learning with double q-learning. In Proceedings of the AAAI conference on artificial intelligence, volume 30, 2016.  
+Vieillard, N., Pietquin, O., and Geist, M. Munchausen reinforcement learning. Advances in Neural Information Processing Systems, 33:4235-4246, 2020.  
+Wang, S., Liu, S., Ye, W., You, J., and Gao, Y. Efficientzero v2: Mastering discrete and continuous control with limited data. In International Conference on Machine Learning, pp. 51041-51062. PMLR, 2024.  
+Wang, Z., Schaul, T., Hessel, M., Hasselt, H., Lanctot, M., and Freitas, N. Dueling network architectures for deep reinforcement learning. In International conference on machine learning, pp. 1995-2003. PMLR, 2016.  
+Watkins, C. J. and Dayan, P. Q-learning. Machine learning, 8:279-292, 1992.
+
+# A. Full Results Tables
+
+Table A1: Maximum scores obtained during training (averaged over 100 episodes and all performed using random seeds) after 200M Frames on the Atari-60 benchmark. Fast & Efficient Rainbow DQN and Munchausen-IQN refer to Schmidt & Schmied (2021) and (Vieillard et al., 2020) respectively. FE-Rainbow uses Life Information (See Appendix I), only 10M frames, and has missing games, so metrics are based on existing games.
+
+<table><tr><td>Game</td><td>Random</td><td>Human</td><td>DQN (Nature)</td><td>Rainbow</td><td>M-QN</td><td>FE-Rainbow</td><td>BTR</td></tr><tr><td>AirRaid</td><td>400</td><td>1277</td><td>12472</td><td>1911</td><td>1258</td><td>1899</td><td>51719</td></tr><tr><td>Aliamar</td><td>5</td><td>1719</td><td>1268</td><td>2390</td><td>1653</td><td>1407</td><td>4690</td></tr><tr><td>Assault</td><td>222</td><td>742</td><td>3490</td><td>6014</td><td>10709</td><td>19064</td><td>2190</td></tr><tr><td>Asterix</td><td>710</td><td>4788</td><td>2803</td><td>16547</td><td>42615</td><td>346758</td><td>60829</td></tr><tr><td>Astereux</td><td>210</td><td>4788</td><td>16547</td><td>42615</td><td>16547</td><td>124758</td><td>60829</td></tr><tr><td>Antlers</td><td>13850</td><td>29028</td><td>84372</td><td>791393</td><td>866810</td><td>812825</td><td>891773</td></tr><tr><td>BankHeist</td><td>14</td><td>753</td><td>560</td><td>1070</td><td>1305</td><td>112652</td><td>168450</td></tr><tr><td>BeetleJack</td><td>123</td><td>160</td><td>30</td><td>43</td><td>23</td><td>40</td><td>63</td></tr><tr><td>BeaconRider</td><td>363</td><td>1927</td><td>560</td><td>1070</td><td>1305</td><td>112652</td><td>168450</td></tr><tr><td>Ballzoon</td><td>23</td><td>160</td><td>30</td><td>43</td><td>23</td><td>40</td><td>63</td></tr><tr><td>BeachRider</td><td>363</td><td>1927</td><td>560</td><td>1070</td><td>1305</td><td>112652</td><td>168450</td></tr><tr><td>BeachZone</td><td>363</td><td>1927</td><td>560</td><td>1070</td><td>1305</td><td>112652</td><td>168450</td></tr><tr><td>BeachZone</td><td>363</td><td>1927</td><td>560</td><td>1070</td><td>1305</td><td>112652</td><td>168450</td></tr><tr><td>BeachZone</td><td>363</td><td>1927</td><td>560</td><td></td><td></td><td></td><td></td></tr><tr><td>BeachZone</td><td>363</td><td>1927</td><td>560</td><td></td><td></td><td></td><td></td></tr><tr><td>BeachZone</td><td>363</td><td>1927</td><td>560</td><td></td><td></td><td></td><td></td></tr><tr><td>BeachZone</td><td>363</td><td>1927</td><td>560</td><td></td><td></td><td></td><td></td></tr><tr><td>BeachZone</td><td>363</td><td>1927</td><td>560</td><td></td><td></td><td></td><td></td></tr><tr><td>BeachZone</td><td>10780</td><td>35829</td><td>103549</td><td>146262</td><td>14619</td><td>140712</td><td>140927</td></tr><tr><td>CrazyClimber</td><td>152</td><td>1911</td><td>5437</td><td>69237</td><td>69237</td><td>69237</td><td>69237</td></tr><tr><td>DemonAttack</td><td>-18</td><td>-16</td><td>-22</td><td>21</td><td>1</td><td>-1</td><td>76941</td></tr><tr><td>Elevator Action</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>Enduro</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>FishingDerby</td><td>-91</td><td>-29</td><td>-26</td><td>33</td><td>34</td><td>34</td><td>34</td></tr><tr><td>Freeway</td><td>65</td><td>4334</td><td>482</td><td>8309</td><td>9419</td><td>5282</td><td>15158</td></tr><tr><td>Grosbie</td><td>257</td><td>2412</td><td>5440</td><td>9987</td><td>2310</td><td>25606</td><td>97879</td></tr><tr><td>Grotchar</td><td>173</td><td>3351</td><td>209</td><td>1249</td><td>1105</td><td>2107</td><td>4753</td></tr><tr><td>Gravit</td><td>1027</td><td>30826</td><td>15766</td><td>46200</td><td>25555</td><td>15377</td><td>25371</td></tr><tr><td>Heron</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>Jamebond</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>Kingsport</td><td>52</td><td>302</td><td>671</td><td>995</td><td>1526</td><td>53302</td><td>53302</td></tr><tr><td>Marathon</td><td>52</td><td>302</td><td>671</td><td>995</td><td>1526</td><td>53302</td><td>53302</td></tr><tr><td>MapleGate</td><td>52</td><td>302</td><td>671</td><td>995</td><td>1526</td><td>53302</td><td>53302</td></tr><tr><td>NameThisGame</td><td>52</td><td>302</td><td>671</td><td>995</td><td>1526</td><td>53302</td><td>53302</td></tr><tr><td>Phenix</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>Pong</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>Poonan</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>PrivateEye</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>RiverRider</td><td>138</td><td>1911</td><td>5749</td><td>8900</td><td>12440</td><td>19819</td><td>27917</td></tr><tr><td>RoadRider</td><td>138</td><td>1911</td><td>5749</td><td>8900</td><td>12440</td><td>19819</td><td>27917</td></tr><tr><td>Robotank</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>Seashrest</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>StarCunner</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>Tennis</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>Tulipin</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>Unpinned</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>VidePinfall</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>WizardsOfWar</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>YardRevenge</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>Zaxxon</td><td>0</td><td>3000</td><td>642</td><td>2165</td><td>2266</td><td>2358</td><td>2358</td></tr><tr><td>IQM(↑)</td><td>0.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td></tr><tr><td>Median(↑)</td><td>0.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td></tr><tr><td>Mean(↑)</td><td>0.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td></tr><tr><td>Optimality Gap(↓)</td><td>0.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td><td>1.000</td></tr><tr><td>Best</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>&gt;Human</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></table>
+
+Table A2: Maximum scores obtained during training (averaged over 100 episodes and all performed 3 random seeds) after 200M Frames on the Atari-5 Environment, compared against other non-recurrent non-distributed algorithms. FE-Rainbow refers to Fast and Efficient Rainbow DQN (Schmidt & Schmied, 2021), and M-IQN refers to Munchausen-IQN (Vieillard et al., 2020). Metrics do not use the recommended regression procedure, as explained in Appendix K.  
+
+<table><tr><td>Game</td><td>Random</td><td>Human</td><td>Rainbow DQN (Dopamine)</td><td>Rainbow DQN (Full)</td><td>M-IQN</td><td>FE-Rainbow</td><td>BTR</td></tr><tr><td>BattleZone</td><td>2360</td><td>37188</td><td>40895</td><td>62010</td><td>52517</td><td>112652</td><td>168340</td></tr><tr><td>DoubleDunk</td><td>-19</td><td>-16</td><td>22</td><td>0</td><td>22</td><td>-1</td><td>23</td></tr><tr><td>NameThisGame</td><td>2292</td><td>8049</td><td>9229</td><td>13136</td><td>12761</td><td>19819</td><td>27917</td></tr><tr><td>Phoenix</td><td>761</td><td>7243</td><td>8605</td><td>108529</td><td>5327</td><td>60955</td><td>427481</td></tr><tr><td>Qbert</td><td>164</td><td>13455</td><td>18503</td><td>33818</td><td>14739</td><td>25712</td><td>42927</td></tr><tr><td>IQM</td><td>0.000</td><td>1.000</td><td>1.265</td><td>3.583</td><td>1.452</td><td>4.070</td><td>7.739</td></tr><tr><td>Median</td><td>0.000</td><td>1.000</td><td>1.21</td><td>2.532</td><td>1.44</td><td>3.167</td><td>4.766</td></tr><tr><td>Mean</td><td>0.000</td><td>1.000</td><td>3.714</td><td>5.817</td><td>3.745</td><td>4.684</td><td>18.453</td></tr></table>
+
+Table A3: Comparison of performance and walltime against PQN (Gallici et al., 2024). PQN only reports results at 400M frames and includes life information, which greatly affects performance (see Appendix I). To provide a fairer comparison, we also report our results using life information but only use 200M frames. Below are Atari-5 IQM and per-game Scores, with BTR averaged over 3 seeds. Human-Normalized scores are reported for individual games, with the raw score in brackets.  
+
+<table><tr><td>Game</td><td>BTR (with life info, 200M frames)</td><td>PQN (with life info, 400M frames)</td></tr><tr><td>Inter-Quartile Mean</td><td>12.18</td><td>3.86</td></tr><tr><td>BattleZone</td><td>12.73 (445,827)</td><td>1.51 (54,791)</td></tr><tr><td>DoubleDunk</td><td>14 (23.0)</td><td>6.03 (-0.92)</td></tr><tr><td>NameThisGame</td><td>4.51 (28,834)</td><td>3.18 (20,603)</td></tr><tr><td>Phoenix</td><td>90.85 (589,662)</td><td>38.79 (252,173)</td></tr><tr><td>QBert</td><td>9.79 (130,348)</td><td>2.37 (31,716)</td></tr><tr><td>Walltime (A100)</td><td>22 Hours</td><td>2 Hours</td></tr><tr><td>Backend</td><td>(PyTorch (non-compiled) + gymnasium async)</td><td>(JAX + envpool)</td></tr></table>
+
+# B. Full Results Graphs
+
+![](images/a2c52b6e84cf49ee49a22e4a9bfcfc962891bda5e9ff37c6196ba043ae23c687.jpg)  
+Figure B1: Performance of BTR (4 Seeds) on each individual game in all 60 Atari games. Shaded areas show  $95\%$  confidence intervals over different seeds. 15
+
+![](images/11f7330efb1cb05a4f813c86a2e02c9abffa3e300f750f4c4168437d09319289.jpg)
+
+![](images/6e040cf689405df8ec8c6b61cebe6930659a4348c4872d02dea0534afd38e478.jpg)
+
+![](images/e241c440e41995caca0b37184c8a63e439f997f31ee8e360cb199a72b6549b76.jpg)  
+Figure B2: Final performance of BTR on Atari-60 (as used in RLiable (Agarwal et al., 2021)), against other popular algorithms. The plot displays performance profiles, with  $95\%$  confidence intervals and 4 seeds for BTR.
+
+![](images/a2256052f146ae805542074009b91eca9c5326e46d15d02e07c96ca6ec972040.jpg)
+
+![](images/4dbb8062743f5298e4ccedb5d9d2e467d12223f60fb20c1785f10af47e02f3f8.jpg)
+
+![](images/acc05add0b7d6d2715cc1b501829e61d49408f7fcfb20768224bcb5c6483d973.jpg)
+
+![](images/8cf2a971f346ec43a055648226d38a9c6cc19734f05e2d25df761398af69592e.jpg)  
+Figure B3: Performance of BTR with different components removed on individual games, and interquartile mean in the top left. Results are averaged over 4 seeds, with shaded areas showing  $95\%$  bootstrapped confidence intervals.
+
+![](images/67ee8172ac99390e494a10f733d1471d67027e4ccbdc2568ad995385020529b1.jpg)
+
+![](images/b5e42d2e5492f5f98cc50e614ec827f4eb1984c5ed7296f8c6ce209a0184db8c.jpg)
+
+![](images/fe918846ad148607670e0c3a698f6337906ea257a529c85fc3ba3e27f348c317.jpg)  
+Figure B4: Figure shows BTR against PPO on Atari 57 (PPO's Atari 60 scores have not been reported). PPO uses the results provided by DreamerV3 (Hafner et al., 2023), which additionally uses vectorization and the impala algorithm. Shaded areas show  $95\%$  confidence intervals with BTR using 4 seeds.
+
+![](images/1327f17ffcd7bb9da47468e0d64221e97d42732f2fa637a6866c93364ffe314e.jpg)  
+Figure B5: Performance of BTR on each individual game in the Progen benchmark. Shaded areas show  $95\%$  confidence intervals. The red dotted line shows the performance of Rainbow DQN + Impala with 4x scaled Impala blocks (Cobbe et al., 2020), after 200M frames.
+
+# C. Additional Ablations
+
+Figure C6: Figures show BTR's human-normalized scores without different components, with shaded areas showing  $95\%$  bootstrapped confidence intervals averaged over 4 seeds. Left: Predicted Atari-60 median score using the regression procedure defined in Aitchison et al. (2023). However, we find the predicted median does not match the true median (see Appendix K). Right: Interquartile mean across the 5 games.  
+![](images/f24776f4d568831529d815aaf2aba9aece96c1e250d1ca0f44345ad8d2f3afba.jpg)  
+Beyond The Rainbow (BTR)  
+BTR - Vectorization
+
+![](images/b332a0f741a086e143f17700e84d532695504a927678b97ec28bd7c65469ec6a.jpg)  
+Rainbow + Vectorization  
+Rainbow + Impala
+
+# D. Hyperparameters
+
+# D.1. Environment Details
+
+Table D4: Environment Details for Atari Experiments.  
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Grey-Scaling</td><td>True</td></tr><tr><td>Observation down-sampling</td><td>84x84</td></tr><tr><td>Frames Stacked</td><td>4</td></tr><tr><td>Reward Clipping</td><td>[-1, 1]</td></tr><tr><td>Terminal on loss of life</td><td>False</td></tr><tr><td>Life Information</td><td>False</td></tr><tr><td>Max frames per episode</td><td>108K</td></tr><tr><td>Sticky Actions</td><td>True</td></tr></table>
+
+Table D5: Environment Details for Procgen Experiments.  
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Grey-Scaling</td><td>False</td></tr><tr><td>Observation Size</td><td>64x64</td></tr><tr><td>Frames Stacked</td><td>1</td></tr><tr><td>Reward Clipping</td><td>False</td></tr><tr><td>Max frames per episode</td><td>108K</td></tr><tr><td>Distribution Mode</td><td>Hard</td></tr><tr><td>Number of Unique Levels (Train &amp; Test)</td><td>Unlimited</td></tr></table>
+
+# D.2. Algorithm Hyperparameters
+
+Table D6: Table showing the hyperparameters used in the BTR algorithm.  
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Learning Rate</td><td>1e-4</td></tr><tr><td>Discount Rate</td><td>0.997</td></tr><tr><td>N-Step</td><td>3</td></tr><tr><td>IQN Taus</td><td>8</td></tr><tr><td>IQN Number Cos&#x27;</td><td>64</td></tr><tr><td>Huber Loss κ</td><td>1.0</td></tr><tr><td>Gradient Clipping Max Norm</td><td>10</td></tr><tr><td>Parallel Environments</td><td>64</td></tr><tr><td>Gradient Step Every</td><td>64 Environment Steps (1 Vectorized Environment Step)</td></tr><tr><td>Replace Target Network Frequency (C)</td><td>500 Gradient Steps (32K Environment Steps)</td></tr><tr><td>Batch Size</td><td>256</td></tr><tr><td>Total Replay Ratio</td><td>1/64</td></tr><tr><td>Impala Width Scale</td><td>2</td></tr><tr><td>Spectral Normalization</td><td>All Convolutional Residual Layers</td></tr><tr><td>Adaptive Maxpooling Size</td><td>6x6</td></tr><tr><td>Linear Size (Per Dueling Layer)</td><td>512</td></tr><tr><td>Noisy Networks σ</td><td>0.5</td></tr><tr><td>Activation Function</td><td>ReLU</td></tr><tr><td>ε-greedy Start</td><td>1.0</td></tr><tr><td>ε-greedy Decay</td><td>8M Frames</td></tr><tr><td>ε-greedy End</td><td>0.01</td></tr><tr><td>ε-greedy Disabled</td><td>100M Frames</td></tr><tr><td>Replay Buffer Size</td><td>1,048,576 Transitions (220)</td></tr><tr><td>Minimum Replay Size for Sampling</td><td>200K Transitions</td></tr><tr><td>PER Alpha</td><td>0.2</td></tr><tr><td>Optimizer</td><td>Adam</td></tr><tr><td>Adam Epsilon Parameter</td><td>1.95e-5 (equal to 0.005 batchsize)</td></tr><tr><td>Adam β1</td><td>0.9</td></tr><tr><td>Adam β2</td><td>0.999</td></tr><tr><td>Munchausen Temperature τ</td><td>0.03</td></tr><tr><td>Munchausen Scaling Term α</td><td>0.9</td></tr><tr><td>Munchausen Clipping Value (l0)</td><td>-1.0</td></tr><tr><td>Evaluation Epsilon</td><td>0.01 until 125M frames, then 0</td></tr><tr><td>Evaluation Episodes</td><td>100</td></tr><tr><td>Evaluation Every</td><td>1M Environment Frames (250K Environment Steps)</td></tr></table>
+
+# D.3. Clarity of the terms Frames, Steps and Transitions
+
+Throughout the Arcade Learning Environment's history (ALE) (Bellemare et al., 2013; Machado et al., 2018), there have been many ambiguities around the terms: frames, steps and transitions, which are sometimes used interchangeably. Frames refer to the number of individual frames the agent plays, including those within repeated actions (also called frame skipping). This is notably different from the number of steps the agent takes, which does not include these skipped frames. When using the standard Atari wrapper, training for 200M frames is equivalent to training for 50M steps. Lastly, transitions refer to the standard tuple  $(s_t, a_t, r_t, s_{t+1})$ , where the timestep  $t$  refers to a steps, not frames. We encourage researchers to make this clear when publishing work, including when mentioning the values of different hyperparameters.
+
+# E. Beyond The Rainbow Architecture & Loss Function
+
+# E.1. Architecture
+
+![](images/91ac0a59bf2298ca076d453a83841f8e39c60e86d2e3ffc6260f44f081c069aa.jpg)  
+Figure E7 shows the neural network architecture of the BTR algorithm. The architecture is highly similar to the Impala architecture (Espeholt et al., 2018), with notable exceptions:  
+Figure E7: Architectural diagram of the BTR algorithm's neural network. The model contains a total of 2.91 million parameters, 2.52 million of which are within linear layers.
+
+- Spectral Normalization Within each Impala CNN block, each residual layer (containing two Conv  $3 \times 3 + \mathrm{ReLU}$ ) has spectral normalization applied, as discussed in Section 3.1.  
+- Maxpooling Following the CNN blocks, a 6x6 adaptive maxpooling layer is added.
+
+- IQN In order to use IQN, it is required to draw Tau samples, which are multiplied by the output of the CNN layers, as shown by the section 'IQN Samples' in figure E7.  
+- **Dueling**. **Dueling (as included in the original Rainbow DQN)** splits the fully connected layers into value and advantage streams, where the advantage stream output has a mean of 0, and is then added to the value stream.  
+- Noisy Networks As included in Rainbow DQN, Noisy Networks replace the linear layers with noisy layers.
+
+Lastly, the sizes of many of the layers given in Figure E7 are dependent upon the Impala width scale, of which we use the value 2. For example, the Impala CNN blocks have  $[16 \times \text{width}, 32 \times \text{width}, 32 \times \text{width}]$  channels respectively. The output size of the convolutional layers (including the maxpooling layer) is  $6 \times 6 \times 32 \times \text{width}$ , as a 6x6 maxpooling layer is used. Lastly, the cos embedding layer, after generating IQN samples, requires the same size as the output of the convolutional layers. Hence, the size is selected accordingly. Another benefit of the 6x6 maxpooling layer is following the product of the convolutional layers and IQN samples, the number of parameters is fixed, regardless of the input size. Figure E8 shows the number of parameters the ablated versions of BTR have.
+
+![](images/f305904ee010a7509f57f9a877f2fa59e4796577abfe0c8db25f5394e5da9683.jpg)  
+Figure E8: Total number of parameters in BTR with different components removed. Those not included in the graph (Munchausen and Spectral Normalization) used the same number of parameters as BTR.
+
+# E.2. Loss Function
+
+The resulting loss function for the BTR algorithm remains the same as that defined in the appendix of the Munchausen paper, which gave a loss function for Munchausen-IQN. As the other components in BTR do not affect the loss, the resulting temporal-difference loss function is the same. For self-containment, we include this loss function below:
+
+$$
+T D _ {B T R} = r _ {t} + \alpha [ \tau \ln \pi (a _ {t} | s _ {t}) ] _ {l _ {0}} ^ {0} + \gamma \sum_ {a \in A} \pi (a | s _ {t + 1}) (z _ {\sigma^ {\prime}} (s _ {t + 1}, a) - \tau \ln \pi (a | s _ {t + 1})) - z _ {\sigma} (s _ {t}, a _ {t}) \tag {E1}
+$$
+
+with  $\pi (\cdot |s) = sm\left(\frac{\tilde{q}(s,\cdot)}{\tau}\right)$  (that is, the policy is softmax with  $\mathbf{q}^*$ , the quantity with respect to which the original policy of IQN is greedy). It is also worth noting here that due to the character conflict of both Munchausen and IQN using  $\tau$  (Munchausen as a temperature parameter, and IQN for drawing samples), we replace IQN's  $\tau$  with  $\sigma$ .  $l_0, \tau$  and  $\alpha$  are hyperparameters set by Munchausen. We use the same values in BTR, also shown in our hyperparameter table in Appendix D.2.
+
+# E.3. Analysis Confidence Intervals
+
+Due to space constraints, we was unable to include confidence intervals for Table 2 in the main paper. A repeat of the main paper Table can be found in Table E7, with the associated confidence intervals in Table E8.
+
+# F. BTR with and without Epsilon Greedy
+
+One of the first observations we made early in the testing process was that the inclusion of using  $\epsilon$ -greedy in addition to NoisyNetworks benefited some environments but not others. Specifically, performance was reduced on BattleZone and Phoenix, both games where the agent reached very high levels of performance with extremely precise control. However, DoubleDunk performed significantly worse, only reaching a score of 0, rather than the score of 23 the final BTR algorithm achieved. Similar findings were also found in the full version of Rainbow DQN, which used only NoisyNetworks, which
+
+Table E7: Repeat of the main paper's Table 2 for reference against the table of  $95\%$  confidence intervals below. Comparison of policy churn, action gaps, actions swaps and evaluation performance with different quantities of  $\epsilon$ -actions and color jitter (both only applied for evaluation). All measurements use the final agent, trained on 200 million frames, for Atari Phoenix, averaged over 3 seeds. Action Gap is the average absolute Q-value difference between the highest two valued actions. % Actions Swap is the percentage of times the agent's argmax action has changed from the last timestep. Policy churn is the percentage of states which the agent's argmax action has changed on after a single gradient step. Color jitter applies a random  $10\%$  change to the brightness, saturation and hue of each frame. For associated error with these values, please see Appendix E.3.
+
+<table><tr><td>Category</td><td>BTR</td><td>w/o Munchausen</td><td>w/o IQN</td><td>w/o SN</td><td>w/o Impala</td><td>w/o Maxpool</td></tr><tr><td>Action Gap</td><td>0.282</td><td>0.055</td><td>0.180</td><td>0.274</td><td>0.215</td><td>0.264</td></tr><tr><td>% Action Swaps</td><td>36.6%</td><td>47.7%</td><td>42.2%</td><td>40.3%</td><td>41.1%</td><td>39.3%</td></tr><tr><td>Policy Churn</td><td>3.8%</td><td>11.0%</td><td>0.5%</td><td>3.3%</td><td>4.5%</td><td>4.2%</td></tr><tr><td>Score ColorJitter</td><td>212k</td><td>85k</td><td>110k</td><td>187k</td><td>19k</td><td>187k</td></tr><tr><td>Score ε = 0.03</td><td>94k</td><td>42k</td><td>62k</td><td>75k</td><td>10k</td><td>86k</td></tr><tr><td>Score ε = 0.01</td><td>194k</td><td>70k</td><td>110k</td><td>132k</td><td>13k</td><td>171k</td></tr><tr><td>Score ε = 0</td><td>330k</td><td>184k</td><td>187k</td><td>296k</td><td>21k</td><td>406k</td></tr></table>
+
+Table E8: 95% confidence intervals for the main paper Table 2, calculated using 6 seeds. A repeat of the original table is shown above in Table E7.  
+
+<table><tr><td>Category</td><td>BTR</td><td>w/o Munchausen</td><td>w/o IQN</td><td>w/o SN</td><td>w/o Impala</td><td>w/o Maxpool</td></tr><tr><td>Action Gap</td><td>[0.25, 0.31]</td><td>[0.05, 0.06]</td><td>[0.17, 0.19]</td><td>[0.25, 0.3]</td><td>[0.1, 0.33]</td><td>[0.23, 0.3]</td></tr><tr><td>% Action Swaps</td><td>[33.6, 39.6]</td><td>[45.8, 49.7]</td><td>[41.0, 43.3]</td><td>[38.5, 42.2]</td><td>[31.4, 50.8]</td><td>[38.2, 40.4]</td></tr><tr><td>Policy Churn</td><td>[3.3, 4.2]</td><td>[10.2, 11.7]</td><td>[0.5, 0.5]</td><td>[2.9, 3.6]</td><td>[3.9, 5.0]</td><td>[3.7, 4.7]</td></tr><tr><td>Score ColorJitter</td><td>[204k, 218k]</td><td>[77k, 92k]</td><td>[90k, 128k]</td><td>[173k, 201k]</td><td>[0k, 40k]</td><td>[158k, 215k]</td></tr><tr><td>Score ε = 0.03</td><td>[88k, 99k]</td><td>[38k, 46k]</td><td>[52k, 72k]</td><td>[68k, 82k]</td><td>[2k, 18k]</td><td>[74k, 97k]</td></tr><tr><td>Score ε = 0.01</td><td>[177k, 211k]</td><td>[62k, 77k]</td><td>[97k, 122k]</td><td>[139k, 176k]</td><td>[0k, 52k]</td><td>[143k, 199k]</td></tr><tr><td>Score ε = 0</td><td>[282k, 377k]</td><td>[116k, 251k]</td><td>[163k, 209k]</td><td>[244k, 348k]</td><td>[0k, 43k]</td><td>[332k, 479k]</td></tr></table>
+
+achieved a best score of -0.3 (Dopamine's "compact" Rainbow DQN, however, which did not use NoisyNetworks achieved 22). From this, we conclude that NoisyNetworks alone failed to sufficiently explore the environment, whereas  $\epsilon$ -greedy did not. From these results, we eventually decided to use both methods, but disable  $\epsilon$ -greedy halfway through training to reap the best of both techniques.
+
+# G. Experiment Compute Resources
+
+# G.1. Our Compute Resources
+
+For running our experiments, we used a mixture of desktop computers and internal clusters. The desktop PCs used an GPU Nvidia RTX4090, CPU intel i9-14900k and 64GB of DDR5 6000mhz RAM. When using internal clusters, we used a mixture of GPUs, including Nvidia A100s, Nvidia Volta V100 and Nvidia Quadro RTX 8000. As for CPUs, we used 2 x 2.4 GHz Intel(R) Xeon(R) Gold 6336Y, 48 Cores. Lastly, we saved the models used to produce our analysis, totalling around 300gb across all of our ablations on the Atari-5 benchmark, saving a model every 1 million frames.
+
+As most of our experiments were performed on desktop PC, in the main body of our paper we reference these speeds. We found that desktop PCs actually outperformed internal clusters, likely due to desktop CPUs being more suited to performing environment steps, outlined in the next subsection.
+
+When testing ideas originally (those mentioned in Appendix H), we only tested them using a single run of the games BattleZone, NameThisGame and Phoenix unless otherwise stated. Whilst this method of evaluation is not statistically significant, for preliminary purposes with computational restrictions, we deemed this the best option.
+
+# G.2. BTR with Different Hardware
+
+In this work, we look to make high-performance RL more accessible to those with fewer computing resources, especially those only with access to desktop computers. Most of our experiments were performed with an RTX4090, we also provide some walltimes for 200M Atari frames for lower-end machines and provide a brief comparison of desktop PCs against internal clusters:
+
+# Desktops:
+
+Original: RTX 4090, Intel i9-13900k (2023), 64GB RAM - 11.5 Hours
+
+RTX 3070, Ryzen 9 3900X (2019), 64GB RAM - 52 Hours
+
+RTX 2080 ti, Intel(R) Xeon(R) Silver 4112 CPU @ 2.60GHz (2018), 128GB RAM - 32 Hours
+
+# Internal Clusters:
+
+Nvidia H100, 48 Core Intel(R) Xeon(R) Platinum 8468 (2023), 2TB RAM - 15 Hours
+
+Nvidia A100, 24 Core Intel(R) Xeon(R) Gold 6336Y (2021), 512GB RAM - 22 Hours
+
+We note that there is significant variability in hardware (processors, memory bus speeds, etc), but the results still show reasonable times compared to not using BTR. Overall, we found that training BTR was very capable of running on lower end machines, with the agent (excluding the environments) using around 15GB of RAM. The main performance bottleneck was running the environment in parallel, making the number of CPU cores and processor speed most important. BTR also provides strong performance long before 200M frames, thus providing practical utility for lower-end machines.
+
+# H. Other Things We Tried
+
+Throughout the development of the BTR algorithm, we experimented with many different components and hyperparameters. A brief list of ideas we tried that performed worse or equivalent to the final algorithm includes:
+
+- Using Exponential Moving Average networks rather than using fixed target networks (this was both computationally slower and performed worse).  
+- Varying the frequency of updating the target network (we tested 250, 500 and 1000, finding 500 to perform best).  
+- Changing the size of maxpool layer following the convolutional layers (we tested 4 and 8, however 6 performed significantly better).  
+- Decaying the learning rate from  $1 \times 10^{-4}$  to 0 over the course of training (this made no significant difference).  
+- Different learning rates, finding  $1 \times 10^{-4}$  to perform best, however  $5 \times 10^{-5}$  also performed similarly as was used in Implicit Quantile Networks (IQN).  
+- Using the AdamW optimizer(Loshchilov & Hutter, 2019) which uses weight decay with the decay parameter  $1e - 4$ , however found this made no significant difference.  
+- Using the GeLu activation instead of ReLu, which drastically reduced performance.
+
+Only testing on a single environment (BattleZone), we also tried:
+
+- Annealing the discount rate from 0.97 to 0.997 throughout training, but found no significant difference.  
+- Applying spectral normalization to the linear layers (dramatically worse performance).  
+- Increasing the number of cos' from IQN (no significant difference on performance).  
+- Using Dopamine's Prioritized Experience Replay buffer which doesn't include a  $\alpha$  value (moderately worse performance).  
+- As discussed in F, we also tried not using  $\epsilon$ -greedy when using noisy nets.
+
+Lastly we also tried removing some of the original components from Rainbow DQN on Atari BattleZone, including Dueling, Prioritized Experience Replay and Noisy Networks. Prioritized Experience Replay and Noisy Networks both proved beneficial, so were kept in the algorithm. Dueling did not seem to make any significant difference, however we did not choose to remove it for a clearer continuation of Rainbow DQN, in addition to potentially being useful in other Atari environments.
+
+Shortly after the submission of this work, we tested BTR with addition of Layer Normalization, and found positive results. Layer Normalization can improve the robustness to a variety of pathologies that cause loss of plasticity (Lyle et al., 2024), and helps to improve the conditioning of the network's gradients in RL (Ball et al., 2023). Below in Figure H9 we show the impact of including layer normalization into BTR.
+
+Figure H9: Graph shows Atari-5 performance with and without layer normalization using Inter-quartile mean and Atari-60 predicted median from Aitchison et al. (2023). Layer normalization uses 3 seeds, with shaded areas showing  $95\%$  confidence intervals.  
+![](images/54c56ebf387e786f469a878fb82daa75aa91f66574ca63d04a1a615bf2395460.jpg)  
+Beyond The Rainbow (BTR) BTR + LayerNorm
+
+![](images/7dc4c39dcd6048ce6a47b22b8c3190a9fcba2c2ee2e4941f4d95ad71aaece104.jpg)
+
+Table H9: Maximum scores obtained during training (averaged over 100 episodes and all performed random seeds) after 200M Frames on the Atari-5 Environment, compared to BTR with Layer Normalization.  
+
+<table><tr><td>Game</td><td>Random</td><td>Human</td><td>BTR + Layer Normalization</td><td>BTR</td></tr><tr><td>BattleZone</td><td>2360</td><td>37188</td><td>183240</td><td>168340</td></tr><tr><td>DoubleDunk</td><td>-19</td><td>-16</td><td>23</td><td>23</td></tr><tr><td>NameThisGame</td><td>2292</td><td>8049</td><td>33258</td><td>27917</td></tr><tr><td>Phoenix</td><td>761</td><td>7243</td><td>493762</td><td>427481</td></tr><tr><td>QBert</td><td>164</td><td>13455</td><td>47384</td><td>42927</td></tr><tr><td>IQM</td><td>0.000</td><td>1.000</td><td>8.191</td><td>7.739</td></tr><tr><td>Median</td><td>0.000</td><td>1.000</td><td>5.379</td><td>4.766</td></tr><tr><td>Mean</td><td>0.000</td><td>1.000</td><td>20.836</td><td>18.453</td></tr></table>
+
+# I. Altered Atari Environment Settings
+
+In order to investigate the impact of the environmental sticky actions parameter and to compare against other works, we include results for it on the Atari-5 benchmark in Figure I10.
+
+![](images/80e6cd11bf6aa0b13b7bf457be79add07d0771b6d169d90e1dbe0b8f666981cc.jpg)  
+Figure I10: Graph shows Atari-5 performance with and without sticky actions (sticky actions is the default) using Inter-quartile mean and Atari-60 predicted median from Aitchison et al. (2023). No Sticky Actions uses a single seed, so this result should be used with caution.
+
+![](images/60696ab5b621398e6d196f553217299bdab04371493b3522e9de013ea5756e77.jpg)  
+Beyond The Rainbow (BTR) BTR no Sticky Actions
+
+Some prior works choose to pass life information to the agent (Schmidt & Schmied, 2021). To clarify, this is different to terminal on loss of life. Life information does not reset the episode upon losing a life, but does pass a terminal to the buffer, allowing the agent to experience further into episodes while also giving the agent a negative signal for losing a life. This setting is not recommended in Machado et al. (2018), and works which use it are not comparable to those which don't. To emphasize this point, we take the three games from the Atari-5 and perform a comparison.
+
+![](images/f55fe40185e080a59896237bbca538be90575b645845ba17d1400fa5bf972d81.jpg)  
+Figure I11: Graph shows Atari-5 performance with and without life information using Inter-quartile mean and Atari-60 predicted median from Aitchison et al. (2023). Life Information uses 3 seeds, with shaded areas showing  $95\%$  confidence intervals. From this we conclude results using life information are invalid for comparison.
+
+![](images/126db6916a08aeafca72aad0d0e5d9041c403602a011f379842ba46b74d64f27.jpg)  
+Beyond The Rainbow (BTR) BTR + Life Info
+
+# J. BTR for Wii Games
+
+BTR interfaces with different Wii Games via the Dolphin Emulator. Specifically, we use a forked repository of Dolphin Emulator to allow Python scripts to interact with the emulator. This includes loading savestates (used to reset episodes),
+
+grabbing the screen as a PIL image at the Wii's internal resolution of 480p (downsampled to 140x114 and grey-scaled, used for all observations), reading the Wii's RAM (used for reward functions and termination conditions) and allowed programmatic input into the emulator (for setting actions). Using Dolphin's portable setting, we are able to run multiple Dolphin Emulators simultaneously on the same machine. Each instance runs as a unique process, and communicates with the agent via Python's multiprocessing library. Similarly to the Atari benchmark, for all games we used a frameskip of 4.
+
+# J.1. Super Mario Galaxy
+
+This environment used Super Mario Galaxy's final level, The Center of the Universe, and had to make it to the final fight at the end of the game. The agent had 6 actions, including None, moving in each direction and jumping. Additionally, if the jump action was performed following a movement, the agent would continue to move in that direction.
+
+Rewards were given via finding many values in the Wii's memory that resembled progress in the level. The agent was then rewarded for this progress value increasing from the last frame. If the agent's position entered a set region, the progress variable would be moved. Additionally, the game uses a life system, where the player has a maximum of 3 lives and can lose or gain lives in many different ways. The agent was given a reward of  $+1$  for gaining a life, and -1 for losing a life. Lastly, episode termination occurred if the agent reached 0 lives, or if the agent made it to the end of the level. For this task, we also allowed the agent to start episodes at many points throughout the level, which rapidly sped up training since the agent could easily experience different areas of the level.
+
+Whilst a difficult task, once the agent first completed the level, it did not take long to start consistently completing it due to the deterministic nature of the game.
+
+# J.2. Mario Kart Wii
+
+The Mario Kart Wii environment had the agent play against the game's internal opponents (on hard mode, with 12 racers including the agent), on the course Rainbow Road (with items on the 150cc speed setting). The agent had to complete 4 laps of the course to finish the race. The agent had just 4 action, including accelerate, drifting left or right, and using its item. While this limited the agent's potential actions substantially, we found using fewer actions to dramatically accelerate training.
+
+Rewards of  $+1$  were given via reaching checkpoints that were scattered throughout the course (100 in total per lap). Additionally, if the agent's speed dropped below a set threshold  $(65\mathrm{km / h})$ , the agent would receive a reward of -0.01 per frame. The agent would be terminated with a reward of -10 if its speed dropped below the threshold for over 80 frames, or with a reward of +10 for finishing the race, with a bonus based on the position the agent finished in. Lastly, the agent was rewarded with a +1 for using its item. Without this reward, we found the agent to often neglect using its item, likely due to many of the items only providing rewards in the long term, such as slowing down other racers or blocking incoming items far in the future. Similarly to Super Mario Galaxy, we had the agent start the episode in multiple positions around the first lap, allowing it to experience the whole track early in training.
+
+This agent took the longest to train, taking around 160M frames to reach consistent completion. In particular, the agent took a long time to consistently complete the race due to the other racers and randomized items making the environment highly stochastic, with many rare scenarios which could cause the episode to terminate.
+
+# J.3. Mortal Kombat
+
+The Mortal Kombat environment put the agent in the game's endurance mode, where the agent would sequentially fight 15 different opponents, but keep retain its health between fights, and only gain health after defeating every 3 opponents. We provided the agent with 14 actions, including: None, Left, Right, Up, Down, Axe Kick, Punch, Snap Kick, Grab, Block, Toggle Weapon, Jump Left, Jump Right, and Crouch. These actions were far from the game's total action space, and limited the agent's ability to perform some of the combos within the game. We limited the agent's actions as the full action space is extremely large.
+
+The agent was positively rewarded for damaging the opponent, and negatively rewarded for taking damage, with one taking one tenth of the health bar equating to  $+1$  reward respectively. The episode was terminated with a reward of -10 for reaching 0 health, and +10 for defeating the 15th and final enemy.
+
+The Mortal Kombat agent learned considerably faster than Super Mario Galaxy and Mario Kart Wii, first completing the
+
+environment in 50M frames, and getting progressively more consistent until training was stopped at 90M frames. The agent quickly learned how to dodge enemy hits, and relied heavily upon this strategy.
+
+# K. Atari-5 Regression Procedure
+
+In our main paper ablation figure (Figure 5), we used the regression procedure recommended in Atari-5 (Aitchison et al., 2023). This procedure is typically used to predict the Median score across the entire 57 game Atari suite, while only needing to use 5 games. While we believe this procedure produces a valid and useful plot, we find that BTR did differ significantly from the predicted value. We opted to use both the predicted median and the IQM across the 5 games to give two easy to interpret averages. Figure K12 shows the 57 game suite's true median, compared to the median predicted by Atari-5.
+
+![](images/2f04d6ddd73dbd58da819011283829968ab71801e756953b6bc35e12bb60d2e9.jpg)  
+Figure K12: BTR's 60 game median against that predicted by Atari-5 using the regression procedure from Aitchison et al. (2023). Shaded areas show  $95\%$  confidence intervals using 4 seeds.

2025/Beyond The Rainbow_ High Performance Deep Reinforcement Learning on a Desktop PC/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:872a2974bcb676080b42e199d22e05d0024424d137a193ef1652eb8ad46a1b4e
+size 2408687

2025/Beyond Topological Self-Explainable GNNs_ A Formal Explainability Perspective/b9cd7a30-96a8-41ab-ad86-4786cf1b9eef_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:312129c396e0712380d64ed6b3efe33f545f5fbb5d405323659499a3474fdae8
+size 1514711

2025/Beyond Topological Self-Explainable GNNs_ A Formal Explainability Perspective/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b9496eb4385b6d8c16fa1ecfceaa6a16bde351c5697f3df866460e64215edede
+size 1653539

2025/Beyond Zero Initialization_ Investigating the Impact of Non-Zero Initialization on LoRA Fine-Tuning Dynamics/9bd2a800-d91c-4f7b-970e-a45ea7881131_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:99a2bc3bbb3df6e03212a696faaabc7b0a3ef9cb9f4704ab2fadc02c96e42287
+size 15208631

2025/Beyond Zero Initialization_ Investigating the Impact of Non-Zero Initialization on LoRA Fine-Tuning Dynamics/full.md

@@ -0,0 +1,596 @@
+# Beyond Zero Initialization: Investigating the Impact of Non-Zero Initialization on LoRA Fine-Tuning Dynamics
+
+Shiwei Li  $^{12*}$  Xiandi Luo  $^{1*}$  Xing Tang  $^{2}$  Haozhao Wang  $^{1}$  Hao Chen  $^{3}$  Weihong Luo  $^{3}$  Yuhua Li  $^{1}$  Xiuqiang He  $^{2}$  Ruixuan Li  $^{1}$
+
+# Abstract
+
+Low-rank adaptation (LoRA) is a widely used parameter-efficient fine-tuning method. In standard LoRA layers, one of the matrices,  $A$  or  $B$ , is initialized to zero, ensuring that fine-tuning starts from the pretrained model. However, there is no theoretical support for this practice. In this paper, we investigate the impact of non-zero initialization on LoRA's fine-tuning dynamics from an infinite-width perspective. Our analysis reveals that, compared to zero initialization, simultaneously initializing  $A$  and  $B$  to non-zero values improves LoRA's robustness to suboptimal learning rates, particularly smaller ones. Further analysis indicates that although the non-zero initialization of  $AB$  introduces random noise into the pretrained weight, it generally does not affect fine-tuning performance. In other words, fine-tuning does not need to strictly start from the pretrained model. The validity of our findings is confirmed through extensive experiments across various models and datasets. The code is available at https://github.com/Leopold1423/non_zero_lora-icml25.
+
+# 1. Introduction
+
+Large language models (LLMs), such as GPT-4 (OpenAI, 2024) and DeepSeek-V3 (DeepSeek-AI, 2024), have achieved unprecedented performance across a broad range of tasks. These state-of-the-art models typically contain hundreds of billions of parameters. Consequently, fully fine-tuning these models is computationally expensive and
+
+*Equal contribution. *Huazhong University of Science and Technology, Wuhan, China *Shenzhen Technology University, Shenzhen, China *FiT, Tencent, Shenzhen, China. Correspondence to: Xing Tang <xing.tang@hotmail.com>, Haozhao Wang <hz_wang@hust.edu.cn>, Ruixuan Li <rxli@hust.edu.cn>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+<table><tr><td>LoRA Structure</td><td colspan="2">Pretrained Weights
+W ∈ Rm×n
+A ∈ Rr×n</td></tr><tr><td>Initialization Strategy</td><td>Init[A]: A ∈ N(0,σ2)
+B = 0</td><td>Init[AB]: A ∈ N(0,σ2)
+B ∈ N(0,σ2)</td></tr><tr><td>Fine-tuning Dynamics</td><td>Efficient Learning with Stability, but Sensitive to learning rate</td><td>Efficiency and Stability, with Robustness to suboptimal learning rate</td></tr></table>
+
+Figure 1. LoRA's fine-tuning dynamics with zero initialization (i.e., Init[A]) and non-zero initialization (i.e., Init[AB]). Compared to Init[A], Init[AB] improves LoRA's robustness to suboptimal learning rates, generally leading to better performance.
+
+storage-intensive, presenting significant challenges for their practical deployment. To address these issues, various parameter-efficient fine-tuning (PEFT) methods have been proposed to reduce the number of trainable parameters, thereby decreasing the cost of fine-tuning (Xu et al., 2023).
+
+Among these PEFT methods, low-rank adaptation (LoRA) (Hu et al., 2022) has gained considerable attention due to its efficiency and minimal impact on inference latency. LoRA approximates the update of the pretrained weight matrix  $W$  using smaller low-rank matrices,  $A$  and  $B$ , as illustrated in Figure 1. The forward pass of LoRA is formulated as  $Y = (W + \alpha / rBA)X$ , where  $r$  is the LoRA rank and  $\alpha$  is a tunable scaling factor. During fine-tuning,  $W$  remains fixed, while only matrices  $A$  and  $B$  are updated, usually via the Adam optimizer (Kingma & Ba, 2015).
+
+In standard LoRA layers, either  $A$  or  $B$  is initialized to zero, ensuring that fine-tuning starts from the pretrained model. Building on this premise, Hayou et al. (2024b) explores the difference between applying Kaiming initialization (He et al., 2016) to  $A$  and  $B$ . However, there is no theoretical justification for the zero initialization of LoRA, which raises
+
+the question: Is it optimal to initialize  $A$  or  $B$  to zero? To answer this question, we explore the impact of initialization on LoRA's fine-tuning dynamics. Let Init[A] denote the case where  $A$  is randomly initialized and  $B$  is set to zero, while Init[AB] denotes the case where both  $A$  and  $B$  are initialized to non-zero values, as illustrated in Figure 1. Our analysis demonstrates that, compared to Init[A], Init[AB] improves LoRA's robustness to suboptimal learning rates, particularly smaller ones. Notably, learning rate decay is commonly applied during fine-tuning, making smaller learning rates more prevalent.
+
+However, non-zero initialization introduces noise to the pretrained model before fine-tuning, which raises another question: Is it necessary for fine-tuning to strictly start from the pretrained model? Note that pretrained weights are usually suboptimal for downstream tasks, suggesting that the pretrained model itself contains inherent noise. Thus, we conjecture that the noise introduced by non-zero initialization can also be corrected through fine-tuning, without additional optimization cost. Extensive verification shows that, with appropriate initialization variance, the noise introduced by non-zero initialization does not degrade the fine-tuning accuracy. Notably, the range of suitable initialization variances is quite broad, encompassing that used in Kaiming initialization. In conclusion, fine-tuning does not have to strictly start from the pretrained model.
+
+The main contributions of this paper can be summarized as two novel insights that challenge the traditional inertial constraints of LoRA fine-tuning:
+
+1. You do not need to initialize LoRA's matrices  $A$  or  $B$  to zero. Compared to zero initialization, non-zero initialization enhances LoRA's robustness to the learning rate and generally achieves better fine-tuning accuracy.  
+2. You do not need to start fine-tuning strictly from a pretrained model. Although non-zero initialization alters the starting point of LoRA fine-tuning, it does not impact the fine-tuning performance, as long as an appropriate initialization variance is employed.  
+3. Experimental results confirm the validity of our theory and findings. LoRA with non-zero initialization consistently achieves superior accuracy compared to zero initialization, particularly at smaller learning rates.
+
+# 2. Preliminaries
+
+# 2.1. Scaling Theory of Neural Networks
+
+The scaling theory of neural networks is widely used to guide the selection of hyperparameters, such as learning rates (Hayou et al., 2024a), initialization strategies (He et al., 2016; Yang, 2019; Hayou et al., 2024b) and depth
+
+parametrizations (Bordelon et al., 2024; Yang, 2020) etc. It typically involves analyzing the statistical properties of key quantities, such as pre-activation values, by extending the network's width or depth to infinity. Subsequently, this analysis is leveraged to refine hyperparameters, ensuring favorable convergence properties under these extreme conditions. For instance, Kaiming initialization (He et al., 2016) suggests that the variance of the weights in hidden layers should be scaled by  $\frac{1}{n}$  to prevent excessively large pre-activations as the model width  $n$  increases. More details on the scaling theory are provided in Appendix A.1.
+
+In this paper, we apply the scaling theory to investigate the fine-tuning dynamics of LoRA from the perspective of infinite width. Specifically, we explore how to set the learning rate and initialization strategy for LoRA to achieve desired performance. In our analysis, we assume that only the model width  $n$  increases, while other factors, such as the LoRA rank, number of layers, sequence length, and number of training steps, remain constant. To characterize the asymptotic behavior of the analyzed variables as  $n$  increases, we further introduce the following notations, which have also been utilized in previous analysis of LoRA, particularly in the context of learning rates (Hayou et al., 2024a) and zero-initialization strategies (Hayou et al., 2024b).
+
+Notations. Let  $c_{n} \in \mathbb{R}$  and  $d_{n} \in \mathbb{R}^{+}$  be sequences. If there exists a constant  $\kappa > 0$  such that  $c_{n} < \kappa d_{n}$ , we write  $c_{n} = \mathcal{O}(d_{n})$ . If  $c_{n} > \kappa d_{n}$ , we write  $c_{n} = \Omega(d_{n})$ . If both  $c_{n} = \mathcal{O}(d_{n})$  and  $c_{n} = \Omega(d_{n})$  hold simultaneously, we write  $c_{n} = \Theta(d_{n})$ . For a vector sequence  $c_{n} = (c_{n}^{i})_{1 \leq i \leq k} \in \mathbb{R}^{k}$  ( $k > 0$ ), we write  $c_{n} = \mathcal{O}(d_{n})$  if and only if  $c_{n}^{i} = \mathcal{O}(d_{n})$  for all  $i \in [k]$ . This similarly applies to  $\Theta$  and  $\Omega$ . Additionally, when the sequence  $c_{n}$  is a vector of random variables, convergence is understood to be convergence of its  $\ell_{2}$  norm. To track the exponent of the asymptotic behaviour of some quantity  $v$ , we further introduce the  $\gamma$  operator, defined as  $v = \Theta(n^{\gamma[v]})$ . For example,  $\gamma[n] = 1$ ,  $\gamma[1] = 0$ , and  $\gamma[0] = -\infty$ . For two real-valued variables  $u$  and  $v$ , we have  $\gamma[v \times u] = \gamma[v] + \gamma[u]$ , and  $\gamma[v + u] = \max\{\gamma[v], \gamma[u]\}$  provided that  $v \neq -u$ . More details about the  $\gamma$ -operator are provided in Appendix A.2.
+
+Next, we focus on the asymptotic behavior of the learning rate,  $\gamma [\eta ]$ , as the network width  $n\to \infty$ , rather than its exact value. Given  $\gamma [\eta ]$ ,  $\eta = \Theta (n^{\gamma [\eta ]})\approx c\cdot n^{\gamma [\eta ]}$ , where  $c > 0$  is a constant and lower-order terms are neglected. As  $n\to \infty$ , the term  $n^{\gamma [\eta ]}$  dominates, making  $\gamma [\eta ]$  the key factor determining the asymptotic behavior of  $\eta$ . While the constant  $c$  is important for exact values, it doesn't influence the asymptotic scaling behavior. Ignoring the influence of  $c$ , perturbations to  $\gamma [\eta ]$  and  $\eta$  are effectively equivalent. Therefore, we analyze the impact of perturbations in  $\gamma [\eta ]$
+
+on fine-tuning dynamics, excluding  $c$  from consideration. This analysis aims to guide the selection of learning rates and initialization strategies based on asymptotic properties, rather than computing precise values that depend on  $c$ .
+
+# 2.2. LoRA and LoRA Features
+
+Based on the previous analysis of LoRA presented in (Hayou et al., 2024a;b), we adopt the same notation and consider a general neural network of the following form:
+
+$$
+\left\{ \begin{array}{l} Y _ {i n} (X) = W _ {i n} X, \\ Y _ {l} (X) = \mathcal {F} _ {l} \left(W _ {l}, Y _ {l - 1} (X)\right), l \in [ L ], \\ Y _ {\text {o u t}} (X) = W _ {\text {o u t}} Y _ {L} (X), \end{array} \right. \tag {1}
+$$
+
+where  $X \in \mathbb{R}^d$  denotes the input vector, and  $L \geq 1$  is the network depth. Let  $n$  denote the network width and  $\mathcal{F}$  denote the nonlinear mapping function.  $W_{l} \in \mathbb{R}^{n \times n}$  is the weight matrix of the  $l$ -th hidden layer.  $W_{in}$  and  $W_{out}$  correspond to the weight matrices of the input and output layers, respectively. This model represents a pretrained neural network that will be fine-tuned for a specific downstream task using the LoRA framework defined below.
+
+Definition 1 (LoRA). Let  $W \in \mathbb{R}^{n_1 \times n_2}$  be a weight matrix from the pretrained model. LoRA approximates the update of  $W$  using low-rank matrices, such that the value of the pretrained weight becomes  $W + \alpha / rBA$ . Here, only  $B \in \mathbb{R}^{n_1 \times r}$  and  $A \in \mathbb{R}^{r \times n_2}$  are trainable. The rank  $r \ll \min(n_1, n_2)$ , and  $\alpha \in \mathbb{R}$  is a tunable scaling factor.
+
+For each LoRA layer within the network, let  $\underline{Z}$  denote its input and let  $\bar{Z}$  denote the output after incorporating the pretrained weights, i.e.,  $\bar{Z} = W\underline{Z} + \alpha / rBA\underline{Z}$ . In this paper, we primarily investigate the statistical properties of the output of the LoRA module, i.e.,  $BA\underline{Z}$ . To this end, we further introduce the LoRA features as follows:
+
+Definition 2 (LoRA Features). Given a general neural architecture and a LoRA (Definition 1) layer, we define the LoRA features  $(Z_A,Z_B)$  as  $Z_{A} = A\underline{Z}$  and  $Z_{B} = BZ_{A} = BA\underline{Z}$ . The superscript  $t$  of  $Z_A^t$ ,  $Z_B^t$  represents the value of the LoRA features in the  $t$ -th step of fine-tuning.
+
+Definitions 1 and 2 are straightforward and have been used in (Hayou et al., 2024a;b). Next, we focus on the training dynamics of the final output  $Z_{B}$  as  $n$  approaches infinity.
+
+# 3. Fine-Tuning Dynamics of LoRA with Adam
+
+In this section, we first introduce two critical statistics of LoRA: stability and efficiency, as discussed in Section 3.2, along with their respective solution sets for the learning rate and initialization. Next, we investigate the sensitivity of these solutions to the learning rate in Section 3.3 and explain how non-zero initialization improves their robustness.
+
+Additionally, a simple toy model is presented in Section 3.4 to validate our theoretical findings. Finally, we explore whether non-zero initialization affects fine-tuning accuracy by deviating from the intended purpose of zero initialization, which is to start fine-tuning from a pretrained model.
+
+# 3.1. Simplified Settings and the Optimizer
+
+To simplify the analysis, we adopt the setup proposed in (Hayou et al., 2024a;b). Specifically, we assume that the fine-tuning dataset comprises a single data point,  $(x,y)$ . The optimization goal is to minimize the loss calculated using the pretrained model that incorporates LoRA.
+
+Moreover, we train only one LoRA layer to isolate the contribution of individual LoRA layers to feature learning. Let  $\underline{Z}$  denote the input to the LoRA layer computed from  $x$ , and let  $d\bar{Z}^t$  represent the gradient of the loss function, evaluated at the data point  $(x,y)$ , with respect to  $\bar{Z}$ . During fine-tuning, the input  $Z$  remains constant as the step  $t$  increases. In contrast,  $d\bar{Z}^t$  varies with  $t$ , as  $\bar{Z}^t$  evolves over time. Next, we examine the behavior of  $Z_B$  (Definition 2) as  $n$  approaches infinity from three perspectives: stability, efficiency, and robustness. Formal definitions of these perspectives will be provided in subsequent sections.
+
+Adam (Kingma & Ba, 2015) is a commonly used optimizer for fine-tuning LLMs. Therefore, we focus on the fine-tuning dynamics of LoRA with the Adam optimizer. A similar analysis for the SGD optimizer (Bottou, 2010) is provided in Appendix C, where we observe that Adam outperforms SGD in enhancing LoRA's feature learning capabilities, primarily due to the gradient normalization process. The conclusions drawn in this section are partially applicable to SGD; further details can be found in Appendix C.
+
+# 3.2. Stability and Efficiency of LoRA
+
+To ensure that the LoRA layers produce stable outputs, it is expected that  $Z_{B}$  remains within a reasonable range throughout the fine-tuning process. Consequently, the concept of stability is introduced and defined as follows:
+
+Definition 3 (Stability). A LoRA fine-tuning process is considered stable if, for all LoRA layers and for all  $t > 1$ ,  $Z_B^t = \Theta(1)$  as the width  $n$  approaches infinity.
+
+Stability indicates that as the width increases, the output of LoRA,  $Z_{B}$ , remains bounded and does not diminish, demonstrating LoRA's effective and positive contribution to fine-tuning outcomes. For the sake of analytical clarity,
+
+we do not consider LoRA's internal stability (i.e.,  $Z_A$ ) in this section. Instead, a detailed analysis and discussion of internal stability are provided in Appendix B.2.
+
+In addition to the stability of  $Z_{B}$ , we are more concerned about whether  $Z_{B}$  is effectively updated during fine-tuning. At the  $t$ -th step, the update to  $Z_{B}$  is given by:
+
+$$
+\Delta Z _ {B} ^ {t} = \underbrace {B _ {t - 1} \Delta Z _ {A} ^ {t}} _ {\delta_ {t} ^ {1}} + \underbrace {\Delta B _ {t} Z _ {A} ^ {t - 1}} _ {\delta_ {t} ^ {2}} + \underbrace {\Delta B _ {t} \Delta Z _ {A} ^ {t}} _ {\delta_ {t} ^ {3}}, \quad (2)
+$$
+
+As discussed in (Hayou et al., 2024a;b), the terms  $\delta_t^1$  and  $\delta_t^2$  denote 'linear' updates, reflecting the effects of training  $A$  and  $B$  independently. The third term,  $\delta_t^3$ , represents a 'multiplicative' update, capturing the combined effect of optimizing both  $A$  and  $B$  simultaneously. Ideally, as  $n$  approaches infinity,  $\Delta Z_B$  should remain  $\Theta(1)$  to ensure effective updates to  $Z_B$  during fine-tuning. A similar condition was proposed in (Yang, 2020) for pretraining and later adopted by (Hayou et al., 2024a;b) as a metric for efficient feature learning in LoRA. According to (Hayou et al., 2024a;b), it is essential for both  $\delta_t^1$  and  $\delta_t^2$  to be  $\Theta(1)$  to ensure that the updates of  $A$  and  $B$  contribute positively to  $\Delta Z_B$ . Based on the above analysis, the efficiency of LoRA's feature learning is defined as follows:
+
+Definition 4 (Efficiency). A LoRA fine-tuning process is considered efficient if it is stable (Definition 3) and, for all LoRA layers and for all  $t > 1$ ,  $\delta_t^1 = \Theta(1)$  and  $\delta_t^2 = \Theta(1)$ .
+
+In contrast to stability, efficiency emphasizes the effective updating of  $Z_{B}$ . We now analyze the effects of initialization and learning rate on both stability and efficiency. At the  $t$ -th step of fine-tuning, the LoRA weights are updated as  $A_{t} = A_{t - 1} - \eta_{A}g_{A}^{t - 1} = A_{0} - \eta_{A}\sum_{i = 0}^{t - 1}g_{A}^{i}$  and  $B_{t} = B_{t - 1} - \eta_{B}g_{B}^{t - 1} = B_{0} - \eta_{B}\sum_{i = 0}^{t - 1}g_{B}^{i}$ , where  $(\eta_A,\eta_B)$  are the learning rates,  $(A_0,B_0)$  are the initialization values, and  $(g_{A},g_{B})$  represent the normalized gradients generated by the Adam optimizer. Note that both  $g_{A}$  and  $g_{B}$  are of order  $\Theta (1)$  due to the gradient normalization process. By substituting  $\Delta Z_A^t = -\eta_A g_A \underline{Z}$  and  $\Delta B^{t} = -\eta_{B} g_{B}$  into Eq.(2), and applying the  $\gamma$ -operator, the conditions for LoRA to achieve stability and efficiency can be expressed as:
+
+$$
+\left\{ \begin{array}{l l} \gamma [ \delta_ {t} ^ {1} ] = \gamma [ - \eta_ {A} B _ {t - 1} g _ {A} ^ {t - 1} \underline {{Z}} ] = 0 & (\delta_ {t} ^ {1} = \Theta (1)) \\ \gamma [ \delta_ {t} ^ {2} ] = \gamma [ - \eta_ {B} g _ {B} ^ {t - 1} A _ {t - 1} \underline {{Z}} ] = 0 & (\delta_ {t} ^ {2} = \Theta (1)) \\ \gamma [ Z _ {B} ^ {t - 1} ] = \gamma [ B _ {t - 1} A _ {t - 1} \underline {{Z}} ] = 0 & (Z _ {B} ^ {t - 1} = \Theta (1)) \end{array} \right. \tag {3}
+$$
+
+Using the properties of the  $\gamma$ -operator, we can express the
+
+above equations in terms of  $(A_0,B_0)$  and  $(\eta_A,\eta_B)$  as:6
+
+$$
+\left\{ \begin{array}{l} \max  \left(\gamma \left[ B _ {0} \right], \gamma \left[ \eta_ {B} \right]\right) + \gamma \left[ \eta_ {A} \right] + 1 = 0, \\ \max  \left(\gamma \left[ A _ {0} \right], \gamma \left[ \eta_ {A} \right]\right) + \gamma \left[ \eta_ {B} \right] + 1 = 0, \\ \max  \left(\gamma \left[ A _ {0} \right], \gamma \left[ \eta_ {A} \right]\right) + \max  \left(\gamma \left[ B _ {0} \right], \gamma \left[ \eta_ {B} \right]\right) + 1 = 0, \end{array} \right. \tag {4}
+$$
+
+which further leads to the following conditions:
+
+$$
+\left\{ \begin{array}{l} \gamma [ \eta_ {A} ] + \gamma [ \eta_ {B} ] = - 1, \\ \gamma [ A _ {0} ] \leq \eta_ {A}, \quad \gamma [ B _ {0} ] \leq \eta_ {B}. \end{array} \right. \tag {5}
+$$
+
+By default, a uniform learning rate is used for the weight matrices  $A$  and  $B$ , i.e.,  $\eta_{A} = \eta_{B}$ . Based on this condition, the solution to the above equations is
+
+$$
+\left\{ \begin{array}{l} \gamma [ \eta_ {A} ] = \gamma [ \eta_ {B} ] = - 1 / 2, \\ \gamma [ A _ {0} ] \leq - 1 / 2, \quad \gamma [ B _ {0} ] \leq - 1 / 2, \end{array} \right. \tag {6}
+$$
+
+where  $\gamma [A_0]$  and  $\gamma [B_0]$  cannot both be equal to  $-\infty$  simultaneously, as this would imply that both  $A$  and  $B$  are zero, preventing optimization via gradient descent algorithms.
+
+In standard LoRA,  $A$  is initialized using Kaiming initialization, while  $B$  is initialized to zero. This initialization strategy is denoted as Init [A]. In contrast, when  $B$  is initialized using Kaiming initialization and  $A$  is initialized to zero, the strategy is referred to as Init [B]. Let  $\sigma_A^2$  and  $\sigma_B^2$  represent the variances of  $A_0$  and  $B_0$ , respectively. For Init [A],  $\sigma_A^2 = 1/n$  and  $\sigma_B^2 = 0$ , which results in  $\gamma[A_0] = -1$  and  $\gamma[B_0] = -\infty$ , thereby satisfying Eq.(6). However, for Init [B],  $\sigma_A^2 = 0$  and  $\sigma_B^2 = 1/r$ , resulting in  $\gamma[A_0] = -\infty$  and  $\gamma[B_0] = 0$ , which fails to meet the desired conditions. Consequently, we conclude that LoRA achieves stable and efficient feature learning when using Adam with Init [A], but not with Init [B]. This conclusion aligns with the findings of (Hayou et al., 2024b).
+
+Similarly, when using the SGD optimizer, solutions to ensure stability and efficiency for LoRA can be derived as:
+
+$$
+\left\{ \begin{array}{l} \gamma [ \eta_ {A} ] = \gamma [ \eta_ {B} ] = - 1 / 2, \\ \gamma [ A _ {0} ] \leq - 3 / 4, \quad \gamma [ B _ {0} ] \leq - 1 / 4, \end{array} \right.
+$$
+
+where at least one of the two inequalities must hold with equality. It is evident that neither Init[A] nor Init[B] can satisfy the desired conditions when using the SGD optimizer. Furthermore, we highlight that Adam offers more flexible initialization options than SGD, a benefit attributed to the gradient normalization process. Detailed proofs and explanations can be found in Appendix C.
+
+# 3.3. Robustness of LoRA
+
+The solution presented in Eq.(6) demonstrates that LoRA's stability and efficiency impose stricter constraints on the learning rate, while the range of feasible initialization values is comparatively broader. Note that strictly setting  $\gamma[\eta_A]$  and  $\gamma[\eta_B]$  to  $-1/2$  is particularly challenging in practical scenarios, as the constant in  $\Theta$  is generally intractable. Consequently, we investigate the impact of deviations in  $\gamma[\eta]$  from  $-\frac{1}{2}$  on LoRA's stability and efficiency. For the case where  $\gamma[\eta] > -1/2$ ,  $\gamma[\delta_1] = \gamma[\delta_2] = \gamma[Z_B] = 2\gamma[\eta] + 1 > 0$ , which inevitably causes instability and inefficiency. Therefore, learning rate decay is commonly applied during fine-tuning to prevent excessively large learning rates. In the following, we primarily focus on the case where  $\gamma[\eta] \leq -1/2$ .
+
+First, when  $\gamma[\eta_A] \leq \gamma[A_0] \leq -1/2$ ,  $\max(\gamma[A_0], \gamma[\eta_A]) = \gamma[A_0]$ . In this case,  $\gamma[\eta_A] < -1/2$  will not affect  $\gamma[\delta_2]$  and  $\gamma[Z_B]$ . Similarly, when  $\gamma[\eta_B] \leq \gamma[B_0] \leq -1/2$ ,  $\gamma[\eta_B] < -1/2$  will not affect  $\gamma[\delta_1]$  and  $\gamma[Z_B]$ . Conversely, when  $\gamma[\eta_A] > \gamma[A_0]$ ,  $\gamma[\delta_1] = \gamma[\eta_A] + \gamma[\eta_B] + 1 = 2\gamma[\eta] + 1$ , which implies that changes in  $\eta$  will result in a quadratic change in  $\delta_1$ . The same holds for  $\delta_2$  and  $Z_B$ . Clearly, in this case where  $\gamma[\eta_A] > \gamma[A_0]$  and  $\gamma[\eta_B] > \gamma[B_0]$ , LoRA is more sensitive to fluctuations in the learning rate. Based on these insights, we formally define the robustness of LoRA's stability and efficiency as follows:
+
+Definition 5 (Robustness). Assume a LoRA fine-tuning process is stable and efficient (Definitions 3-4) with  $\gamma[\eta] = \gamma^{*}$ . The process is considered robust with respect to the learning rates  $\gamma[\eta_A] \leq \gamma^*$  and  $\gamma[\eta_B] \leq \gamma^*$  if, for all LoRA layers and for all  $t > 1$ , the following relations hold:  $P(\gamma[\delta_1] \mid \gamma[\eta_B]) = P(\gamma[\delta_1])$ ,  $P(\gamma[\delta_2] \mid \gamma[\eta_A]) = P(\gamma[\delta_2])$ , and  $P(\gamma[Z_B] \mid \gamma[\eta_A]) = P(\gamma[Z_B] \mid \gamma[\eta_B]) = P(\gamma[Z_B])$ .
+
+Robustness focuses on the impact of suboptimal learning rates, particularly those lower than the optimal value, on the stability and efficiency (i.e.,  $\delta_1, \delta_2$  and  $Z_B$ ). Specifically,  $\delta_1$  reflects the effect of training  $A$  independently, and robustness seeks its independence from  $\eta_B$ . The same principle applies to  $\delta_2$ . To achieve this, both  $\gamma[A_0]$  and  $\gamma[B_0]$  should be set to  $-1/2$ , which leads to the following relationships for all learning rates satisfying  $\gamma[\eta_A] \leq -1/2$  and  $\gamma[\eta_B] \leq -1/2$ .
+
+$$
+\left\{ \begin{array}{l} \gamma [ \delta_ {1} ] = \gamma [ B _ {0} ] + \gamma [ \eta_ {A} ] + 1 = \gamma [ \eta_ {A} ] + 1 / 2, \\ \gamma [ \delta_ {2} ] = \gamma [ A _ {0} ] + \gamma [ \eta_ {B} ] + 1 = \gamma [ \eta_ {B} ] + 1 / 2, \\ \gamma [ Z _ {B} ] = \gamma [ A _ {0} ] + \gamma [ B _ {0} ] = 1. \end{array} \right.
+$$
+
+Clearly,  $\delta_{1}$  and  $\delta_{2}$  vary linearly with changes in the learning rates  $\eta_{A}$  and  $\eta_{B}$ , respectively, while  $\gamma [Z_B] = 1$  remains
+
+unaffected, thereby ensuring consistent stability. Based on the above analysis, we derive Theorem 1, which presents the initialization strategy that optimally supports robustness (Definition 5) while maintaining stability and efficiency (Definitions 3 and 4) for LoRA fine-tuning.
+
+Theorem 1 (Robust LoRA (Informal)). A LoRA fine-tuning process optimized with Adam exhibits optimal robustness (Definition 5) to the learning rate when  $A_0 = \Theta(n^{-1/2})$  and  $B_0 = \Theta(n^{-1/2})$ .
+
+This initialization scheme is referred to as Init[AB], where it is observed that  $A_0B_0 = \Theta(n^{-1})$ . This is equivalent to treating  $AB$  as a whole and then performing Kaiming initialization, rather than directly performing Kaiming initialization for  $A$  or  $B$  as done in Init[A] or Init[B]. As shown in Table 1, Init[AB] can simultaneously achieve stability, efficiency, and robustness, whereas Init[A] and Init[B] cannot.
+
+Table 1. A comparison between various initialization strategies.  
+
+<table><tr><td>Initialization</td><td>Stability</td><td>Efficiency</td><td>Robustness</td></tr><tr><td>Init [B]</td><td>✓</td><td>×</td><td>×</td></tr><tr><td>Init [A]</td><td>✓</td><td>✓</td><td>×</td></tr><tr><td>Init [AB]</td><td>✓</td><td>✓</td><td>✓</td></tr></table>
+
+However, enforcing  $A_0$  and  $B_0$  to be exactly  $\Theta(n^{-1/2})$  is challenging (similar to the learning rate), as the constant within  $\Theta$  is typically intractable. Therefore, we do not impose a strict requirement on  $A_0$  and  $B_0$  to be of order  $\Theta(n^{-1/2})$ . Instead, our main focus is ensuring that  $A_0 / B_0 = \Theta(1)$ , which guarantees that  $A$  and  $B$  are initialized to nonzero values simultaneously. For simplicity, we ignore the constant in  $\Theta(1)$ , implying that the variances of  $A_0$  and  $B_0$  are equal. Compared to initializing  $B$  to zero, this at least enhances the robustness of LoRA fine-tuning with respect to  $\eta_B$ . Finally, we formally define the non-zero initialization for LoRA as follows:
+
+Definition 6 (Non-Zero Initialization). A LoRA layer is non-zero initialized if the weight matrices  $A$  and  $B$  are initialized with the same variance, i.e.,  $\sigma_A^2 = \sigma_B^2$ .
+
+# 3.4. Toy Model
+
+In this section, we verify the advantages of non-zero initialization in LoRA by fine-tuning the following toy model, which is actually the general model in Eq.(1) with  $L = 1$ .
+
+$$
+f (X) = W _ {o u t} \mathcal {F} ((W _ {0} + B A) \mathcal {F} (W _ {i n} X)),
+$$
+
+where  $X \in \mathbb{R}^d$  is the input vector,  $W_{in} \in \mathbb{R}^{n \times d}$ ,  $W_{out} \in \mathbb{R}^{10 \times n}$ , and  $W_0 \in \mathbb{R}^{n \times n}$  are the pretrained weights.  $A \in \mathbb{R}^{r \times n}$  and  $B \in \mathbb{R}^{n \times r}$  are the LoRA weights. The dimensions  $(d, n, r)$  are set to 784, 4096, and 32. The MNIST (Deng, 2012) and FashionMNIST (Xiao et al., 2017)
+
+datasets are used for pretraining and fine-tuning, respectively. During fine-tuning, only matrices  $A$  and  $B$  are trainable, while  $W_{in}$ ,  $W_{0}$  and  $W_{out}$  remain fixed. Adam is used as the optimizer. More details about the toy model can be found in Appendix D.1. Subsequently, we compare the test loss of LoRA under zero and non-zero initializations, varying learning rates and initialization variances. The results, presented in Figure 2, demonstrate that the performance of LoRA with zero initialization deteriorates significantly at lower learning rates, whereas the non-zero initialization consistently yields better performance, aligning with our theoretical expectations.
+
+![](images/e4e4fc3a2814de2110cc7d3f26596024f4367c23086bd7293167fb832ce0aad8.jpg)
+
+![](images/58f965dec84bcc7832eb5ba98450d242d1e17c6356c0e0204694a516d7f9772d.jpg)  
+Figure 2. Test loss of LoRA fine-tuned with zero initialization (Init[A]) and non-zero initialization (Init[AB]). The lower-left corner presents the improvement ratio of Init[AB] relative to Init[A], while the lower-right corner presents the convergence curves of Init[AB] and Init[A] under specific settings.
+
+# 3.5. Starting Point of LoRA Fine-Tuning
+
+Previous studies have unintentionally employed the nonzero initialization for LoRA (Li et al., 2024; Meng et al., 2024; Wang et al., 2024). LoftQ (Li et al., 2024) fine-tunes a quantized pretrained model and uses quantization errors to initialize the LoRA module. Specifically, it generates the initialization values for LoRA by performing truncated Singular Value Decomposition (SVD) on the quantization errors, thereby minimizing these errors in the pretrained model. PISSA (Meng et al., 2024) suggests performing SVD on the pretrained weights and initializing  $A$  and  $B$  using the singular vectors and largest singular values. It believes that updating the principal components of the pretrained model accelerates convergence and enhances performance. LoRA-GA (Wang et al., 2024) proposes initializing
+
+$A$  and  $B$  based on gradient information. This involves computing the gradient of the pretrained model over several steps of gradient descent and applying truncated SVD on the gradient to derive the initialization values for matrices  $A$  and  $B$ . Although these previous methods utilize non-zero initialization for LoRA, they do not discuss the benefits nor provide a theoretical foundation for such initialization. This study bridges these gaps and offers theoretical support for non-zero initialization techniques.
+
+Standard LoRA employs zero initialization to ensure that fine-tuning starts from the pretrained model. However, nonzero initialized LoRA introduces noise to the pretrained model before fine-tuning begins. A straightforward solution to this problem is to subtract  $\frac{\alpha}{r} A_0 B_0$  from the pretrained weights, as proposed by Wang et al. (2024). However, this requires modifications to the pretrained model, complicating its reuse. This challenge becomes particularly evident in multi-LoRA scenarios, where the pretrained model is shared across multiple LoRA modules for different tasks (Sheng et al., 2023). In such cases, it is required that the initialization of different LoRA modules remains consistent.
+
+![](images/f5fb33bb484ca167d74286b130d4f2b45214d9a981100529283a02345fee2f49.jpg)  
+Figure 3. Test loss of un-fine-tuned T5-Base on different dataset with various initialization size  $(\beta^{2})$ . Note that the initialization variances of matrices  $A$  and  $B$  are both set to  $(\beta \sigma_{k})^{2}$ , where  $\sigma_{k}^{2}$  is the variances of  $A$  when initialized with Kaiming initialization.
+
+Above discussion raises the question of whether it is necessary to subtract  $\frac{\alpha}{r} A_0 B_0$  from the pretrained model. Note that pretrained weights are typically not optimal for downstream tasks. Therefore, we conjecture that introducing small perturbations to the pretrained weights will not significantly affect their performance on downstream tasks, nor will it substantially impact convergence during fin-tuning. To test this hypothesis, we conducted a preliminary evaluation using the T5-Base model (Raffel et al., 2020). Specifically, we compared the accuracy of the un-fine-tuned T5-Base on several datasets of the GLUE benchmark (Wang et al., 2019), utilizing non-zero initialized LoRA with various initialization variances. As shown in Figure 3, the non-zero initialization does not impact the initial performance of the pretrained model on downstream tasks unless the initialization variance is exceedingly large. Therefore, we conclude that, with appropriate initialization variances,
+
+![](images/11b182ac07c893beeaff22cee5b5cf0e60b2c9881c392a3ca25823558305355a.jpg)
+
+![](images/1ff823c15bc354fcba22be97d8c758dbc74e346ad23cb33e423f74f00ad8c069.jpg)
+
+![](images/b170e78157bb6c37652e8a944ca8b083fc278b622500605bad1e0b91d90a2083.jpg)
+
+![](images/9bad6b60bd290c42c33805e4023e1bcf642b97ec54bd611dcef96d71adb567f8.jpg)
+
+![](images/8b20c9fa5c4d4f297e62a9c44bef22f66c2d896582c7452c78f49f7a5e6a0c3f.jpg)  
+Figure 4. Fine-tuning accuracy of T5-Base on different datasets with various learning rates  $(\eta)$  and initialization sizes  $(\beta)$ . Note that the initialization variance is set to  $(\beta \sigma_k)^2$ , where  $\sigma_k^2$  is the variance of matrix  $A$  when initialized with Kaiming initialization. Init[AB]-Init[A] denotes the accuracy improvement achieved by non-zero initialization, where  $B$  is initialized with the same variance as  $A$ .
+
+![](images/b93f42d6e2211c70f40a392237cb909d7a05b322c0a9a514a48a98e4ab8eb4ef.jpg)
+
+![](images/f210fe73202bb291ca25a00a65fe2b05a1ee561a9b63f6faa8983f04a3b6baab.jpg)
+
+![](images/e8647424e7dad642e1670def69ee7d1893d710ffec5729879c16103acad5edea.jpg)
+
+there is no need to subtract the non-zero initialized LoRA from the pretrained model. Further verification will be conducted in subsequent experiments.
+
+# 4. Experiments
+
+In this section, we evaluate our findings across three benchmarks. First, we evaluate natural language understanding tasks using a subset of the GLUE benchmark (Wang et al., 2019). Next, we evaluate natural language generation tasks using the commonsense reasoning and arithmetic reasoning benchmarks, as proposed in (Hu et al., 2023). Our primary objective is to examine the impact of zero and non-zero initialization on fine-tuning performance across different learning rates and initialization variances. Let  $\sigma_k^2$  represent the variance of matrix  $A$  under Kaiming initialization (i.e.,  $\sigma_k^2 = \frac{1}{n}$ ). In our experiments, the initialization variance is set to  $(\beta \sigma_k)^2$ , where  $\beta$  controls the initialization size. For non-zero initialization,  $B$  is initialized with the same variance as  $A$ . By default, we subtract the non-zero initialized LoRA from the pretrained weights. The effects of omitting this subtraction process will be discussed in Section 4.3. More experimental details and results can be found in Appendices D and E. It is worth noting that the standard deviation for the GLUE dataset ranges from 0.01 to 0.4, while for commonsense and arithmetic reasoning tasks, it spans from 0.2 to 0.4. In our experiments, smaller learning rates tend to converge less effectively and exhibit higher standard deviations. However, this standard deviation
+
+is generally minimal compared to the performance gains achieved by non-zero initialization.
+
+# 4.1. Natural Language Understanding Tasks
+
+Models and Datasets. In this section, we fine-tune the T5-Base model (Raffel et al., 2020) on several datasets from the GLUE benchmark, including MRPC, CoLA, SST-2, and QNLI. The experiment is conducted based on the code provided in (Wang et al., 2024). Each experiment is run three times, and the average test accuracy is reported.
+
+Results. Figure 4 presents the test accuracy of T5-Base on the GLUE benchmark. As the initialization variance increases, Init[A] exhibits enhanced robustness at smaller learning rates, aligning with our theoretical predictions. Notably,  $\beta = 1$  corresponds to the Kaiming initialization  $(\gamma [A_0] = -1)$ , which is smaller than the theoretically optimal value of  $\gamma [A_0] = -\frac{1}{2}$ . Moreover, initializing  $B$  with the same variance as  $A$  (i.e., Init[AB]) further improves performance over Init[A] at smaller learning rates, resulting in up to a  $10\%$  accuracy improvement. Figure 5 illustrates the convergence curves of Init[A] and Init[AB] on the QNLI dataset with a learning rate of 3e-7, where Init[AB] accelerates convergence by nearly a factor of two. Additionally, even when the learning rate is near optimal, Init[AB] still yields a  $1\%$  accuracy improvement. A heatmap for Init[AB] is provided in Appendix E.
+
+![](images/52810876baa1e1c8ace348a8accea1735f4798fa0faa1290bee4f8cf18d78104.jpg)  
+Figure 5. Test accuracy of T5-Base finetuned on the QNLI dataset with learning rate  $\eta = 3e - 7$  and the initialization size  $\beta = 16$ .
+
+# 4.2. Natural Language Generation Tasks.
+
+Models and Datasets. In this section, we fine-tune the Llama 3-8B model (Dubey et al., 2024) on commonsense and arithmetic reasoning benchmarks. The former contains eight subtasks, while the latter contains six subtasks. Each subtask is associated with a predefined training and test set. The experiment is conducted using the code provided in (Hu et al., 2023). Following the procedure outlined in (Hu et al., 2023), we combine the training datasets of all subtasks to create a final training dataset and evaluate each task using a single test dataset. The average test accuracy is reported, and task-specific accuracies are provided in Appendix E.
+
+Results. Figure 6 illustrates the average test accuracy of Llama 3-8B on commonsense and arithmetic reasoning benchmarks. The results are consistent with those observed on the GLUE benchmark, where Init[AB] generally outperforms Init[A], particularly at smaller learning rates. Moreover, the performance improvement observed with Init[AB] on these two benchmarks is notably greater than that on the GLUE dataset. This is likely due to the increased complexity of these two benchmarks, which impose higher demands on the learning rate robustness.
+
+# 4.3. Ablation on the Starting Point of Fine-Tuning
+
+By default, we subtract the non-zero initialized  $AB$  from the pretrained weights  $W$  to ensure that fine-tuning starts from the pretrained model. In this section, we investigate whether this subtraction is necessary. The strategy of nonzero initialization without this subtraction process is denoted as Init[AB+] . The advantage of Init[AB+] is that it avoids modifying the pretrained model, rather than improving accuracy. Figure 7 illustrates the differences between Init[AB+] and Init[AB] under varying initialization variances. When the initialization variance exceeds a certain threshold  $(\beta \geq 8)$ , Init[AB+] leads to a significant accu
+
+![](images/c18800ba647adf38c9dd1c4cdb07f94692f35ed5310c6e63c41e4f879eb72ca1.jpg)
+
+![](images/111f42f05733f07f4fcb770030d2bc02ed4f2ce7cb0d83b1fb882b1582ba79a3.jpg)  
+Figure 6. Fine-tuning accuracy of Llama 3-8B.
+
+racy drop due to excessive noise. However, when suitable initialization variances are applied, there is no discernible difference between Init[AB+] and Init[AB]. Furthermore, Figure 7 demonstrates that the range of suitable initialization variances is relatively wide, encompassing the variance used in the Kaiming initialization. More ablation results for different models and datasets are in Appendix E.
+
+![](images/a03779ccf38a4caa21a51a18ce75e4febc65d02ced436238140c7efd1b1057f1.jpg)  
+Figure 7. Fine-tuning accuracy with different starting points. Init[AB+] denotes the variant which do not subtract non-zero initialized  $AB$  from  $W$ , and Init[AB-] - Init[AB] indicates the accuracy gap between these two initialization strategies.
+
+# 5. Related Works
+
+# 5.1. Low-Rank Adaptation
+
+The fine-tuning performance of LoRA is influenced by several factors, such as rank, gradient, learning rate, and initialization. AdaLoRA (Zhang et al., 2023) adaptively adjusts the rank across different LoRA layers to improve efficiency. Zeng & Lee (2024) provide theoretical guidelines for select-
+
+ing an appropriate rank in LoRA. BoRA (Li et al., 2025) adopts a block-wise mechanism in LoRA, leveraging block matrix multiplication with block-wise diagonal matrices to enhance its expressivity. LoRA-Pro (Wang et al., 2025) improves LoRA by adjusting low-rank gradients to better match full fine-tuning, reducing the performance gap. Additionally, LoRA+ (Hayou et al., 2024a) analyzes how a larger learning rate for the matrix  $B$  promotes more stable and efficient LoRA fine-tuning. A key advantage of LoRA+ is its ability to achieve internal stability, i.e.,  $Z_A = \Theta(1)$ , which cannot be achieved using a uniform learning rate for  $A$  and  $B$ . Notably, LoRA+ can also be applied to non-zero initialization to ensure internal stability, as further discussed in Appendix B.2.
+
+Hayou et al. (2024b) have analyzed the effect of initialization on LoRA's fine-tuning dynamics, though their analysis is limited to zero initialization. In contrast, we further investigate and demonstrate that non-zero initialization can enhance LoRA's robustness to suboptimal learning rates. In fact, non-zero initialization has several practical applications, such as initializing the matrices  $A$  and  $B$  with quantization errors (Li et al., 2024), the main components of pretrained weights (Meng et al., 2024), or the gradients of pretrained weights derived from several data samples (Wang et al., 2024). Recent work (Ponkshe et al., 2024) has highlighted the benefits of using approximate gradients for LoRA initialization. LoRA-One (Zhang et al., 2025) leverages spectral initialization to achieve subspace alignment and generalization guarantees at initialization, enabling efficient convergence from the very start.
+
+However, these studies overlook a crucial distinction between their approaches and standard LoRA, namely, the difference between zero and non-zero initialization. Unlike LoRA-GA and PiSSA, which were motivated by intuition, our approach is motivated by the theoretical analysis of LoRA's fine-tuning dynamics. From the solution set in Eqs.(5-6), we observe that stable and efficient learning imposes stricter constraints on learning rates, whereas the initialization space is more flexible. Traditional zero initialization  $(\gamma [B_0] = -\infty)$  is merely an extreme case. This motivates us to reconsider the necessity of zero initialization and explore the potential benefits of non-zero initialization. Notably, our motivation and claims are not in competition with prior non-zero initialization methods, such as LoRA-GA and PiSSA. Instead, our findings offer a theoretical foundation for their effectiveness and provide an explanation for the observed performance improvements.
+
+# 5.2. Other Parameter-Efficient Fine-Tuning Methods
+
+In addition to LoRA, two widely adopted PEFT strategies are adapter-based and soft prompt-based approaches. Adapter-based methods (Houlsby et al., 2019; He et al.,
+
+2022; Wang et al., 2022) introduce small trainable modules into the pretrained model and update only these components, significantly reducing training costs. However, the added layers can increase inference latency. In contrast, soft prompt-based methods (Lester et al., 2021; Li & Liang, 2021; Razdaibiedina et al., 2023) prepend task-specific, learnable tokens to the input, leveraging the pretrained model's existing knowledge for adaptation. Despite their efficiency, these methods also incur additional computational overhead and inference delays. By comparison, LoRA supports manual integration of weight updates into the pretrained weights post fine-tuning, thereby avoiding extra inference latency. Additionally, sparse fine-tuning methods such as BitFit (Zaken et al., 2022) update only the model's bias terms, offering another lightweight alternative.
+
+# 6. Conclusion
+
+In this paper, we revisit the fine-tuning dynamics of LoRA and identify that stable and efficient fine-tuning requires specific conditions on both the learning rate and initialization. Compared to initialization, LoRA imposes stricter requirements on the learning rate. Further analysis reveals that initializing both  $A$  and  $B$  of LoRA to non-zero values (i.e., non-zero initialization) can improve its robustness to suboptimal learning rates, particularly small ones. Note that this is a zero-cost adjustment of LoRA. Additionally, we examine the noise introduced by non-zero initialization for pretraining weights. Extensive experiments demonstrate that with appropriate initialization variances, the noise from non-zero initialization can be effectively corrected through fine-tuning, without negatively impacting fine-tuning performance. Our experiments reveal that the range of suitable initialization variances is relatively broad, encompassing commonly used methods like Kaiming initialization.
+
+# Acknowledgements
+
+This work is supported by the National Key Research and Development Program of China under grant 2024YFC3307900; the National Natural Science Foundation of China under grants 62376103, 62302184, 62436003, and 62206102; Major Science and Technology Project of Hubei Province under grant 2024BAA008; Hubei Science and Technology Talent Service Project under grant 2024DJC078; and Ant Group through CCF-Ant Research Fund. The computation is completed in the HPC Platform of Huazhong University of Science and Technology.
+
+# Impact Statement
+
+This paper explores the non-zero initialization of LoRA, which has the potential to improve the performance of LoRA fine-tuning. Our work builds upon LoRA, and we assert that it does not introduce any negative social implications that would necessitate further discussion.
+
+# References
+
+Bordelon, B., Noci, L., Li, M. B., Hanin, B., and Pehlevan, C. Depthwise hyperparameter transfer in residual networks: Dynamics and scaling limit. In International Conference on Learning Representations (ICLR), 2024.  
+Bottou, L. Large-scale machine learning with stochastic gradient descent. In International Conference on Computational Statistics (COMPSTAT), 2010.  
+DeepSeek-AI. Deepseek-v3 technical report. CoRR, 2024.  
+Deng, L. The mnist database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine, 29(6):141-142, 2012.  
+Dubey, A., Jauhri, A., Pandey, A., Kadian, A., Al-Dahle, A., et al. The llama 3 herd of models. CoRR, 2024.  
+Glorot, X., Bordes, A., and Bengio, Y. Deep sparse rectifier neural networks. In International Conference on Artificial Intelligence and Statistics (AISTATS), pp. 315-323, 2011.  
+Hayou, S., Doucet, A., and Rousseau, J. On the impact of the activation function on deep neural networks training. In International Conference on Machine Learning (ICML), 2019.  
+Hayou, S., Ghosh, N., and Yu, B. Lora+: Efficient low rank adaptation of large models. In International Conference on Machine Learning (ICML), 2024a.  
+Hayou, S., Ghosh, N., and Yu, B. The impact of initialization on lora finetuning dynamics. In Advances in Neural Information Processing Systems (NeurIPS), 2024b.  
+He, J., Zhou, C., Ma, X., Berg-Kirkpatrick, T., and Neubig, G. Towards a unified view of parameter-efficient transfer learning. In International Conference on Learning Representations (ICLR), 2022.  
+He, K., Zhang, X., Ren, S., and Sun, J. Delving deep into rectifiers: Surpassing human-level performance on imagelet classification. In IEEE International Conference on Computer Vision, (ICCV), 2015.  
+He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learning for image recognition. In IEEE Conference on Computer Vision and Pattern Recognition, (CVPR), 2016.  
+Hoffmann, J., Borgeaud, S., Mensch, A., Buchatskaya, E., Cai, T., et al. Training compute-optimal large language models. CoRR, 2022.  
+Houlsby, N., Giurgiu, A., Jastrzebski, S., Morrone, B., de Laroussilhe, Q., Gesmundo, A., Attariyan, M., and Gelly, S. Parameter-efficient transfer learning for NLP. In International Conference on Machine Learning (ICML), 2019.
+
+Hu, E. J., Shen, Y., Wallis, P., Allen-Zhu, Z., Li, Y., Wang, S., Wang, L., and Chen, W. Lora: Low-rank adaptation of large language models. In International Conference on Learning Representations (ICLR), 2022.  
+Hu, Z., Wang, L., Lan, Y., Xu, W., Lim, E., Bing, L., Xu, X., Poria, S., and Lee, R. K. Llm-adapters: An adapter family for parameter-efficient fine-tuning of large language models. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing (EMNLP), 2023.  
+Kaplan, J., McCandlish, S., Henighan, T., Brown, T. B., Chess, B., et al. Scaling laws for neural language models. CoRR, 2020.  
+Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. In Bengio, Y. and LeCun, Y. (eds.), International Conference on Learning Representations (ICLR), 2015.  
+Lester, B., Al-Rfou, R., and Constant, N. The power of scale for parameter-efficient prompt tuning. In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 3045-3059, 2021.  
+Li, S., Luo, X., Wang, H., Tang, X., Cui, Z., Liu, D., Li, Y., xiuqiang He, and Li, R. Bora: Towards more expressive low-rank adaptation with block diversity. CoRR, 2025.  
+Li, X. L. and Liang, P. Prefix-tuning: Optimizing continuous prompts for generation. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing, (ACL/IJCNLP), 2021.  
+Li, Y., Yu, Y., Liang, C., Karampatziakis, N., He, P., Chen, W., and Zhao, T. Loftq: Lora-fine-tuning-aware quantization for large language models. In International Conference on Learning Representations (ICLR), 2024.  
+Loshchilov, I. and Hutter, F. Decoupled weight decay regularization. In International Conference on Learning Representations (ICLR), 2019.  
+Meng, F., Wang, Z., and Zhang, M. Pissa: Principal singular values and singular vectors adaptation of large language models. In Advances in Neural Information Processing Systems (NeurIPS), 2024.  
+OpenAI. Gpt-4 technical report. CoRR, 2024.  
+Ponkshe, K., Singhal, R., Gorbunov, E., Tumanov, A., Horvath, S., and Vepakomma, P. Initialization using update approximation is a silver bullet for extremely efficient low-rank fine-tuning. CoRR, 2024.
+
+Raffel, C., Shazeer, N., Roberts, A., Lee, K., Narang, S., Matena, M., Zhou, Y., Li, W., and Liu, P. J. Exploring the limits of transfer learning with a unified text-to-text transformer. J. Mach. Learn. Res., 21:140:1-140:67, 2020.  
+Razdaibiedina, A., Mao, Y., Khabsa, M., Lewis, M., Hou, R., Ba, J., and Almahairi, A. Residual prompt tuning: improving prompt tuning with residual reparameterization. In Findings of the Association for Computational Linguistics (ACL), 2023.  
+Schoenholz, S. S., Gilmer, J., Ganguli, S., and Sohl-Dickstein, J. Deep information propagation. In International Conference on Learning Representations (ICLR), 2017.  
+Sheng, Y., Cao, S., Li, D., Hooper, C., Lee, N., Yang, S., Chou, C., Zhu, B., Zheng, L., Keutzer, K., Gonzalez, J. E., and Stoica, I. S-lora: Serving thousands of concurrent lora adapters. CoRR, 2023.  
+Wang, A., Singh, A., Michael, J., Hill, F., Levy, O., and Bowman, S. R. GLUE: A multi-task benchmark and analysis platform for natural language understanding. In International Conference on Learning Representations (ICLR), 2019.  
+Wang, S., Yu, L., and Li, J. Lora-ga: Low-rank adaptation with gradient approximation. In Advances in Neural Information Processing Systems (NeurIPS), 2024.  
+Wang, Y., Agarwal, S., Mukherjee, S., Liu, X., Gao, J., Awadallah, A. H., and Gao, J. Adamix: Mixture-of-adaptations for parameter-efficient model tuning. In Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing (EMNLP), 2022.  
+Wang, Z., Liang, J., He, R., Wang, Z., and Tan, T. Lora-pro: Are low-rank adapters properly optimized? In International Conference on Learning Representations (ICLR), 2025.  
+Xiao, H., Rasul, K., and Vollgraf, R. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. CoRR, 2017.  
+Xu, L., Xie, H., Qin, S. J., Tao, X., and Wang, F. L. Parameter-efficient fine-tuning methods for pretrained language models: A critical review and assessment. CoRR, 2023.  
+Yang, G. Scaling limits of wide neural networks with weight sharing: Gaussian process behavior, gradient independence, and neural tangent kernel derivation. CoRR, 2019.  
+Yang, G. Tensor programs III: neural matrix laws. CoRR, 2020.
+
+Yang, G., Hu, E. J., Babuschkin, I., Sidor, S., Liu, X., et al. Tensor programs V: tuning large neural networks via zero-shot hyperparameter transfer. CoRR, 2022.  
+Yang, G., Yu, D., Zhu, C., and Hayou, S. Tensor programs VI: feature learning in infinite depth neural networks. In International Conference on Learning Representations (ICLR), 2024.  
+Zaken, E. B., Goldberg, Y., and Ravfogel, S. Bitfit: Simple parameter-efficient fine-tuning for transformer-based masked language-models. In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (ACL), 2022.  
+Zeng, Y. and Lee, K. The expressive power of low-rank adaptation. In International Conference on Learning Representations (ICLR), 2024.  
+Zhang, Q., Chen, M., Bukharin, A., He, P., Cheng, Y., Chen, W., and Zhao, T. Adaptive budget allocation for parameter-efficient fine-tuning. In International Conference on Learning Representations (ICLR), 2023.  
+Zhang, Y., Liu, F., and Chen, Y. One-step full gradient suffices for low-rank fine-tuning, provably and efficiently. CoRR, abs/2502.01235, 2025.
+
+# A. Background
+
+# A.1. Scaling Theory of Neural Networks
+
+In neural networks, scaling refers to enhancing the model's representational capacity by increasing the scale of a specific structure within the model (Kaplan et al., 2020; Hoffmann et al., 2022). It involves analyzing and optimizing several factors, such as model depth, width, training sample size, and the number of training steps. This paper focuses on scaling the model width, which holds practical significance, as the most advanced LLMs today employ extremely large widths, with hidden layer dimensions often reaching tens of thousands (OpenAI, 2024; DeepSeek-AI, 2024).
+
+As the width  $n$  increases, the synergistic adaptation of parameter initialization and training dynamics becomes increasingly critical. Previous studies have demonstrated that the variance of parameters should be scaled by  $1/n$  to prevent gradient instability during optimization (He et al., 2015). These scaling rules are typically derived from asymptotic analyses of key quantities (e.g., pre-activation values), which theoretically guide adjustments to parameter initialization schemes, learning rates, and even network architectures (Hayou et al., 2019; Schoenholz et al., 2017; Yang, 2019).
+
+Building on this foundation, Yang et al. (2022) proposed the Maximal Update Parameterization  $(\mu P)$ , which constructs scaling rules for initialization schemes, learning rates, and network architectures to theoretically guarantee stability and maximal feature learning in the limit of infinite network width. The key conditions for  $\mu P$  are as follows: 1) Stability: for all layers  $l$ ,  $Y_{l} = \Theta(1)$ , where the asymptotic notation  $\Theta(\cdot)$  is defined in Section 2.1; 2) Feature learning:  $\Delta Y_{l} = \Theta(1)$ , where  $\Delta$  represents the parameter update after one gradient descent step. In general,  $\mu P$  specifies the parameter weights to be randomly initialized at a scale of  $\Theta(n^{-1/2})$ , with weight updates at a scale of  $\Theta(n^{-1})$ . The random initialization and update scales for input and output weights should be  $\Theta(1)$  and  $\Theta(n^{-1})$ , respectively. This scaling rule successfully resolves the instability issues inherent in traditional parameterization schemes, making it a key tool in the theoretical analysis and practical application of large models.
+
+# A.2. The  $\gamma$ -operator
+
+In neural network scaling theory, the asymptotic behavior of key quantities is typically analyzed when scaling specific model components. For example, when scaling the network width, the objective is to quantify how certain network properties evolve as the width  $n$  grows. In this context, asymptotic notations, such as  $\Theta(\cdot)$ , are particularly useful, as defined in Section 2.1. This standard approach has been employed to derive scaling rules for various components, such as initialization (Schoenholz et al., 2017), activation functions (Hayou et al., 2019), and network parameterization (Yang et al., 2024).
+
+Assuming that the weights are initialized as  $\Theta (n^{-\beta})$ $(\beta \geq 0)$ , and that the learning rate scales polynomially with respect to the width  $n$ , it follows that key quantities, such as pre-activations, gradients, and weight updates, are also asymptotically polynomial in  $n$ . This regularity arises from the aggregate nature of neural network operations, where tensor operations involve linear combinations of numerous parameters. Consequently, we introduce the  $\gamma$ -operator to express the polynomial behavior as  $v = \Theta (\gamma [v])$ . We now define some basic operations using the  $\gamma$ -operator.
+
+Zero:  $\gamma [0] = -\infty$  , corresponding to the limit of  $\gamma [n^{\beta}]$  when  $\beta \rightarrow -\infty$
+
+Product: For two real-valued variables  $v, v' \in \mathbb{R}$ , we have  $\gamma[vv'] = \gamma[v] + \gamma[v']$ .
+
+Addition: For two real-valued variables  $v, v' \in \mathbb{R}$  and  $v \neq -v'$ , we have  $\gamma[v + v'] = \max(\gamma[v], \gamma[v'])$ .
+
+# B. Fine-Tuning Dynamics of LoRA with Adam
+
+# B.1. Detailed Proof
+
+In this section, we present a detailed explanation of the process from Eq. (3) to Eq. (5). To facilitate the analysis, we first introduce Lemmas 1 and 2.
+
+Lemma 1. Let  $U \in \mathbb{R}^{1 \times r}$  and  $V \in \mathbb{R}^r$ , where  $r$  is independent of  $n$ . If  $U = \Theta(n^\alpha)$  and  $V = \Theta(n^\beta)$ , then  $UV = \Theta(n^{\alpha + \beta})$ .
+
+Proof of Lemma 1. By the definition of the notation  $\Theta$ , each element of  $U$  and  $V$  satisfies the following relations:
+
+$$
+U _ {i} = \Theta \left(n ^ {\alpha}\right), \quad V _ {i} = \Theta \left(n ^ {\beta}\right), \quad i \in [ r ] \tag {7}
+$$
+
+There exist positive constants  $\kappa_u^l, \kappa_u^h, \kappa_v^l$  and  $\kappa_v^h$  such that
+
+$$
+\kappa_ {u} ^ {l} n ^ {\alpha} <   U _ {i} <   \kappa_ {u} ^ {h} n ^ {\alpha}, \quad \kappa_ {v} ^ {l} n ^ {\beta} <   V _ {i} <   \kappa_ {v} ^ {h} n ^ {\beta}, \tag {8}
+$$
+
+Multiplying  $U_{i}$  and  $V_{i}$ , we obtain
+
+$$
+\kappa_ {u} ^ {l} \kappa_ {v} ^ {l} n ^ {\alpha + \beta} <   U _ {i} V _ {i} <   \kappa_ {u} ^ {h} \kappa_ {v} ^ {h} n ^ {\alpha + \beta} \tag {9}
+$$
+
+Thus, the product  $UV$  satisfies the following inequality:
+
+$$
+r \kappa_ {u} ^ {l} \kappa_ {v} ^ {l} n ^ {\alpha + \beta} <   U V = \sum_ {i = 1} ^ {r} U _ {i} V _ {i} <   r \kappa_ {u} ^ {h} \kappa_ {v} ^ {h} n ^ {\alpha + \beta} \tag {10}
+$$
+
+Therefore, by the definition of the notation  $\Theta$ , we conclude that
+
+$$
+U V = \Theta (\alpha + \beta), \quad \gamma [ U V ] = \alpha + \beta \tag {11}
+$$
+
+Note that the same reasoning and result hold for the matrix multiplication of  $U \in \mathbb{R}^{a \times r}$  and  $U \in \mathbb{R}^{r \times b}$ .
+
+Lemma 2. Let  $U \in \mathbb{R}^{1 \times n}$  and  $V \in \mathbb{R}^n$ . If  $U = \Theta(n^\alpha)$  and  $V = \Theta(n^\beta)$ , then  $UV = \Theta(n^{\alpha + \beta + 1})$ .
+
+Proof of Lemma 2. Similar to the proof of Lemma 1, the product  $UV$  satisfies the following inequality:
+
+$$
+\kappa_ {u} ^ {l} \kappa_ {v} ^ {l} n ^ {\alpha + \beta + 1} <   U V = \sum_ {i = 1} ^ {n} U _ {i} V _ {i} <   \kappa_ {u} ^ {h} \kappa_ {v} ^ {h} n ^ {\alpha + \beta + 1} \tag {12}
+$$
+
+Thus, by the definition of the notation  $\Theta$ , we conclude that
+
+$$
+U V = \Theta (\alpha + \beta + 1), \quad \gamma [ U V ] = \alpha + \beta + 1 \tag {13}
+$$
+
+Note that the same reasoning and result hold for the matrix multiplication of  $U \in \mathbb{R}^{a \times n}$  and  $U \in \mathbb{R}^{n \times b}$ .
+
+Proof of the process from Eq.(3) and Eq.(5). Eq. (3) outlines the conditions for LoRA to achieve stability and efficiency, as follows:
+
+$$
+\left\{ \begin{array}{l l} \gamma \left[ \delta_ {t} ^ {1} \right] = \gamma \left[ - \eta_ {A} B _ {t - 1} g _ {A} ^ {t - 1} \underline {{Z}} \right] = 0 & \left(\delta_ {t} ^ {1} = \Theta (1)\right) \\ \gamma \left[ \delta_ {t} ^ {2} \right] = \gamma \left[ - \eta_ {B} g _ {B} ^ {t - 1} A _ {t - 1} \underline {{Z}} \right] = 0 & \left(\delta_ {t} ^ {2} = \Theta (1)\right) \\ \gamma \left[ Z _ {B} ^ {t - 1} \right] = \gamma \left[ B _ {t - 1} A _ {t - 1} \underline {{Z}} \right] = 0 & \left(Z _ {B} ^ {t - 1} = \Theta (1)\right) \end{array} \right. \tag {14}
+$$
+
+Taking  $\delta_t^1$  as an example, where  $\eta_A\in \mathbb{R}$ ,  $B_{t - 1}\in \mathbb{R}^{n\times r}$ ,  $g_{A}\in \mathbb{R}^{r\times n}$ , and  $\underline{Z}\in \mathbb{R}^n$ , we expand  $\gamma [\delta_t^1 ]$  as follows:
+
+$$
+\begin{array}{l} \gamma [ \delta_ {t} ^ {1} ] = \gamma [ - \eta_ {A} B _ {t - 1} g _ {A} ^ {t - 1} \underline {{Z}} ] \\ \stackrel {(a)} {\leq} \gamma [ \eta_ {A} ] + \gamma [ B _ {t - 1} g _ {A} ^ {t - 1} \underline {{Z}} ] \\ \stackrel {(b)} {\leq} \gamma [ \eta_ {A} ] + \gamma [ B _ {t - 1} ] + \gamma [ g _ {A} ^ {t - 1} \underline {{Z}} ] \\ \stackrel {(c)} {\leq} \gamma [ \eta_ {A} ] + \gamma [ B _ {t - 1} ] + \gamma [ g _ {A} ^ {t - 1} ] + \gamma [ \underline {{Z}} ] + 1 \tag {15} \\ = \gamma [ \eta_ {A} ] + \gamma [ B _ {0} + \eta_ {B} \sum_ {i = 0} ^ {t - 2} g _ {B} ^ {i} ] + \gamma [ g _ {A} ^ {t - 1} ] + \gamma [ \underline {{Z}} ] + 1 \\ \stackrel {(d)} {\leq} \gamma [ \eta_ {A} ] + \max  (\gamma [ B _ {0} ], \gamma [ \eta_ {B} ]) + 1, \\ \end{array}
+$$
+
+where (a) follows from the multiplication properties of the  $\gamma$ -operator, (b) follows from Lemma 1, (c) follows from Lemma 2, and (d) follows from the fact that  $\gamma[g_A] = \gamma[g_B] = \gamma[\underline{Z}] = 0$ .
+
+Similarly, performing the same analysis on  $\delta_t^2$  and  $Z_B$  yields the following relationships:
+
+$$
+\left\{ \begin{array}{l} \gamma \left[ \delta_ {t} ^ {1} \right] = \max  \left(\gamma \left[ B _ {0} \right], \gamma \left[ \eta_ {B} \right]\right) + \gamma \left[ \eta_ {A} \right] + 1 = 0, \\ \gamma \left[ \delta_ {t} ^ {2} \right] = \max  \left(\gamma \left[ A _ {0} \right], \gamma \left[ \eta_ {A} \right]\right) + \gamma \left[ \eta_ {B} \right] + 1 = 0, \\ \gamma \left[ Z _ {B} ^ {t - 1} \right] = \max  \left(\gamma \left[ A _ {0} \right], \gamma \left[ \eta_ {A} \right]\right) + \max  \left(\gamma \left[ B _ {0} \right], \gamma \left[ \eta_ {B} \right]\right) + 1 = 0. \end{array} \right. \tag {16}
+$$
+
+Furthermore, we obtain:
+
+$$
+\gamma [ \delta_ {t} ^ {1} ] + \gamma [ \delta_ {t} ^ {2} ] - \gamma [ Z _ {B} ^ {t - 1} ] = \gamma [ \eta_ {A} ] + \gamma [ \eta_ {B} ] + 1 = 0. \tag {17}
+$$
+
+Substituting this equation back into  $\gamma [\delta_t^1 ]$  and  $\gamma [\delta_t^2 ]$  yields:
+
+$$
+\left\{ \begin{array}{l} \max  \left(\gamma \left[ A _ {0} \right], \gamma \left[ \eta_ {A} \right]\right) = \gamma \left[ \eta_ {A} \right], \\ \max  \left(\gamma \left[ B _ {0} \right], \gamma \left[ \eta_ {B} \right]\right) = \gamma \left[ \eta_ {B} \right]. \end{array} \right. \tag {18}
+$$
+
+Thus, we conclude that:
+
+$$
+\left\{ \begin{array}{l} \gamma [ \eta_ {A} ] + \gamma [ \eta_ {B} ] = - 1, \\ \gamma [ A _ {0} ] \leq \gamma [ \eta_ {A} ], \quad \gamma [ B _ {0} ] \leq \gamma [ \eta_ {B} ]. \end{array} \right. \tag {19}
+$$
+
+# B.2. Internal Stability
+
+In this paper, we focus solely on the stability of LoRA's final output,  $Z_{B}$ , while excluding the stability of LoRA's internal output,  $Z_{A}$ . Hu et al. (2022) discusses  $Z_{A}$  and demonstrates that init[B] ensures stability for both  $Z_{A}$  and  $Z_{B}$ , but fails to facilitate efficient updates of  $Z_{B}$ , as defined in Definition 4. In contrast, init[A] achieves stability and efficient updating of  $Z_{B}$ , but does not ensure the stability of  $Z_{A}$ . Here, we additionally consider the internal stability by introducing the condition  $Z_{A}^{t - 1} = A_{t - 1}\underline{Z} = \Theta (1)$ , as a requirement in Eq. (3).
+
+$$
+\left\{ \begin{array}{l l} \gamma \left[ \delta_ {t} ^ {1} \right] = \gamma \left[ - \eta_ {A} B _ {t - 1} g _ {A} ^ {t - 1} \underline {{Z}} \right] = 0 & \left(\delta_ {t} ^ {1} = \Theta (1)\right) \\ \gamma \left[ \delta_ {t} ^ {2} \right] = \gamma \left[ - \eta_ {B} g _ {B} ^ {t - 1} A _ {t - 1} \underline {{Z}} \right] = 0 & \left(\delta_ {t} ^ {2} = \Theta (1)\right) \\ \gamma \left[ Z _ {B} ^ {t - 1} \right] = \gamma \left[ B _ {t - 1} A _ {t - 1} \underline {{Z}} \right] = 0 & \left(Z _ {B} ^ {t - 1} = \Theta (1)\right) \\ \gamma \left[ Z _ {A} ^ {t - 1} \right] = \gamma \left[ A _ {t - 1} \underline {{Z}} \right] = 0 & \left(Z _ {A} ^ {t - 1} = \Theta (1)\right) \end{array} \right. \tag {20}
+$$
+
+Similar to the proof in Section B.1, we can express the above equations in terms of  $(A_0,B_0)$  and  $(\eta_A,\eta_B)$  as follows:
+
+$$
+\left\{ \begin{array}{l} \max (\gamma [ B _ {0} ], \gamma [ \eta_ {B} ]) + \gamma [ \eta_ {A} ] + 1 = 0, \\ \max (\gamma [ A _ {0} ], \gamma [ \eta_ {A} ]) + \gamma [ \eta_ {B} ] + 1 = 0, \\ \max (\gamma [ A _ {0} ], \gamma [ \eta_ {A} ]) + \max (\gamma [ B _ {0} ], \gamma [ \eta_ {B} ]) + 1 = 0, \\ \max (\gamma [ A _ {0} ], \gamma [ \eta_ {A} ]) + 1 = 0. \end{array} \right.
+$$
+
+Based on the derivation steps outlined in Appendix B.1, the solution set can be obtained as follows:
+
+$$
+\left\{ \begin{array}{l} \gamma [ \eta_ {A} ] = - 1, \gamma [ \eta_ {B} ] = 0, \\ \gamma [ A _ {0} ] \leq \gamma [ \eta_ {A} ] = - 1, \quad \gamma [ B _ {0} ] \leq \gamma [ \eta_ {B} ] = 0, \end{array} \right. \tag {21}
+$$
+
+where  $\gamma [A_0]$  and  $\gamma [B_0]$  cannot both be equal to  $-\infty$  simultaneously, as this would imply that both  $A$  and  $B$  are zero, preventing optimization via gradient descent algorithms.
+
+This is actually the solution proposed in LoRA+ (Hayou et al., 2024a), which sets  $\gamma[\eta_A] = -1$ ,  $\gamma[\eta_B] = 0$ . Similar to the analysis in Section 3.3, setting  $\gamma[A_0] = -1$  and  $\gamma[B_0] = 0$  can achieve the Robust LoRA defined in Theorem 1. Thus, non-zero initialization can also improve the robustness of LoRA+ in relation to the learning rate.
+
+# C. Fine-Tuning Dynamics of LoRA with SGD
+
+In this section, we examine the fine-tuning dynamics of LoRA using the SGD optimizer. For simplicity, we consider the model described in LoRA+ (Hayou et al., 2024a), which can be expressed as:
+
+$$
+f (x) = b a ^ {\top} x, \tag {22}
+$$
+
+where the pretrained weights are assumed to be zero,  $x \in \mathbb{R}^n$  is the input vector, and  $b \in \mathbb{R}, a \in \mathbb{R}^n$  are the LoRA weights. Additionally, we assume that the fine-tuning dataset consists of a single data point,  $(x,y)$ , with  $x = \Theta(1)$ . The objective is to minimize the loss function  $\mathcal{L}(x) = \frac{1}{2} (f(x) - y)^2$ . The corresponding gradients are as follows:
+
+$$
+g _ {a} ^ {t} = b _ {t} \left(f _ {t} (x) - y\right) x, \quad g _ {b} ^ {t} = a _ {t} ^ {\top} x (f (x) - y). \tag {23}
+$$
+
+Using Lemmas 1 and 2, we have
+
+$$
+\gamma \left[ g _ {a} ^ {t} \right] = \gamma \left[ b ^ {t} \right], \quad \gamma \left[ g _ {b} ^ {t} \right] = \gamma \left[ a ^ {t} \right] + 1. \tag {24}
+$$
+
+Notably, the Adam optimizer guarantees that  $g_{A} = \Theta(1)$  and  $g_{B} = \Theta(1)$  through gradient normalization, whereas SGD does not ensure this. Let  $U_{t} = f_{t}(x) - y$ . At step  $t$ , the update of LoRA's output is given by:
+
+$$
+\Delta f _ {t} = f _ {t} (x) - f _ {t - 1} (x) = - \underbrace {\eta_ {a} b _ {t - 1} ^ {2} U _ {t - 1} \| x \| ^ {2}} _ {\delta_ {t} ^ {1}} - \underbrace {\eta_ {b} \left(a _ {t - 1} ^ {\top} x\right) ^ {2} U _ {t - 1}} _ {\delta_ {t} ^ {2}} + \underbrace {\eta_ {a} \eta_ {b} U _ {t - 1} ^ {2} b _ {t - 1} \left(a _ {t - 1} ^ {\top} x\right) \| x \| ^ {2}} _ {\delta_ {t} ^ {3}}. \tag {25}
+$$
+
+To ensure stability and efficiency as defined in Definitions 3 and 4, the following conditions must hold for  $t > 1$ :
+
+$$
+\left\{ \begin{array}{l l} \gamma \left[ \delta_ {t} ^ {1} \right] = \gamma \left[ \eta_ {a} b _ {t - 1} ^ {2} U _ {t - 1} \| x \| ^ {2} \right] = \gamma \left[ \eta_ {a} \right] + 2 \gamma \left[ b _ {t - 1} \right] + 1 = 0 & \left(\delta_ {t} ^ {1} = \Theta (1)\right) \\ \gamma \left[ \delta_ {t} ^ {2} \right] = \gamma \left[ \eta_ {b} \left(a _ {t - 1} ^ {\top} x\right) ^ {2} U _ {t - 1} \right] = \gamma \left[ \eta_ {b} \right] + 2 \gamma \left[ a _ {t - 1} ^ {\top} \right] + 2 = 0 & \left(\delta_ {t} ^ {2} = \Theta (1)\right) \\ \gamma \left[ f _ {t - 1} (x) \right] = \gamma \left[ b _ {t - 1} a _ {t - 1} ^ {\top} x \right] = \gamma \left[ b _ {t - 1} \right] + \gamma \left[ a _ {t - 1} \right] + 1 = 0 & \left(f _ {t - 1} (x) = \Theta (1)\right) \end{array} \right. \tag {26}
+$$
+
+Further, we have the following relation:
+
+$$
+\gamma \left[ \delta_ {t} ^ {1} \right] + \gamma \left[ \delta_ {t} ^ {2} \right] - 2 \gamma \left[ f _ {t - 1} (x) \right] = \gamma \left[ \eta_ {a} \right] + \gamma \left[ \eta_ {b} \right] + 1 = 0, \tag {27}
+$$
+
+which is the same as the result obtained with the Adam optimizer. Using a uniform learning rate for both  $a$  and  $b$ , we have
+
+$$
+\left\{ \begin{array}{l} \gamma [ \eta_ {a} ] = \gamma [ \eta_ {b} ] = - \frac {1}{2}, \\ \gamma [ a _ {t - 1} ] = - \frac {3}{4}, \quad \gamma [ b _ {t - 1} ] = - \frac {1}{4}, \quad (t > 1). \end{array} \right. \tag {28}
+$$
+
+However, the values of  $\gamma [a_0]$  and  $\gamma [b_0]$  are unknown. Therefore, we take  $a_1 = a_0 + \eta_a g_a^0$  and  $b_{1} = b_{0} + \eta_{b}g_{b}^{0}$ , leading to
+
+$$
+\left\{ \begin{array}{l} \gamma [ a _ {1} ] = \max  \left\{\gamma [ a _ {0} ], \gamma [ \eta_ {a} ] + \gamma [ g _ {a} ^ {0} ] \right\} = \max  \left\{\gamma [ a _ {0} ], - \frac {1}{2} + \gamma [ b ^ {0} ] \right\} = - \frac {3}{4}, \\ \gamma [ b _ {1} ] = \max  \left\{\gamma [ b _ {0} ], \gamma [ \eta_ {b} ] + \gamma [ g _ {b} ^ {0} ] \right\} = \max  \left\{\gamma [ b _ {0} ], - \frac {1}{2} + \gamma [ a ^ {0} ] + 1 \right\} = - \frac {1}{4}. \end{array} \right. \tag {29}
+$$
+
+Solving this system of equations yields the following conditions:
+
+$$
+\left\{ \begin{array}{l} \gamma \left[ a _ {0} \right] \leq - \frac {3}{4}, \\ \gamma \left[ b _ {0} \right] \leq - \frac {1}{4}, \end{array} \right. \tag {30}
+$$
+
+where at least one of the inequalities must hold with equality. Notably, when both inequalities hold with equality, the conditions for robust LoRA, as defined in Theorem 1, are satisfied. Specifically, when  $\gamma [a_0] = -\frac{3}{4}$ , and  $\gamma [b_0] = -\frac{1}{4}$ , for all  $\gamma [\eta_a] < -1/2$ ,  $\gamma [\eta_b] < -1/2$ , we have the following result:
+
+$$
+\left\{ \begin{array}{l} \gamma \left[ \delta_ {t} ^ {1} \right] = \gamma \left[ \eta_ {a} \right] + \frac {1}{2}, \\ \gamma \left[ \delta_ {t} ^ {2} \right] = \gamma \left[ \eta_ {b} \right] + \frac {1}{2}, \\ \gamma \left[ f _ {t - 1} (x) \right] = 0, \end{array} \right. \tag {31}
+$$
+
+where  $\gamma[\delta_t^1]$  depends only on  $\gamma[\eta_a]$ ,  $\gamma[\delta_t^2]$  depends only on  $\gamma[\eta_b]$ , and  $\gamma[f_{t-1}(x)]$  is independent of both  $\gamma[\eta_a]$  and  $\gamma[\eta_b]$ .
+
+# D. Experimental Settings
+
+# D.1. Toy Model
+
+- Model and Dataset: The toy model used in Section 3.4 is defined as follows:
+
+$$
+f (X) = W _ {o u t} \mathcal {F} ((W _ {0} + B A) \mathcal {F} (W _ {i n} X)),
+$$
+
+where  $X \in \mathbb{R}^d$  is the input data,  $W_{in} \in \mathbb{R}^{n \times d}$ ,  $W_0 \in \mathbb{R}^{n \times n}$ , and  $W_{out} \in \mathbb{R}^{10 \times n}$  are the pretrained weights,  $A \in \mathbb{R}^{r \times n}$  and  $B \in \mathbb{R}^{n \times r}$  are the LoRA weights, and  $\mathcal{F}$  denotes the ReLU activation function(Glorot et al., 2011). The MNIST (Deng, 2012) and FashionMNIST (Xiao et al., 2017) datasets are used for pretraining and fine-tuning, with input images flattened into one-dimensional vectors. The model dimensions  $(d, n, r)$  are set to (784, 4096, 32).
+
+- Training Algorithm: Both pretraining and fine-tuning employ the AdamW optimizer (Loshchilov & Hutter, 2019), with  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$ ,  $\epsilon = 1e - 8$ , and a weight decay of 0. The batch size is set to 64. During pretraining, the learning rate is set to  $1e - 3$ , with 2000 training steps. During fine-tuning, the learning rate is tuned among the set  $\{3e - 4, 4e - 4, 5e - 4, 6e - 4, 7e - 4, 8e - 4, 9e - 4, 1e - 3, 2e - 3, 3e - 3\}$ , with  $W_{0}, W_{in}, W_{out}$  frozen. Each learning rate is trained for 100 steps. All experiments are conducted with 10 independent repetitions using different random seeds, and the final results are averaged over the test accuracy.  
+- Initialization Strategy: During pretraining,  $W_0$ ,  $W_{in}$ ,  $W_{out}$  are initialized using Kaiming initialization (He et al., 2015). During fine-tuning,  $A$  and  $B$  are initialized with a Gaussian distribution, with mean 0 and standard deviations  $\sigma_A$  and  $\sigma_B$ , respectively. For Init [A],  $\sigma_B$  is set to 0, and  $\sigma_A$  is tuned among the set  $\{2e - 5, 5e - 5, 8e - 5, 1e - 4, 4e - 4, 7e - 4, 1e - 3, 3e - 3, 6e - 3, 9e - 3, 2e - 2, 5e - 2\}$ . For Init [AB],  $\sigma_A$  and  $\sigma_B$  are kept equal, and both standard deviations are tuned among the same set. Note that the non-zero initialized  $AB$  is not abstracted from  $W_0$ .
+
+# D.2. Experiments on Natural Language Understanding.
+
+- Model and Dataset: T5-Base (Raffel et al., 2020), with sequence length  $T = 128$ , is fine-tuned on several datasets from the GLUE benchmark (Wang et al., 2019), including MRPC, CoLA, SST-2, and QNLI.  
+- LoRA Hyperparameters: The rank is set to  $r = 16$  and  $\alpha = 16$ . LoRA is applied to the 'query' and 'value' weights, using full precision (FP32).  
+- Training Algorithm: AdamW (Loshchilov & Hutter, 2019) with  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$ ,  $\epsilon = 1e - 8$  and weight decay of 0. The learning rate is tuned among the set  $\{1e - 7, 3e - 7, 1e - 6, 3e - 6, 1e - 5, 3e - 5, 1e - 4, 3e - 4\}$ , a warmup ratio of 0.03 is employed. Number of train epochs  $E = 1$ .  
+- Initialization Strategy: The initialization variance is set to  $(\beta \sigma_k)^2$ , where  $\sigma_k^2$  represent the variance of matrix  $A$  under Kaiming initialization (i.e.,  $\sigma_k^2 = \frac{1}{n}$ ).  $\beta$  is tuned among  $\{1, 2, 4, 8, 16\}$ . For non-zero initialization,  $B$  is initialized with the same variance as  $A$ . By default, we subtract the non-zero initialized LoRA from the pretrained weights.
+
+# D.3. Experiments on Natural Language Generation.
+
+- Model and Dataset: Llama 3-8B (Dubey et al., 2024), with sequence length  $T = 256$ , is fine-tuned on the commonsense reasoning and arithmetic reasoning benchmarks, as proposed in (Hu et al., 2023).  
+- LoRA Hyperparameters: The rank is set to  $r = 16$  and  $\alpha = 16$ . LoRA is applied to the 'q-Proj', 'k-Proj', 'v-Proj', 'up-Proj', and 'down-Proj' weights, using full precision (FP32).  
+- Training Algorithm: AdamW (Loshchilov & Hutter, 2019) with  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$ ,  $\epsilon = 1e - 8$  and weight decay of 0. The learning rate is tuned among the set  $\{3e - 7, 1e - 6, 3e - 6, 1e - 5, 3e - 5, 1e - 4, 3e - 4\}$ , a warmup ratio of 0.03 is employed, and linear decay are employed. Number of train epochs  $E = 1$ .  
+- Initialization Strategy: The initialization variance is set to  $(\beta \sigma_k)^2$ , where  $\sigma_k^2$  represent the variance of matrix  $A$  under Kaiming initialization.  $\beta$  is tuned among  $\{0.25, 0.5, 1, 2, 4\}$ . For non-zero initialization,  $B$  is initialized with the same variance as  $A$ . By default, we subtract the non-zero initialized LoRA from the pretrained weights.
+
+# E. Additional Experimental Results
+
+In Section 4, we present the accuracy heatmaps for Init[A] and Init[AB]-Init[A]. Here, we directly display the heatmaps for Init[AB], including for the T5-Base model, as shown in Figure 8, and for the Llama 3-8B model, as depicted in Figure 9. Additionally, we provide the accuracy gap between Init[AB+] and Init[AB] on the MRPC and CoLA datasets. As shown in Figure 10, there is no significant difference between Init[AB+] and Init[AB] under appropriate initialization variance. To verify the impact of non-zero initialization on LoRA+ (Hayou et al., 2024a), we compared the accuracy of Init[AB] and Init[A] under various settings. Figure 11 that non-zero initialization can also improve LoRA+’s robustness and accuracy.
+
+![](images/61716eff088635f679e98dd7d53cbee8002462816635e144e2b4880d1c1dba48.jpg)  
+Figure 8. Fine-tuning accuracy of T5-Base on the GLUE benchmark with non-zero initialization.
+
+![](images/4c6a48feba61ad5be1a2e68e6d611a7f20fffc13237672733e5fb1fc9aa3fbc1.jpg)
+
+![](images/49ea0e6e79fa9eeea2ab0502f13be160110044fbcb99443c93c2a7b2bae46162.jpg)
+
+![](images/cbe502b2664e8911c5a510abf271200fb8cb88833ed12052c51423a4cf378376.jpg)
+
+![](images/d82d2d3af9272f602b92f6415882e09551ee64ca7e8b825c2a75f44155d7fad3.jpg)  
+Figure 9. Fine-tuning accuracy of Llama 3-8B on the commonsense and arithmetic reasoning benchmarks with non-zero initialization.
+
+![](images/0b65f33ba7dd05f5befda53136e6b60a04401c6f30d7f4fc3265ccd224905a9b.jpg)  
+Figure 10. Fine-tuning accuracy with different starting points. Init[AB+] denotes the variant which do not subtract non-zero initialized  $A_0B_0$  from W, and Init[AB+]-Init[AB] indicates the accuracy gap between these two initialization strategies.
+
+![](images/f19c3fa3c70854e7711553abaf5e514a3dbfaabe4dd5f5a50ffdcb8ce2807550.jpg)  
+Figure 11. Experimental results with LoRA+. The hyperparameter  $\lambda$  represents the value of  $\eta_B / \eta_A$  (i.e.,  $\eta_A = \eta$ ,  $\eta_B = \lambda \eta$ ). Both Init[A] and Init[AB] use the initialization size of  $\beta = 1$ , that is,  $\sigma_A^2 = \sigma_B^2 = 1/n$ , where  $n$  is the network width.
+
+![](images/fc5a75be71eb1be069a90ba7de3fb3967eec6c9b511a53bea4ae50d88fe7f60e.jpg)

2025/Beyond Zero Initialization_ Investigating the Impact of Non-Zero Initialization on LoRA Fine-Tuning Dynamics/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c2f0d38a5d0593ba22b52524d082da7ce29afe15c5cfb390d5c5a014d840518d
+size 942982

2025/Beyond the Permutation Symmetry of Transformers_ The Role of Rotation for Model Fusion/37b9cc0d-806d-4c07-8dae-3066093d0e30_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:55c8bff475025124faedaceeb89e7371a2bea49e6a6b9339171db1dfcbab67a1
+size 1308678

2025/Beyond the Permutation Symmetry of Transformers_ The Role of Rotation for Model Fusion/full.md

@@ -0,0 +1,518 @@
+# Beyond the Permutation Symmetry of Transformers: The Role of Rotation for Model Fusion
+
+Binchi Zhang\*1 Zaiyi Zheng\*1 Zhengzhang Chen 2 Jundong Li
+
+# Abstract
+
+Symmetry in the parameter space of deep neural networks (DNNs) has proven beneficial for various deep learning applications. A well-known example is the permutation symmetry in MultiLayer Perceptrons (MLPs), where permuting the rows of weight matrices in one layer and applying the inverse permutation to adjacent layers yields a functionally equivalent model. While permutation symmetry fully characterizes the equivalence set for MLPs, its discrete nature limits its utility for transformers. In this paper, we introduce rotation symmetry, a novel form of parameter space symmetry for transformers that generalizes permutation symmetry by rotating parameter matrices in self-attention layers. Unlike permutation symmetry, rotation symmetry operates in a continuous domain, thereby significantly expanding the equivalence set for transformers. Based on this property, we propose a theoretically optimal parameter matching algorithm as a plug-and-play module to enhance model fusion. We evaluate our approach using pre-trained transformers across diverse natural language and vision tasks. Experimental results demonstrate that our rotation symmetry-based matching algorithm substantially improves model fusion, highlighting the potential of parameter space symmetry to facilitate model fusion. Our code is available on https://github.com/zhengzaiyi/RotationSymmetry.
+
+# 1. Introduction
+
+Parameter space symmetry is an intriguing property of neural networks that has garnered increasing attention in recent years (Du et al., 2018; Armenta & Jodoin, 2021; Kunin et al.,
+
+*Equal contribution  $^{1}$ University of Virginia  $^{2}$ NEC Laboratories America. Correspondence to: Jundong Li <jundong@virginia.edu>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+2021; Simsek et al., 2021; Entezari et al., 2022; Grigsby et al., 2023; Lim et al., 2024b; Ziyin et al., 2024; 2025). One of the most studied forms of parameter space symmetry is permutation symmetry (Ainsworth et al., 2023; Entezari et al., 2022). For instance, in a two-layer MLP, permuting the rows of the weight matrix in the first layer and applying the corresponding inverse permutation to the second layer results in a functionally equivalent model, i.e., the outputs of the original and permuted models remain identical for any given input (Ainsworth et al., 2023). All functionally equivalent models corresponding to weight permutations form an equivalence set, which provides theoretical insights into neural network optimization, such as the linear mode connectivity of loss landscapes (Entezari et al., 2022; Zhou et al., 2023c; Ferbach et al., 2024). In addition, permutation symmetry has also proven helpful in advancing neural network applications, such as model fusion (Singh & Jaggi, 2020; Ainsworth et al., 2023) and optimization (Zhao et al., 2024; Zamir et al., 2025).
+
+Although parameter space symmetry has been extensively studied in classical neural network architectures, such as MLPs and CNNs, the understanding of its application in transformers (Vaswani et al., 2017) remains limited. Transformers have seen rapid advancements in recent years, achieving remarkable success in a wide range of applications (Yun et al., 2019; Lewis et al., 2020; Raffel et al., 2020; Clark et al., 2020; Zhou et al., 2021; He et al., 2024; Zhu et al., 2024; Zheng et al., 2024; Zhang et al., 2025; Wei & Liu, 2025). The transformer architecture is built upon two primary submodules: feedforward networks and self-attention layers. The feedforward network, which is structurally similar to MLPs, naturally inherits the permutation symmetry that has been extensively studied in the existing literature. Self-attention layers, on the other hand, involve a unique attention mechanism powered by matrix products of queries, keys, and values, which introduce additional potentials for symmetry beyond permutations. Permutation symmetries limit the equivalence set of neural networks to discrete operations, which aligns well with MLPs due to their element-wise activations (e.g., ReLU (Glorot et al., 2011)). In contrast, the continuous nature of the matrix operations in self-attention layers necessitates more flexible operations to fully characterize their equivalence set.
+
+In this paper, we introduce rotation symmetry, a novel form of parameter space symmetry for self-attention layers in transformers. Specifically, we analyze the query and key matrices jointly and demonstrate that applying a rotation to the query matrix, followed by the corresponding inverse rotation to the key matrix, preserves the query-key product. Additionally, we find that the same rotation rule can be applied to the value and output matrices. Our findings provide novel insights into the functional invariance of the attention mechanism and extend the permutation symmetry (Singh & Jaggi, 2020; Wang et al., 2020; Tatro et al., 2020; Entezari et al., 2022; Ainsworth et al., 2023; Imfeld et al., 2024) from discrete spaces to continuous spaces, which significantly extends the scope of parameter space symmetries for transformers.
+
+To further demonstrate the benefits of rotation symmetry for transformers, we explore its utility in model fusion. The goal of model fusion is to merge multiple well-trained end models in the parameter space to produce a single merged model with improved overall utility. Model fusion is widely adopted across various settings, such as hyperparameter tuning (where end models are trained on the same benchmark) and multi-task learning (where end models are trained with different tasks) (Jin et al., 2023). Unlike ensemble learning (Dietterich, 2000; Lakshminarayanan et al., 2017; Sagi & Rokach, 2018; Dong et al., 2020), model fusion can work in a data-agnostic manner, making it suitable for privacy-sensitive scenarios such as federated learning (Yurochkin et al., 2019; Wang et al., 2020).
+
+The existing literature has demonstrated that the performance of model fusion is closely tied to the distance between the end models (Wortsman et al., 2022). Inspired by this finding, we propose a parameter matching algorithm that selects functionally equivalent end models from the equivalence class determined by rotation symmetry. This approach ensures that the selected representative models are closer in parameter space, resulting in a smaller inner distance. To achieve this, we formulate the problem of parameter matching as an optimization problem with orthogonal constraints. Leveraging the continuous nature of rotation symmetry, we propose a closed-form solution to this problem. Our parameter matching algorithm is highly efficient, easy to implement, and can be seamlessly incorporated as a plug-and-play module for model fusion. To evaluate its effectiveness, we conduct extensive experiments with pre-trained transformers on real-world NLP and vision tasks. The experimental results demonstrate that incorporating rotation symmetry into parameter matching improves model fusion effectively and efficiently. Furthermore, additional experiments reveal that even matching a subset of parameters can lead to notable performance improvements, highlighting the practical utility of our approach. Our contributions are threefold:
+
+- We introduce a novel rotation symmetry for the attention mechanism in transformers, extending the concept of symmetry to a continuous space.  
+- Building on rotation symmetry, we propose a theoretically optimal parameter matching algorithm that improves the effectiveness of model fusion in transformers.  
+- Through extensive experiments, we validate the efficacy of our proposed parameter matching algorithm, demonstrating its potential to advance model fusion through parameter space symmetry.
+
+# 2. Preliminaries
+
+To facilitate further discussion, we begin by defining the notations. Let  $\mathbf{A}[i,:], \mathbf{A}(:, j]$ , and  $\mathbf{A}[i,j]$  denote the  $i$ -th row vector, the  $j$ -th column vector, and the  $(i,j)$ -th element of the matrix  $\mathbf{A}$ , respectively. Additionally, let  $\mathbf{1}$  denote an all-ones vector. We now illustrate the concept of permutation symmetry in neural networks, using an MLP as an example. Consider an  $L$ -layer MLP model defined as:
+
+$$
+f _ {\boldsymbol {W}} (\boldsymbol {X}) = \boldsymbol {Z} ^ {(L)}, \boldsymbol {Z} ^ {(l)} = \sigma \left(\boldsymbol {Z} ^ {(l - 1)} \left(\boldsymbol {W} ^ {(l)}\right) ^ {\top} + \boldsymbol {b} ^ {(l)}\right), \tag {1}
+$$
+
+where  $Z^{(0)} = X$  is the input feature matrix,  $\pmb{b}^{(l)}$  is the bias vector  $(\pmb{b}^{(l)})$  is a row vector and can be seen as broadcast over all rows in Equation (1)),  $\pmb{W} = \{\pmb{W}^{(l)},\pmb{b}^{(l)}\}_{l = 1,\dots,L}$  collects all learnable parameters, and  $\sigma (\cdot)$  stands for a nonlinear activation function, such as ReLU (Nair & Hinton, 2010). A loss function  $\mathcal{L}(f_W(\pmb {X}),\pmb {Y})$  is used to measure the distance between the model prediction and the ground truth label  $\pmb{Y}$ .
+
+To analyze the permutation symmetry in the MLP model, let  $P \in \mathcal{P}$  be a permutation matrix, where  $P[i,j] \in \{0,1\}$  and  $P[i,:]\mathbf{1} = P(:,j]^{\top}\mathbf{1} = 1$  for any  $i,j$ . All permutation matrices are orthogonal (Strang, 1976), satisfying  $P^{\top} = P^{-1}$ . Consequently, for layer  $l$  and  $l + 1$ , we have:
+
+$$
+\begin{array}{l} \boldsymbol {Z} ^ {(l + 1)} = \sigma \left(\sigma (\boldsymbol {Z} ^ {(l - 1)} (\boldsymbol {W} ^ {(l)}) ^ {\top} + \boldsymbol {b} ^ {(l)}) (\boldsymbol {W} ^ {(l + 1)}) ^ {\top} + \boldsymbol {b} ^ {(l + 1)}\right) \\ = \sigma \left(\sigma \left(\boldsymbol {Z} ^ {(l - 1)} \left(\boldsymbol {P} ^ {\top} \boldsymbol {W} ^ {(l)}\right) ^ {\top} + \boldsymbol {b} ^ {(l)} \boldsymbol {P}\right) \left(\boldsymbol {W} ^ {(l + 1)} \boldsymbol {P}\right) ^ {\top} + \boldsymbol {b} ^ {(l + 1)}\right). \tag {2} \\ \end{array}
+$$
+
+The third equal sign holds because the element-wise activation function  $\sigma$  is decoupled from column permutation (i.e., being multiplied by  $P$ ). Based on Equation (2), it follows that for layers  $l$  and  $l + 1$ , the mappings  $W^{(l)} \to P^{\top}W^{(l)}$ ,  $b^{(l)} \to b^{(l)}P$ ,  $W^{(l + 1)} \to W^{(l + 1)}P$  preserves the output  $Z^{(l + 1)}$  for any input  $Z^{(l - 1)}$ . For each pair of adjacent layers, a similar mapping exists independently based on a specific permutation matrix, denoted as  $P^{(l)}$  for layers  $l$  and  $l + 1$ . Consequently, for the entire  $L$ -layer MLP, there exists a mapping that preserves the model's predictions for any input  $X$ :  $W^{(l)} \to (P^{(l)})^{\top}W^{(l)}P^{(l - 1)}$ ,  $b^{(l)} \to b^{(l)}P^{(l)}$ . Equivalently, if we define  $W' = \{(P^{(l)})^{\top}W^{(l)}P^{(l - 1)}, b^{(l)}P^{(l)}\}_{l = 1,\dots,L}$  for any  $P^{(l)} \in \mathcal{P}$
+
+$(l = 1,\dots ,L - 1)$  and  $\pmb{P}^{(0)} = \pmb{P}^{(L)} = \pmb{I}$ , then we have  $f_{W^{\prime}}(\pmb {X}) = f_{\pmb{W}}(\pmb {X})$  for any  $\pmb{X}$ . This phenomenon is referred to as the permutation symmetry of the parameter space (Godfrey et al., 2022; Hecht-Nielsen, 1990; Navon et al., 2023; Rossi et al., 2023; Simsek et al., 2021). Leveraging permutation symmetry allows us to identify an equivalence class of functionally equivalent model parameters, which is known as permutation invariance (Ainsworth et al., 2023; Entezari et al., 2022; Lubana et al., 2023). We denote the equivalence relation induced by permutation invariance as  $\pi$ , where, for example,  $\pmb{W}^{\prime} = \pi (\pmb {W})$  represents the equivalence between the original parameters  $\pmb{W}$  and the permuted parameters  $\pmb{W}^{\prime}$ .
+
+# 3. Parameter Space Symmetry of Transformers
+
+Transformers (Vaswani et al., 2017) have revolutionized deep learning with their ability to handle sequential data effectively, particularly in natural language processing (NLP) and other fields (Brown et al., 2020; Devlin et al., 2019; Liu et al., 2021; Radford et al., 2021). Their success is primarily driven by two key components<sup>1</sup>: feedforward networks and self-attention layers (Vaswani et al., 2017). To better understand the parameter space symmetry of transformers, we examine these two core modules individually in the following sections.
+
+# 3.1. Permutation Symmetry of Feedforward Networks
+
+We first look at the feedforward networks. The feedforward network adopted in the transformer blocks is a two-layer MLP model which can be written as
+
+$$
+F F N (\boldsymbol {X}) = L N (R e L U (\boldsymbol {X} \boldsymbol {W} _ {i} ^ {\top} + \boldsymbol {b} _ {i}) \boldsymbol {W} _ {o} ^ {\top} + \boldsymbol {b} _ {o} + \boldsymbol {X}), \tag {3}
+$$
+
+where  $LN$  denotes the Layer Normalization operator (Ba et al., 2016). Different from (Vaswani et al., 2017), we include the residual connection (He et al., 2016) and layer normalization modules into the formula of the feedforward network (and the self-attention layer mentioned later). According to Equation (2) and the analysis in Preliminaries, the feedforward networks have the permutation symmetry property and the equivalence class determined by permutation invariance is defined as
+
+$$
+\boldsymbol {W} _ {i} \rightarrow \boldsymbol {P} ^ {\top} \boldsymbol {W} _ {i}, \boldsymbol {b} _ {i} \rightarrow \boldsymbol {b} _ {i} \boldsymbol {P}, \boldsymbol {W} _ {o} \rightarrow \boldsymbol {W} _ {o} \boldsymbol {P}, \boldsymbol {b} _ {o} \rightarrow \boldsymbol {b} _ {o}, \tag {4}
+$$
+
+where  $P \in \mathcal{P}$  is a permutation matrix. It is worth noting that the permutations of different feedforward networks in a transformer are independent due to the scalable modular design. Consequently, we are able to flexibly compute the permutation invariance equivalence class of each  $FFN$  module in a transformer model.
+
+# 3.2. Rotation Symmetry of Self-attention Layers
+
+We then focus on the self-attention layers. In this paper, we introduce the rotation symmetry of self-attention layers. For MLPs, we have to switch the order of multiplying  $P$  and passing the activation function  $\sigma$  (the third equal sign in Equation (2)), requiring the matrix  $P$  to be a permutation matrix. In contrast, the self-attention layer does not contain an element-wise activation function, enabling a wider range of the matrix  $P$  in self-attention layers. The self-attention layer can be written as
+
+$$
+A T T N (\boldsymbol {X}) = L N \left(C a t _ {h = 1} ^ {H} \left\{\boldsymbol {X} _ {Q K V} ^ {h} \right\} \boldsymbol {W} _ {O} ^ {\top} + \boldsymbol {b} _ {O} + \boldsymbol {X}\right),
+$$
+
+$$
+\boldsymbol {X} _ {Q K V} ^ {h} = S f t m x \left(\boldsymbol {X} _ {Q} ^ {h} \left(\boldsymbol {X} _ {K} ^ {h}\right) ^ {\top} / \sqrt {d _ {k}}\right) \cdot \boldsymbol {X} _ {V} ^ {h},
+$$
+
+where  $Cat$  stands for the operator concatenating the outputs of multi-head attention,  $Sftmx(\cdot)$  denotes the softmax operator,  $H$  denotes the number of multi-heads, and the subscripts  $Q$ ,  $K$ ,  $V$ , and  $O$  denote Query, Key, Value, and Output, respectively.
+
+We first transform the query and key matrices as
+
+$$
+\begin{array}{l} \boldsymbol {X} _ {Q} ^ {h} \left(\boldsymbol {X} _ {K} ^ {h}\right) ^ {\top} = \left(\boldsymbol {X} \left(\boldsymbol {W} _ {Q} ^ {h}\right) ^ {\top} + \boldsymbol {b} _ {Q} ^ {h}\right) \left(\boldsymbol {X} \left(\boldsymbol {W} _ {K} ^ {h}\right) ^ {\top} + \boldsymbol {b} _ {K} ^ {h}\right) ^ {\top} \\ = \left(\boldsymbol {X} \left(\boldsymbol {W} _ {Q} ^ {h}\right) ^ {\top} + \boldsymbol {b} _ {Q} ^ {h}\right) \boldsymbol {R R} ^ {\top} \left(\boldsymbol {X} \left(\boldsymbol {W} _ {K} ^ {h}\right) ^ {\top} + \boldsymbol {b} _ {K} ^ {h}\right) ^ {\top} \\ = \left(\boldsymbol {X} \left(\boldsymbol {R} ^ {\top} \boldsymbol {W} _ {Q} ^ {h}\right) ^ {\top} + \boldsymbol {b} _ {Q} ^ {h} \boldsymbol {R}\right) \left(\boldsymbol {X} \left(\boldsymbol {R} ^ {\top} \boldsymbol {W} _ {K} ^ {h}\right) ^ {\top} + \boldsymbol {b} _ {K} ^ {h} \boldsymbol {R}\right) ^ {\top}, \\ \end{array}
+$$
+
+where  $\pmb{R}$  is a rotation matrix, i.e.,  $\pmb{RR}^{\top} = \pmb{I}$ . It is worth noting that each multi-head corresponds to a specific rotation matrix  $\pmb{R}$ . Let  $\pmb{W}_{O} = [\pmb{W}_{O}^{1}\pmb{W}_{O}^{2}\dots \pmb{W}_{O}^{H}]$  and we then rewrite the concatenating operation (with the product by  $\pmb{W}_{O}$ ) as  $\sum_{h=1}^{H}S(\cdots)(\pmb{X}(\pmb{W}_{V}^{h})^{\top} + \pmb{b}_{V}^{h})(\pmb{W}_{O}^{h})^{\top}$ . Similar to the Q-K case, we can transform the value and output matrices as  $(\pmb{X}(\pmb{W}_{V}^{h})^{\top} + \pmb{b}_{V}^{h})(\pmb{W}_{O}^{h})^{\top} = (\pmb{X}(\pmb{W}_{V}^{h})^{\top} + \pmb{b}_{V}^{h})\pmb{R}\pmb{R}^{\top}(\pmb{W}_{O}^{h})^{\top} = (\pmb{X}(\pmb{R}^{\top}\pmb{W}_{V}^{h})^{\top} + \pmb{b}_{V}^{h}\pmb{R})(\pmb{W}_{O}^{h}\pmb{R})^{\top}$  for each multi-head  $h$ , where  $\pmb{R}$  is a rotation matrix. Finally, we derive an equivalence class of the parameters in a self-attention layer determined by rotation invariance as
+
+$$
+\boldsymbol {W} _ {Q} ^ {h} \rightarrow \left(\boldsymbol {R} _ {q k} ^ {h}\right) ^ {\top} \boldsymbol {W} _ {Q} ^ {h}, \boldsymbol {b} _ {Q} ^ {h} \rightarrow \boldsymbol {b} _ {Q} ^ {h} \boldsymbol {R} _ {q k} ^ {h},
+$$
+
+$$
+\boldsymbol {W} _ {K} ^ {h} \rightarrow (\boldsymbol {R} _ {q k} ^ {h}) ^ {\top} \boldsymbol {W} _ {K} ^ {h}, \boldsymbol {b} _ {K} ^ {h} \rightarrow \boldsymbol {b} _ {K} ^ {h} \boldsymbol {R} _ {q k} ^ {h}, \boldsymbol {W} _ {V} ^ {h} \rightarrow (\boldsymbol {R} _ {v o} ^ {h}) ^ {\top} \boldsymbol {W} _ {V} ^ {h},
+$$
+
+$$
+\boldsymbol {b} _ {V} ^ {h} \rightarrow \boldsymbol {b} _ {V} ^ {h} \boldsymbol {R} _ {v o} ^ {h}, \boldsymbol {W} _ {O} ^ {h} \rightarrow \boldsymbol {W} _ {O} ^ {h} \boldsymbol {R} _ {v o} ^ {h}, \boldsymbol {b} _ {O} \rightarrow \boldsymbol {b} _ {O}, \tag {5}
+$$
+
+where  $\pmb{R}_{qk}^{h}$  and  $\pmb{R}_{vo}^{h}$  are rotation matrices for  $h = 1, \dots, H$ . The progress from permutation to rotation extends the symmetry of transformers to a continuous space and enhances our understanding of the parameter space symmetry of attention mechanism. The denseness of the symmetry allows for the choice of better invariant models to analyze transformers' loss landscapes. For better clarity, we provide an intuitive illustration of the rotation symmetry in Figure 1.
+
+# 4. Symmetry for Model Fusion
+
+In this section, we explore the benefit of the rotation symmetry of transformers in model fusion (Li et al., 2023; Matena
+
+![](images/aeaf67a8993e6e8219761bd7778a1440dc17a49e694fe018a813dc5fae27b323.jpg)  
+Figure 1. The rotation symmetry of self-attention layers.
+
+& Raffel, 2022; Wortsman et al., 2022; Yadav et al., 2023; Jin et al., 2023; Daheim et al., 2024; Yang et al., 2024). Model fusion is proposed to merge multiple given end models trained in different settings (e.g., upon different datasets and hyperparameter settings) in the parameter space to improve model utility and robustness. Compared with ensemble learning (Dietterich, 2000; Sagi & Rokach, 2018; Lakshminarayanan et al., 2017), model fusion has a lower inference-stage complexity without requiring access to the training data. Most existing methods of model fusion conduct a weighted averaging of different end models, e.g., direct averaging (Wortsman et al., 2022), Fisher-weighted averaging (Matena & Raffel, 2022), and regression-mean averaging (Jin et al., 2023). We next show the potential of exploiting the permutation and rotation symmetry as a plug-and-play module to improve the model fusion techniques.
+
+# 4.1. Background and Motivation
+
+Let  $W_{1}, \ldots, W_{k}$  denote  $k$  different end models (after training or pre-training) with the same architecture. The goal of model fusion is to merge the given  $k$  end models in the parameter space and obtain a single model. If the given models are trained over different datasets, we can expect the merged model to have better utility and out-of-distribution robustness (Jin et al., 2023). The theoretical results in previous literature (Wortsman et al., 2022) have shown that strong convexity and closer end models can boost the utility of direct model fusion. Consequently, the primary advantage of permutation (and rotation) symmetry applying to model fusion is that we can substitute the end models with corresponding carefully chosen equivalent models (in the equivalence class determined by permutation and rotation invariance) to make the selected end models more concentrated, i.e., closer to each other. This step is usually called parameter matching (Singh & Jaggi, 2020; Wang et al., 2020; Ainsworth et al., 2023), aka parameter or neuron alignment. Parameter matching, which aims to reduce the distance between end models, can naturally yield closer end models and improve model fusion performance. On the other hand, different end models can lie in the basins of different local optimums regarding the highly non-convex nature of transformers. Previous studies have verified that parameter matching can merge different end models toward a single low-loss basin, i.e., the loss value along the linear interpolation between matched models shows an approxi
+
+![](images/35fd897e691d5974bdc3aff6e68e908e43254ac90b01fafd50534933ea294065.jpg)  
+Figure 2. An intuitive example of the usage of parameter space symmetry for model fusion. The background shows the contour map of the loss landscape in the model parameter space. A and B are the original end models to be merged, AB is the result of naive model fusion, and AB' is the result of model fusion with parameter matching.
+
+mately flat or convex curve (Entezari et al., 2022; Ainsworth et al., 2023). This property is called linear mode connectivity and is regarded as a weak form of convexity (Ainsworth et al., 2023). As a result, parameter matching can also help improve the convexity of the objective in the area adjacent to the end models. We showcase an intuitive example of using parameter space symmetry to improve model fusion in Figure 2 to illustrate the intuition behind parameter matching. We can observe that the merged model after parameter matching (AB') has a lower loss value, i.e., better utility than the naive merged model (AB).
+
+# 4.2. Parameter Matching
+
+Next, our primary goal is to develop a practical parameter matching algorithm to minimize the distance between different end models. We begin by merging two models  $\mathbf{W}_1$  and  $\mathbf{W}_2$ .
+
+Matching two FFNs. Let  $\{W_{i_k}, b_{i_k}, W_{o_k}, b_{o_k}\}_{k=1,2}$  denote the model parameters in the two feedforward networks to be merged where  $k$  denotes the index of the networks. We have already derived the equivalence relation of parameters in the feedforward networks determined by permutation invariance as Equation (4). Based on Equation (4), we can formulate parameter matching as an optimization problem as follows.
+
+$$
+\begin{array}{l} \operatorname {a r g m i n} _ {\boldsymbol {P} _ {1}, \boldsymbol {P} _ {2} \in S} \| \boldsymbol {P} _ {1} ^ {\top} \boldsymbol {W} _ {i _ {1}} - \boldsymbol {P} _ {2} ^ {\top} \boldsymbol {W} _ {i _ {2}} \| _ {F} ^ {2} \\ + \left\| \boldsymbol {b} _ {i _ {1}} \boldsymbol {P} _ {1} - \boldsymbol {b} _ {i _ {2}} \boldsymbol {P} _ {2} \right\| _ {F} ^ {2} + \left\| \boldsymbol {W} _ {o _ {1}} \boldsymbol {P} _ {1} - \boldsymbol {W} _ {o _ {2}} \boldsymbol {P} _ {2} \right\| _ {F} ^ {2}. \\ \end{array}
+$$
+
+As shown in Equation (6), the goal is to find the functionally equivalent models from the equivalence classes  $\pi(W_1)$  and  $\pi(W_2)$  determined by  $P_1$  and  $P_2$ , which has the smallest  $\ell-2$  distance in the parameter space. Following the method (Ainsworth et al., 2023), we reformulate the optimization problem shown in Equation (6) as a linear assignment problem:
+
+$$
+\operatorname {a r g m a x} _ {\boldsymbol {P} \in \mathcal {S}} \left\langle \boldsymbol {P}, \boldsymbol {W} _ {i _ {1}} \boldsymbol {W} _ {i _ {2}} ^ {\top} + \boldsymbol {b} _ {i _ {1}} ^ {\top} \boldsymbol {b} _ {i _ {2}} + \boldsymbol {W} _ {o _ {1}} ^ {\top} \boldsymbol {W} _ {o _ {2}} \right\rangle_ {F}. \tag {7}
+$$
+
+Linear assignment problems such as Equation (7) have been well-studied in previous literature (Martello & Toth, 1987; Burkard & Cela, 1999) and can be solved precisely by Hungarian Algorithm (Martello & Toth, 1987). After calculating the value of  $P$ , we substitute  $P = P_{1}P_{2}^{-1}$  back to Equation (4) and let  $P_{2} = I$  (for simplicity), then we obtain the matched parameter as  $W_{i_1} \rightarrow P^\top W_{i_1}$ ,  $b_{i_1} \rightarrow b_{i_1}P$ ,  $W_{o_1} \rightarrow W_{o_1}P$ ,  $b_{o_1} \rightarrow b_{o_1}$ . Parameters of the second model  $\{W_{i_2}, b_{i_2}, W_{o_2}, b_{o_2}\}$  remain unchanged, which makes the second model an anchor model for matching.
+
+Matching two ATTNs. Let the parameters in the two self-attention layers to be merged be  $\{W_{Q_k}^i, b_{Q_k}^i, W_{K_k}^i, b_{K_k}^i, W_{V_k}^i, b_{V_k}^i, W_{O_k}^i, b_{O_k}^i\}_{k=1,2;i=1\dots H}$  where  $k$  denotes the index of the layers. The equivalence relation of these parameters in the self-attention layers in terms of rotation invariance is shown in Equation (5). Similar to matching FFNs, our goal is to match the parameters of one (source) attention layer to the other (target) attention layer where the matched parameters are supposed to be closest to the target parameters while being functionally equivalent when feeding with any input data, i.e., in the equivalence class determined by rotation invariance. The goal can be formulated as an optimization problem.
+
+$$
+\begin{array}{l} \min  _ {\boldsymbol {R} _ {q k} ^ {h}, \boldsymbol {R} _ {v o} ^ {h} \in \mathcal {R}} \left\| \left[ \begin{array}{c c} (\boldsymbol {W} _ {Q _ {1}} ^ {h}) ^ {\top} & (\boldsymbol {W} _ {Q _ {2}} ^ {h}) ^ {\top} \\ \boldsymbol {b} _ {Q _ {1}} ^ {h} & \boldsymbol {b} _ {Q _ {2}} ^ {h} \end{array} \right] \left[ \begin{array}{c} \boldsymbol {R} _ {q k 1} ^ {h} \\ - \boldsymbol {R} _ {q k 2} ^ {h} \end{array} \right] \right\| _ {F} ^ {2} \\ + \left\| \left[ \begin{array}{c c} \left(\boldsymbol {W} _ {K _ {1}} ^ {h}\right) ^ {\top} & \left(\boldsymbol {W} _ {K _ {2}} ^ {h}\right) ^ {\top} \\ \boldsymbol {b} _ {K _ {1}} ^ {h} & \boldsymbol {b} _ {K _ {2}} ^ {h} \end{array} \right] \left[ \begin{array}{c} \boldsymbol {R} _ {q k 1} ^ {h} \\ - \boldsymbol {R} _ {q k 2} ^ {h} \end{array} \right] \right\| _ {F} ^ {2} \tag {8} \\ + \left\| \left[ \begin{array}{c c} (\boldsymbol {W} _ {V _ {1}} ^ {h}) ^ {\top} & (\boldsymbol {W} _ {V _ {2}} ^ {h}) ^ {\top} \\ \boldsymbol {b} _ {V _ {1}} ^ {h} & \boldsymbol {b} _ {V _ {2}} ^ {h} \end{array} \right] \left[ \begin{array}{c} \boldsymbol {R} _ {v o 1} ^ {h} \\ - \boldsymbol {R} _ {v o 2} ^ {h} \end{array} \right] \right\| _ {F} ^ {2} \\ + \left\| \left[ \begin{array}{c c} \boldsymbol {W} _ {O _ {1}} ^ {h} & \boldsymbol {W} _ {O _ {2}} ^ {h} \\ \boldsymbol {0} & \boldsymbol {0} \end{array} \right] \left[ \begin{array}{c} \boldsymbol {R} _ {v o 1} ^ {h} \\ - \boldsymbol {R} _ {v o 2} ^ {h} \end{array} \right] \right\| _ {F} ^ {2}, \\ \end{array}
+$$
+
+where  $\mathcal{R}$  denotes the set of rotation matrices. To solve this problem, we divide Equation (8) into two separate optimization problems (the first line in terms of  $R_{qk1}^{h}, R_{qk2}^{h}$  as the first objective and the second line in terms of  $R_{vo1}^{h}, R_{vo2}^{h}$  as the second objective). Considering the similar formulation of these two optimization problems, in this paper, we propose the following theorem to solve both problems.
+
+Theorem 4.1. The following optimization problem has a closed-form solution.
+
+$$
+\begin{array}{l} \min  _ {\boldsymbol {R} _ {1}, \boldsymbol {R} _ {2} \in \mathcal {R}} \left\| \left[ \begin{array}{c c} \boldsymbol {W} _ {Q _ {1}} ^ {\top} & \boldsymbol {W} _ {Q _ {2}} ^ {\top} \\ \boldsymbol {b} _ {Q _ {1}} & \boldsymbol {b} _ {Q _ {2}} \end{array} \right] \left[ \begin{array}{c} \boldsymbol {R} _ {1} \\ - \boldsymbol {R} _ {2} \end{array} \right] \right\| _ {F} ^ {2} \tag {9} \\ + \left\| \left[ \begin{array}{c c} \boldsymbol {W} _ {K _ {1}} ^ {\top} & \boldsymbol {W} _ {K _ {2}} ^ {\top} \\ \boldsymbol {b} _ {K _ {1}} & \boldsymbol {b} _ {K _ {2}} \end{array} \right] \left[ \begin{array}{c} \boldsymbol {R} _ {1} \\ - \boldsymbol {R} _ {2} \end{array} \right] \right\| _ {F} ^ {2}. \\ \end{array}
+$$
+
+The solution is given by
+
+$$
+\boldsymbol {R} _ {1} = \boldsymbol {U} \boldsymbol {V} ^ {\top}, \boldsymbol {R} _ {2} = \boldsymbol {I}, \tag {10}
+$$
+
+Algorithm 1 Matching two self-attention layers.
+
+Input: Model parameters of the attention layers  $\{W_{Q_k}^h, b_{Q_k}^h, W_{K_k}^h, b_{K_k}^h, W_{V_k}^h, b_{V_k}^h, W_{O_k}^h, b_{O_k}^h\}_{k = 1,2; h = 1,\dots,H}$
+
+Output: The optimally matched parameters from source parameters  $(k = 1)$  to anchor parameters  $(k = 2$ , maintain unchanged).
+
+1: QK solution:  $\pmb{R}_{qk1}^{h} = \pmb{U}_{qk}^{h}(\pmb{V}_{qk}^{h})^{\top},\pmb{R}_{qk2}^{h} = \pmb{I}$ , where  $\pmb{U}_{qk}^{h}\pmb{\Sigma}_{qk}^{h}(\pmb{V}_{qk}^{h})^{\top} = \pmb{W}_{Q_{1}}^{h}(\pmb{W}_{Q_{2}}^{h})^{\top} + \pmb{W}_{K_{1}}^{h}(\pmb{W}_{K_{2}}^{h})^{\top} + (\pmb{b}_{Q_{1}}^{h})^{\top}\pmb{b}_{Q_{2}}^{h} + (\pmb{b}_{K_{1}}^{h})^{\top}\pmb{b}_{K_{2}}^{h}$ .  
+2: QK matching:  $\pmb{W}_{Q_1}^h \rightarrow (\pmb{R}_{qk1}^h)^\top \pmb{W}_{Q_1}^h$ ,  $\pmb{W}_{K_1}^h \rightarrow (\pmb{R}_{qk1}^h)^\top \pmb{W}_{K_1}^h$ ,  $\pmb{b}_{Q_1}^h \rightarrow \pmb{b}_{Q_1}^h \pmb{R}_{qk1}^h$ ,  $\pmb{b}_{K_1}^h \rightarrow \pmb{b}_{K_1}^h \pmb{R}_{qk1}^h$ .  
+3: VO solution:  $\pmb{R}_{vo1}^{h} = \pmb{U}_{vo}^{h}(\pmb{V}_{vo}^{h})^{\top},\pmb{R}_{vo2}^{h} = \pmb{I}$ , where  $\pmb{U}_{vo}^{h}\pmb{\Sigma}_{vo}^{h}(\pmb{V}_{vo}^{h})^{\top} = \pmb{W}_{V_{1}}^{h}(\pmb{W}_{V_{2}}^{h})^{\top} + (\pmb{W}_{O_{1}}^{h})^{\top}\pmb{W}_{O_{2}}^{h} + (\pmb{b}_{V_{1}}^{h})^{\top}\pmb{b}_{V_{2}}^{h}$ .  
+4: VO matching:  $\pmb{W}_{V_1}^h \rightarrow (\pmb{R}_{vo1}^h)^\top \pmb{W}_{V_1}^h$ ,  $\pmb{W}_{O_1}^h \rightarrow \pmb{W}_{O_1}^h \pmb{R}_{vo1}^h$ ,  $\pmb{b}_{V_1}^h \rightarrow \pmb{b}_{V_1}^h \pmb{R}_{vo1}^h$ .
+
+where  $\pmb{I}$  is the identity matrix and  $\pmb{U} \pmb{\Sigma} \pmb{V}^{\top} = \pmb{W}_{Q_1} \pmb{W}_{Q_2}^{\top} + \pmb{W}_{K_1} \pmb{W}_{K_2}^{\top} + \pmb{b}_{Q_1}^{\top} \pmb{b}_{Q_2} + \pmb{b}_{K_1}^{\top} \pmb{b}_{K_2}$  is the result of eigendecomposition.
+
+We leave a detailed proof of Theorem 4.1 in Appendix A. According to Theorem 4.1, we can obtain the algorithm to match two self-attention layers shown in Algorithm 1. Algorithm 1 can be seen as an adaptation of the Kabsch algorithm (Kabsch, 1976; Umeyama, 1991). Without loss of generality, we let the parameters in the second  $(k = 2)$  self-attention layer be the anchor and conduct rotation for the first layer  $(k = 1)$ . The denseness of rotation symmetry helps reduce the distance between the end models after parameter matching as Algorithm 1 shows.
+
+Rescaling Matching. We find that the rescaling symmetry (Neyshabur et al., 2015; Meng et al., 2019; Godfrey et al., 2022; Kalogeropoulos et al., 2024) can be integrated with our proposed rotation symmetry in the parameter matching algorithm. For instance, for the (simplified) Q-K product  $\mathbf{W}_Q\mathbf{W}_K$ , we can find a rescaling operation that preserves the functionality of the Q-K product  $a\mathbf{W}_Q \cdot \frac{1}{a}\mathbf{W}_K$  where  $a \neq 0$  is a real number. By adding a scalar variable to each parameter matrix, we can formulate the objective of rescaling symmetry as follows (taking the Q-K product as an example).
+
+$$
+\begin{array}{l} \min  _ {a} \left\| a \boldsymbol {W} _ {Q _ {1}} ^ {\prime} - \boldsymbol {W} _ {Q _ {2}} ^ {\prime} \right\| ^ {2} + \left\| a \boldsymbol {b} _ {Q _ {1}} ^ {\prime} - \boldsymbol {b} _ {Q _ {2}} ^ {\prime} \right\| ^ {2} \tag {11} \\ + \left\| \boldsymbol {W} _ {K _ {1}} ^ {\prime} / a - \boldsymbol {W} _ {K _ {2}} ^ {\prime} \right\| ^ {2} + \left\| \boldsymbol {b} _ {K _ {1}} ^ {\prime} / a - \boldsymbol {b} _ {K _ {2}} ^ {\prime} \right\| ^ {2}, \\ \end{array}
+$$
+
+where  $a$  is the rescaling variable and  ${}^{\prime}$  denotes the parameters after rotation symmetry-based matching. Here, we still set model 2 as the anchor model and conduct the rescaling operation on model 1 to align with model 2. We can easily
+
+Table 1. Experimental results of in-domain (Emotion and NER) and out-of-domain (NER-CoNLL) model fusion for two base language models over Emotion and NER tasks.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">Emotion</td><td colspan="2">NER</td><td colspan="2">NER-CoNLL</td></tr><tr><td>Roberta</td><td>Deberta</td><td>Roberta</td><td>Deberta</td><td>Roberta</td><td>Deberta</td></tr><tr><td>Simple</td><td>35.87</td><td>2.99</td><td>60.88</td><td>27.54</td><td>26.86</td><td>10.80</td></tr><tr><td>+match</td><td>35.87</td><td>2.99</td><td>60.88</td><td>31.31</td><td>26.87</td><td>23.30</td></tr><tr><td>Fisher</td><td>44.02</td><td>35.95</td><td>54.55</td><td>33.20</td><td>23.06</td><td>12.53</td></tr><tr><td>+match</td><td>44.05</td><td>35.98</td><td>54.58</td><td>33.83</td><td>23.05</td><td>18.33</td></tr><tr><td>Regmean</td><td>35.87</td><td>2.99</td><td>60.88</td><td>27.54</td><td>26.86</td><td>10.80</td></tr><tr><td>+match</td><td>39.95</td><td>2.99</td><td>60.88</td><td>31.31</td><td>26.87</td><td>14.06</td></tr></table>
+
+solve the optimality condition of Equation (11) as
+
+$$
+\begin{array}{l} \left(\left\| \boldsymbol {W} _ {Q _ {1}} ^ {\prime} \right\| ^ {2} + \left\| \boldsymbol {b} _ {Q _ {1}} ^ {\prime} \right\| ^ {2}\right) a ^ {4} - \left\| \boldsymbol {W} _ {K _ {1}} ^ {\prime} \right\| ^ {2} - \left\| \boldsymbol {b} _ {K _ {1}} ^ {\prime} \right\| ^ {2} \\ - \left(\left\langle \boldsymbol {W} _ {Q _ {1}} ^ {\prime}, \boldsymbol {W} _ {Q _ {2}} ^ {\prime} \right\rangle + \left\langle \boldsymbol {b} _ {Q _ {1}} ^ {\prime}, \boldsymbol {b} _ {Q _ {2}} ^ {\prime} \right\rangle\right) a ^ {3} \\ + \left(\left\langle \boldsymbol {W} _ {K _ {1}} ^ {\prime}, \boldsymbol {W} _ {K _ {2}} ^ {\prime} \right\rangle + \left\langle \boldsymbol {b} _ {K _ {1}} ^ {\prime}, \boldsymbol {b} _ {K _ {2}} ^ {\prime} \right\rangle\right) a = 0. \tag {12} \\ \end{array}
+$$
+
+The roots of Equation (12) can be derived using numerical methods as the value of  $a$ . We then conduct the rescaling operation to model 1 as  $W_{Q_1}' \rightarrow aW_{Q_1}', b_{Q_1}' \rightarrow ab_{Q_1}', W_{K_1}' \rightarrow \frac{1}{a} W_{K_1}', b_{K_1}' \rightarrow \frac{1}{a} b_{K_1}'$ . It is worth noting that the rescaling symmetry-based matching is conducted after the rotation symmetry-based matching. Using the rescaling operation, we extend the rotation matrices to orthogonal matrices with larger norms. Nevertheless, the end models are close to each other in practical scenarios so the value of  $a$  is usually close to 1.
+
+Complexity. We next provide a brief analysis of the complexity of our proposed parameter matching algorithm. We let the hidden dimension of the target transformer be  $d$ , the layer of the target transformer be  $L$ , and the number of attention heads be  $H$ . For the parameter matching of feedforward networks, the linear assignment problem can be solved in  $O(d^{3})$  by the Hungarian algorithm (Kuhn, 1955; Martello & Toth, 1987). Additionally, to solve the optimization problem in Equation (8), Algorithm 1 requires the complexity of  $O(d^{3})$  for eigendecomposition. Hence, the complexity of our proposed full parameter matching algorithm of a transformer is  $O(d^{3}LH)$ , similar to the complexity of the feedforward. It is worth noting that the complexity can be further reduced in two ways. The first way is to match a subset of layers instead of all. In this way, the complexity of our proposed parameter matching algorithm is  $O(d^{3}L_{s}H)$  where  $L_{s}$  denotes the number of selected layers. The second way is to match each unit module (a single feedforward network or a single attention layer) in parallel. The decoupling of matching different modules makes it easy to implement multiprocessing. Consequently, the overall complexity becomes to  $O(d^{3}LH / p)$  where  $p$  is the number of processes in parallel.
+
+Table 2. Experimental results of ViT merging over the image classification task.  
+
+<table><tr><td></td><td>w/o match</td><td>w/o rescale</td><td>match</td></tr><tr><td>Simple</td><td>7.60</td><td>10.22</td><td>10.19</td></tr><tr><td>Fisher</td><td>17.96</td><td>18.61</td><td>18.58</td></tr><tr><td>Regmean</td><td>14.24</td><td>15.31</td><td>15.35</td></tr><tr><td>OT-ACTS-EMD</td><td>32.08</td><td>32.50</td><td>32.53</td></tr><tr><td>OT-ACTS</td><td>61.15</td><td>61.23</td><td>61.25</td></tr><tr><td>OT-WTS</td><td>57.11</td><td>57.16</td><td>57.17</td></tr></table>
+
+# 5. Experiments
+
+In this section, we conduct experiments to evaluate the effectiveness of our proposed parameter matching algorithm based on rotation symmetry. Specifically, we aim to answer the following research questions: RQ1: Can our proposed parameter matching algorithm enhance the performance of model fusion for transformer-based models? RQ2: How does rotation symmetry contribute to parameter matching for self-attention layers? RQ3: How does parameter matching influence the loss landscape between independently trained models? RQ4: As a plugin module, does our algorithm introduce significant additional computational overhead? RQ5: Is matching all transformer layers equally important? Can we improve efficiency without compromising utility by matching only a subset of layers?
+
+# 5.1. Experimental Settings
+
+Platform. Our implementation is based on Python 3.10 and Pytorch 1.13. All fine-tuning, parameter matching, and model merging processes are conducted on a cluster equipped with Nvidia A100 80GB GPUs.
+
+Models. To evaluate the effectiveness of our approach, we use two widely adopted transformer models: RoBERTa (Liu et al., 2019) and DeBERTa (He et al., 2021). We obtain pretrained models of RoBERTa (roberta-base with 12 attention layers) and DeBERTa-Large (microsoft/deberta-v3-large with 24 attention layers) from the Hugging Face library. For vision transformers (ViTs) (Dosovitskiy et al., 2021), we directly use the pretrained models following (Imfeld et al., 2024). By selecting these three type of models, we assess the effectiveness of our method across different model scales and downstream tasks.
+
+Finetuning and Matching. In the experiments, each pretrained language model is fine-tuned for 20 epochs on each dataset individually. We set the learning rate at 1e-5, the batch size at 16, and the warmup ratio at 0.06 for each model. After fine-tuning, we perform parameter matching and model merging, considering both in-domain (pairwise
+
+![](images/93cea4e6c6ab0554f0066db0b12151ae1032653d3f58de6a696e5acba6a33bb0.jpg)
+
+![](images/f24d124edc62be3ea476a935a999e701f9b3d7d83b3019bc4601aab344c179ed.jpg)
+
+![](images/b69b7082beb2919b7ca31c9bc58a3f1fde1cbd24c6d4f59de838165436be1ea6.jpg)
+
+![](images/b819b538c1371ac41e7b8ab6e437bdf4880f12436e740d8d3b737bc207462f98.jpg)  
+Figure 3. Ablation Study of ViT merging over the image classification task. "ATTN" is short for "attention". We compare our matching algorithm with its three variants (w/o ATTN/FFN/rescaling) and the original performance (w/o match) on all six merging baselines.
+
+![](images/7f6f3481c486a52dd44bb94088981386c70b3444ea8fc63c45b97fc7db02e64a.jpg)
+
+![](images/794725d522029c1a37e861c567e49a902fd9a01f255a233bc3e62f694667cd69.jpg)
+
+![](images/cf1c52053114187444accb3ae1652f1cf3baa39ee15a9c8ab020afd733579041.jpg)  
+Figure 4. The Euclidean Distance of end ViT models after different parameter matching algorithms.
+
+fine-tuned models) and out-of-domain (grouped models) experiments. Notably, we match only the parameters within the attention layers, while the classifier module is directly copied from the model fine-tuned on the corresponding downstream task. For additional details on datasets and baseline methods, please refer to Appendix C.
+
+# 5.2. Performance of Model Fusion
+
+To answer RQ1, we compare the performance of three model fusion baselines with and without our proposed parameter matching algorithm.
+
+In-Domain Settings. We evaluate model fusion performance on Emotion Classification and Named Entity Recognition (NER) tasks, where our approach (+match) is integrated as a plugin module for three model fusion baselines. For each task, we fine-tune the language models on each in-domain dataset (5 for Emotion, 6 for NER) respectively and merge them pairwise. The first two columns in Table 1 present the average macro F1-score (for Emotion) and micro F1-score (for NER) of the merged models following (Jin et al., 2023).
+
+Out-of-Domain Settings. To assess the generalization ability, we merge models trained on all in-domain NER datasets and evaluate their performance on CoNLL datasets, which serve as out-of-domain (OOD) test sets. The third column in Table 1 reports the OOD performance of the merged models.
+
+ViT Settings. We employ three OT (optimal transportation)-based methods (OT-ACTS-EMD, OT-ACTS, and OT-WTS) from (Imfeld et al., 2024) as additional baselines. Two pretrained models are merged, and we evaluate their performance on the CIFAR10 (Krizhevsky et al., 2009) dataset. Table 2 presents the classification accuracy of the merged models. To improve the flexibility of parameter matching, we match each layer separately before conducting model fusion. The merged model achieving the highest validation performance is selected as the final test model.
+
+From the results in Table 1 and Table 2, we make the following observations: (1) Our parameter matching algorithm consistently improves the performance of different model fusion methods. The reason is that parameter matching helps align distant models, bringing them closer in parameter space. As a result, the merged model is more likely to approach an overall minimum. (2) Compared to RoBERTa-base, our method brings larger improvement for DeBERTa-large, suggesting that larger models benefit more from parameter matching. We explain this as larger models have more parameters that can be aligned, and in high-dimensional spaces, models tend to be more spread out. Subsequently, parameter matching helps bridge this gap more effectively. (3) Among all the fusion methods, simple fusion shows the largest improvement after parameter matching. This is likely because simple fusion directly averages model weights, so reducing the distance between model parameters brings a clear benefit. In contrast, advanced fusion methods work by aligning outputs of the end models based on input data, which is equivalent to weighted averaging. Unlike direct averaging, these methods benefit from parameter matching in a more implicit way.
+
+# 5.3. Ablation Study
+
+To answer RQ2, we conduct ablation studies to evaluate the contributions of rotation symmetry in matching different model components. These experiments are performed on the same pair of pretrained ViT models used in the previous
+
+![](images/0970077fb1c553f8e8eb8d5804a31dd6af101405d6ff892fb2b7d9e1d1853265.jpg)  
+Figure 5. Loss landscapes and barriers between the two pretrained ViT models under four distinct matching settings. "LB" is short for "Loss Barrier".
+
+Table 3. The average runtime in seconds of the fine-tuning (top), matching (middle), and merging (bottom) stage.  
+
+<table><tr><td></td><td>Deberta</td><td>Roberta</td><td>ViT</td></tr><tr><td>Fine-tuning</td><td>12983.71</td><td>3071.49</td><td>-</td></tr><tr><td>Matching</td><td>1.59</td><td>1.71</td><td>3.48</td></tr><tr><td>Simple Merging</td><td>0.13</td><td>0.09</td><td>0.22</td></tr><tr><td>Fisher Merging</td><td>197.47</td><td>69.57</td><td>83.67</td></tr><tr><td>Regmean Merging</td><td>137.67</td><td>36.44</td><td>71.02</td></tr></table>
+
+section. We design three ablation settings: (1)  $\underline{w / o} \, ATTN$ : Matches only the FFN modules, excluding self-attention modules. (2)  $\underline{w / o} \, FFN$ : Matches only the self-attention modules, excluding FFN modules. (3)  $\underline{w / o} \, rescaling$ : Enables the rescaling symmetry in our algorithm. Ablation results in Figure 3 demonstrate the following observations: (1) In Fisher and RegMean, attention matching contributes more significantly to the fusion process. (2) In contrast, OTFusion benefits more from FFN matching. These distinctions demonstrate the nuanced interactions between the fusion mechanisms and corresponding component alignment strategies. (3)  $\underline{w / o} \, rescaling$  results in performance degradation across most settings, validating the effectiveness of rescaling symmetry integrated with rotation symmetry.
+
+To further quantify the influence of rotation symmetry, we measure the Euclidean distance between the matched model and the anchor model in the parameter space. Instead of aggregating distances at the layer or module level, we analyze the entire model holistically to provide a more comprehensive assessment of alignment. We also compare our method against two external baselines: Git Re-Basin (Ainsworth et al., 2023) and OT-Fusion (Imfeld et al., 2024). Following the original paper, OT-Fusion is applied only to FFN modules. The results in Figure 4 indicate that our rotation symmetry-based parameter matching algorithm consistently reduces the distance between end models more effectively than permutation symmetry-based methods. This improvement can be attributed to the continuous nature of rotation symmetry, which allows for smoother and more precise alignment, especially in self-attention layers. Furthermore, the integration of rescaling symmetry leads to additional distance reductions of end models, further enhancing parameter alignment.
+
+![](images/56d50f7696a408fd6a085ad1d3add16bdf889b4fe2ca661909aaade7125548a8.jpg)  
+(a) Single-Layer Matching  
+Figure 6. Evaluation loss of matching a subset of layers.
+
+![](images/ce9df61833944bd1edfa324ee6f2660fcbc25e8a1ffe57e3f08ac404ce047dc2.jpg)  
+(b) Tail-Layers Matching
+
+# 5.4. Loss Landscape Study
+
+To address RQ3, we examine the loss landscape between the two pretrained ViT models introduced in Section 5.2, both with and without parameter matching. Specifically, we first analyze the cross entropy loss values  $\mathcal{L}(\theta (\alpha))$  along the linear interpolation path between the end model parameters  $\theta (\alpha) = \alpha \cdot \theta_{A} + (1 - \alpha)\cdot \theta_{B},\quad \alpha \in [0,1]$ , where  $\theta_{A}$  and  $\theta_{B}$  denote the end models. Next, we separately apply attention-only matching (w/o  $FFN$ ), FFN-only matching (w/o  $ATTN$ ), and complete matching (w/ match) to  $\theta_{B}$  before constructing the same linear interpolation path with respect to  $\theta_{A}$ . The results in Figure 5 give rise to the following observations: (1) All model pairs exhibit positive loss barriers, i.e., the range of loss value along the interpolation path, highlighting the strong inherent non-convexity of ViTs (Park & Kim, 2022). (2) Complete matching yields the lowest loss barrier, indicating that aligning parameters through rotation symmetry leads to smoother connectivity between model pairs. (3) While FFN-only matching (w/o  $ATTN$ ) has been proven effective for MLPs and CNNs (Ainsworth et al., 2023), its effectiveness remains limited for Transformers. Their intricate loss landscape brings additional challenges for parameter matching.
+
+# 5.5. Complexity Study
+
+To answer RQ4, we evaluate the computational overhead introduced by our parameter matching algorithm. Specifically, we measure the average runtime for fine-tuning (per dataset), matching (per model pair), and merging methods (per model pair) from the main experiments, as shown in Table 3. For ViTs, we only use pre-trained models following (Imfeld et al., 2024) without fine-tuning. The results show that our parameter matching module incurs an overhead of less than  $5\%$  compared to Fisher / RegMean merging across all models, with negligible impact on overall complexity.
+
+# 5.6. Matching a Subset of Layers
+
+To answer RQ5, we investigate the effect of matching different subsets of layers. We fine-tune DeBERTa models on CoLA and STS-B from the GLUE benchmark and evaluate the performance of matched models under different subsets
+
+of layers. Specifically, we cross-validate the importance of each attention layer in matching through two settings:
+
+Single-Layer Matching. In this setting, we match only a single attention layer while leaving all other layers unchanged. The evaluation loss corresponding to different matched layer indices is shown in Figure 6(a).
+
+Tail-Layers Matching. Here, we match a certain number of trailing attention layers (e.g., when matching three attention layers, we align only the last three layers). The evaluation loss for different numbers of matched layers is presented in Figure 6(b).
+
+The experimental results in Figure 6 demonstrate that matching head layers yields greater improvements in model utility compared to tail layers. Notably, the loss value drops sharply when the first 5 layers are matched. Based on this observation, we can further improve efficiency by dropping tail layers without significantly compromising utility.
+
+# 6. Related Work
+
+Parameter Space Symmetry. Parameter space symmetry refers to a set of models with different parameter values but functionally equivalent. This concept has been extensively studied in the context of deep neural networks, as it plays a crucial role in understanding model behavior and training dynamics. Examples of parameter space symmetries include rescaling symmetry (Neyshabur et al., 2015; Badrinarayanan et al., 2015; Du et al., 2018; Meng et al., 2019), scaling symmetry (Kunin et al., 2021), and translation symmetry (Kunin et al., 2021). These symmetries have been identified in conventional deep neural networks to provide deeper insights into training dynamics and to accelerate the optimization process (Zhao et al., 2022; 2023; 2024). Another important type of parameter space symmetry is permutation symmetry, which has been shown to closely relate to the manifold of global minima and critical points (Fukumizu & Amari, 2000; Brea et al., 2019; Simsek et al., 2021; Benton et al., 2021; Entezari et al., 2022; Ainsworth et al., 2023). The permutation symmetry can also be used to align (match) the outputs or model parameters of different end models with the same architecture (Singh & Jaggi, 2020; Wang et al., 2020; Ainsworth et al., 2023; Pena et al., 2023; Imfeld et al., 2024; Navon et al., 2024). Complementary to parameter space symmetry, recent advances in neural functionals and metanetworks (Navon et al., 2023; Zhou et al., 2023b;a; Lim et al., 2024a; Tran et al., 2024) explore permutation-equivariant functionals that operate directly on model weights across diverse architectures. Some concurrent works (Liu, 2024; Tran et al., 2024) investigate similar forms of rotation symmetry in neural networks. Liu (2024) shows that the mirror symmetry leads
+
+to low-rankness. Meanwhile, Tran et al. (2024) leverages the rotation symmetry to construct transformer-based neural functional networks. In comparison, our study focuses on the role of rotation symmetry in model fusion and proposes a theoretically optimal parameter matching approach based on the properties of rotation symmetries.
+
+Model Fusion. The goal of model fusion (Li et al., 2023) is to merge multiple available end models (with the same architecture) to obtain a stronger model. The scenarios of model fusion can be flexible. When training on the same dataset, model fusion can be used to improve the model utility or generalization by merging models trained with different configurations or in different stages (Izmailov et al., 2018; Gupta et al., 2020; Cha et al., 2021; Wortsman et al., 2022; Rame et al., 2022; Arpit et al., 2022; Huang et al., 2024; Yadav et al., 2024; Hammoud et al., 2024). As a representative method in this setting, ModelSoup (Wortsman et al., 2022) greedily averages the models fine-tuned with different hyperparameter configurations to improve the utility and robustness of the model. In addition, when training on different datasets or tasks, model fusion can be used to improve out-of-domain generalization or multitasking of the model (Matena & Raffel, 2022; Choshen et al., 2022; Li et al., 2022; Jin et al., 2023; Zhou et al., 2024), especially for language models. A state-of-the-art merging algorithm, RegMean (Jin et al., 2023), successfully merges language models fine-tuned over different tasks and improves the model's out-of-distribution generalization. Moreover, model fusion plays a pivotal role in federated learning (Konečný et al., 2016; McMahan et al., 2017; Wang et al., 2020) when the local updates are collected to make a global update. FedAvg (McMahan et al., 2017) is a classical merging algorithm that directly computes the average of the local models as the updated global model. Recent studies propose to incorporate the permutation symmetry to align the neurons of different end models (Wang et al., 2020; Singh & Jaggi, 2020; Ainsworth et al., 2023). However, these methods fail to achieve a desirable performance when tackling transformer-based models (Jin et al., 2023).
+
+# 7. Conclusion
+
+In this paper, we introduced rotation symmetry as a novel type of parameter space symmetry for transformers, extending the concept of permutation symmetry to continuous spaces. Building on this foundation, we proposed a theoretically optimal parameter matching algorithm to enhance the fusion of transformer models in a plug-and-play manner. To validate our approach, we conducted extensive experiments on real-world NLP and vision benchmarks. The results demonstrated that incorporating rotation symmetry effectively and efficiently facilitates transformer model fusion, showcasing our method's practical utility.
+
+# Impact Statement
+
+Our study provides novel insights into the parameter space symmetry of transformers and establishes a practical framework for advancing model fusion techniques. Bridging theoretical innovations with practical applications, this work reveals the potential for leveraging parameter space symmetries in deep learning research. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
+# Acknowledgment
+
+This work is supported in part by the National Science Foundation (NSF) under grants IIS-2006844, IIS-2144209, IIS-2223769, CNS-2154962, BCS-2228534, and CMMI-2411248; the Office of Naval Research (ONR) under grant N000142412636; and the Commonwealth Cyber Initiative (CCI) under grant VV-1Q24-011.
+
+# References
+
+Ainsworth, S., Hayase, J., and Srinivasa, S. Git re-basin: Merging models modulo permutation symmetries. In International Conference on Learning Representations, 2023.  
+Armenta, M. and Jodoin, P.-M. The representation theory of neural networks. Mathematics, 9(24):3216, 2021.  
+Arpit, D., Wang, H., Zhou, Y., and Xiong, C. Ensemble of averages: Improving model selection and boosting performance in domain generalization. Advances in Neural Information Processing Systems, 35:8265-8277, 2022.  
+Ba, J. L., Kiros, J. R., and Hinton, G. E. Layer normalization. arXiv preprint arXiv:1607.06450, 2016.  
+Badrinarayanan, V., Mishra, B., and Cipolla, R. Symmetry-invariant optimization in deep networks. arXiv preprint arXiv:1511.01754, 2015.  
+Benton, G., Maddox, W., Lotfi, S., and Wilson, A. G. G. Loss surface simplexes for mode connecting volumes and fast ensembling. In International Conference on Machine Learning, pp. 769-779, 2021.  
+Bostan, L.-A.-M. and Klinger, R. An analysis of annotated corpora for emotion classification in text. In Bender, E. M., Derczynski, L., and Isabelle, P. (eds.), Proceedings of the 27th International Conference on Computational Linguistics, pp. 2104-2119, Santa Fe, New Mexico, USA, August 2018. Association for Computational Linguistics. URL https://aclanthology.org/C18-1179.  
+Brea, J., Simsek, B., Illing, B., and Gerstner, W. Weight-space symmetry in deep networks gives rise to permuta
+
+tion saddles, connected by equal-loss valleys across the loss landscape. arXiv preprint arXiv:1907.02911, 2019.  
+Brown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J. D., Dhariwal, P., Neelakantan, A., Shyam, P., Sastry, G., Askell, A., et al. Language models are few-shot learners. Advances in neural information processing systems, 33: 1877-1901, 2020.  
+Burkard, R. E. and Cela, E. Linear assignment problems and extensions. In Handbook of combinatorial optimization: Supplement volume A, pp. 75-149. 1999.  
+Cha, J., Chun, S., Lee, K., Cho, H.-C., Park, S., Lee, Y., and Park, S. Swad: Domain generalization by seeking flat minima. Advances in Neural Information Processing Systems, 34:22405-22418, 2021.  
+Choshen, L., Venezuelan, E., Slonim, N., and Katz, Y. Fusing finetuned models for better pretraining. arXiv preprint arXiv:2204.03044, 2022.  
+Clark, K., Luong, M.-T., Le, Q. V., and Manning, C. D. Electra: Pre-training text encoders as discriminators rather than generators. In International Conference on Learning Representations, 2020.  
+Daheim, N., Möllenhoff, T., Ponti, E., Gurevych, I., and Khan, M. E. Model merging by uncertainty-based gradient matching. In International Conference on Learning Representations, 2024.  
+Devlin, J., Chang, M.-W., Lee, K., and Toutanova, K. Bert: Pre-training of deep bidirectional transformers for language understanding. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pp. 4171-4186, 2019.  
+Dietterich, T. G. Ensemble methods in machine learning. In International workshop on multiple classifier systems, pp. 1-15, 2000.  
+Dong, X., Yu, Z., Cao, W., Shi, Y., and Ma, Q. A survey on ensemble learning. Frontiers of Computer Science, 14: 241-258, 2020.  
+Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., Uszkoreit, J., and Houlsby, N. An image is worth 16x16 words: Transformers for image recognition at scale, 2021. URL https://arxiv.org/abs/2010.11929.  
+Du, S. S., Hu, W., and Lee, J. D. Algorithmic regularization in learning deep homogeneous models: Layers are automatically balanced. Advances in neural information processing systems, 31, 2018.
+
+Entezari, R., Sedghi, H., Saukh, O., and Neyshabur, B. The role of permutation invariance in linear mode connectivity of neural networks. In International Conference on Learning Representations, 2022.  
+Ferbach, D., Goujaud, B., Gidel, G., and Dieuleveut, A. Proving linear mode connectivity of neural networks via optimal transport. In International Conference on Artificial Intelligence and Statistics, pp. 3853-3861, 2024.  
+Fukumizu, K. and Amari, S.-i. Local minima and plateaus in hierarchical structures of multilayer perceptrons. Neural networks, 13(3):317-327, 2000.  
+Glorot, X., Bordes, A., and Bengio, Y. Deep sparse rectifier neural networks. In Proceedings of the fourteenth international conference on artificial intelligence and statistics, pp. 315-323, 2011.  
+Godfrey, C., Brown, D., Emerson, T., and Kvinge, H. On the symmetries of deep learning models and their internal representations. Advances in Neural Information Processing Systems, 35:11893-11905, 2022.  
+Gower, J. C. and Dijksterhuis, G. B. Procrustes problems, volume 30. OUP Oxford, 2004.  
+Grigsby, E., Lindsey, K., and Rolnick, D. Hidden symmetries of relu networks. In International Conference on Machine Learning, pp. 11734-11760, 2023.  
+Gupta, V., Serrano, S. A., and DeCoste, D. Stochastic weight averaging in parallel: Large-batch training that generalizes well. In International Conference on Learning Representations, 2020.  
+Hammoud, H., Michieli, U., Pizzati, F., Torr, P., Bibi, A., Ghanem, B., and Ozay, M. Model merging and safety alignment: One bad model spoils the bunch. In Findings of the Association for Computational Linguistics: EMNLP 2024, pp. 13033-13046, 2024.  
+He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016.  
+He, P., Liu, X., Gao, J., and Chen, W. Deberta: Decoding-enhanced bert with disentangled attention, 2021.  
+He, Y., Zheng, Z., Soga, P., Zhu, Y., Dong, Y., and Li, J. Explaining graph neural networks with large language models: A counterfactual perspective on molecule graphs. In Findings of the Association for Computational Linguistics: EMNLP 2024, pp. 7079-7096, 2024.  
+Hecht-Nielsen, R. On the algebraic structure of feedforward network weight spaces. In Advanced Neural Computers, pp. 129-135. 1990.
+
+Hovy, E., Marcus, M., Palmer, M., Ramshaw, L., and Weischedel, R. OntoNotes: The  $90\%$  solution. In Moore, R. C., Bilmes, J., Chu-Carroll, J., and Sanderson, M. (eds.), Proceedings of the Human Language Technology Conference of the NAACL, Companion Volume: Short Papers, pp. 57-60, New York City, USA, June 2006. Association for Computational Linguistics. URL https://aclanthology.org/N06-2015.  
+Huang, C., Ye, P., Chen, T., He, T., Yue, X., and Ouyang, W. Emr-merging: Tuning-free high-performance model merging. Advances in Neural Information Processing Systems, 37:122741-122769, 2024.  
+Imfeld, M., Graldi, J., Giordano, M., Hofmann, T., Anagnostidis, S., and Singh, S. P. Transformer fusion with optimal transport. In International Conference on Learning Representations, 2024.  
+Izmailov, P., Podoprikhin, D., Garipov, T., Vetrov, D., and Wilson, A. G. Averaging weights leads to wider optima and better generalization. In 34th Conference on Uncertainty in Artificial Intelligence (UAI), pp. 876-885, 2018.  
+Jin, X., Ren, X., Preotiuc-Pietro, D., and Cheng, P. Dataless knowledge fusion by merging weights of language models. In International Conference on Learning Representations, 2023.  
+Kabsch, W. A solution for the best rotation to relate two sets of vectors. Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography, 32(5):922-923, 1976.  
+Kalogeropoulos, I., Bouritsas, G., and Panagakis, Y. Scale equivariant graph metanetworks. Advances in neural information processing systems, 37:106800-106840, 2024.  
+Konečny, J., McMahan, H. B., Yu, F. X., Richtárik, P., Suresh, A. T., and Bacon, D. Federated learning: Strategies for improving communication efficiency. arXiv preprint arXiv:1610.05492, 2016.  
+Krizhevsky, A., Hinton, G., et al. Learning multiple layers of features from tiny images. 2009.  
+Kuhn, H. W. The hungarian method for the assignment problem. Naval research logistics quarterly, 2(1-2):83-97, 1955.  
+Kunin, D., Sagastuy-Brena, J., Ganguli, S., Yamins, D. L., and Tanaka, H. Neural mechanics: Symmetry and broken conservation laws in deep learning dynamics. In International Conference on Learning Representations, 2021.
+
+Lakshminarayanan, B., Pritzel, A., and Blundell, C. Simple and scalable predictive uncertainty estimation using deep ensembles. Advances in neural information processing systems, 30, 2017.  
+Lewis, M., Liu, Y., Goyal, N., Ghazvininejad, M., Mohamed, A., Levy, O., Stoyanov, V., and Zettlemoyer, L. Bart: Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pp. 7871-7880, 2020.  
+Li, M., Gururangan, S., Dettmers, T., Lewis, M., Althoff, T., Smith, N. A., and Zettlemoyer, L. Branch-train-merge: Embarrassingly parallel training of expert language models. In First Workshop on Interpolation Regularizers and Beyond at NeurIPS 2022, 2022.  
+Li, W., Peng, Y., Zhang, M., Ding, L., Hu, H., and Shen, L. Deep model fusion: A survey. arXiv preprint arXiv:2309.15698, 2023.  
+Lim, D., Maron, H., Law, M. T., Lorraine, J., and Lucas, J. Graph metanetworks for processing diverse neural architectures. In International Conference on Learning Representations, 2024a.  
+Lim, D., Putterman, T., Walters, R., Maron, H., and Jegelka, S. The empirical impact of neural parameter symmetries, or lack thereof. Advances in Neural Information Processing Systems, 37:28322-28358, 2024b.  
+Liu, Y., Ott, M., Goyal, N., Du, J., Joshi, M., Chen, D., Levy, O., Lewis, M., Zettlemoyer, L., and Stoyanov, V. Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692, 2019.  
+Liu, Z. Symmetry induces structure and constraint of learning. In International Conference on Machine Learning, 2024.  
+Liu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., Zhang, Z., Lin, S., and Guo, B. Swin transformer: Hierarchical vision transformer using shifted windows. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 10012-10022, 2021.  
+Lubana, E. S., Bigelow, E. J., Dick, R. P., Krueger, D., and Tanaka, H. Mechanistic mode connectivity. In International Conference on Machine Learning, pp. 22965-23004, 2023.  
+Martello, S. and Toth, P. Linear assignment problems. In North-Holland Mathematics Studies, volume 132, pp. 259-282. 1987.
+
+Matena, M. S. and Raffel, C. A. Merging models with fisher-weighted averaging. Advances in Neural Information Processing Systems, 35:17703-17716, 2022.  
+McMahan, B., Moore, E., Ramage, D., Hampson, S., and y Arcas, B. A. Communication-efficient learning of deep networks from decentralized data. In Artificial intelligence and statistics, pp. 1273-1282, 2017.  
+Meng, Q., Zheng, S., Zhang, H., Chen, W., Ye, Q., Ma, Z.-M., Yu, N., and Liu, T.-Y. G-sgd: Optimizing relu neural networks in its positively scale-invariant space. In International Conference on Learning Representations, 2019.  
+Nair, V. and Hinton, G. E. Rectified linear units improve restricted boltzmann machines. In International Conference on Machine Learning, pp. 807-814, 2010.  
+Navon, A., Shamsian, A., Achituve, I., Fetaya, E., Chechik, G., and Maron, H. Equivariant architectures for learning in deep weight spaces. In International Conference on Machine Learning, pp. 25790-25816, 2023.  
+Navon, A., Shamsian, A., Fetaya, E., Chechik, G., Dym, N., and Maron, H. Equivariant deep weight space alignment. In International Conference on Machine Learning, pp. 37376-37395, 2024.  
+Neyshabur, B., Salakhutdinov, R. R., and Srebro, N. Pathsgd: Path-normalized optimization in deep neural networks. Advances in neural information processing systems, 28, 2015.  
+Park, N. and Kim, S. How do vision transformers work?, 2022. URL https://arxiv.org/abs/2202.06709.  
+Pena, F. A. G., Medeiros, H. R., Dubail, T., Aminbeidokhti, M., Granger, E., and Pedersoli, M. Re-basin via implicit sinkhorn differentiation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 20237-20246, 2023.  
+Radford, A., Kim, J. W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pp. 8748-8763, 2021.  
+Raffel, C., Shazeer, N., Roberts, A., Lee, K., Narang, S., Matena, M., Zhou, Y., Li, W., and Liu, P. J. Exploring the limits of transfer learning with a unified text-to-text transformer. Journal of machine learning research, 21 (140):1-67, 2020.  
+Rame, A., Kirchmeyer, M., Rahier, T., Rakotomamonjy, A., Gallinari, P., and Cord, M. Diverse weight averaging for
+
+out-of-distribution generalization. Advances in Neural Information Processing Systems, 35:10821-10836, 2022.  
+Rossi, S., Singh, A., and Hannagan, T. On permutation symmetries in bayesian neural network posteriors: a variational perspective. Advances in Neural Information Processing Systems, 36, 2023.  
+Sagi, O. and Rokach, L. Ensemble learning: A survey. Wiley interdisciplinary reviews: data mining and knowledge discovery, 8(4):e1249, 2018.  
+Schönemann, P. H. A generalized solution of the orthogonal procrustes problem. Psychometrika, 31(1):1-10, 1966.  
+Simsek, B., Ged, F., Jacot, A., Spadaro, F., Hongler, C., Gerstner, W., and Brea, J. Geometry of the loss landscape in overparameterized neural networks: Symmetries and invariances. In International Conference on Machine Learning, pp. 9722-9732, 2021.  
+Singh, S. P. and Jaggi, M. Model fusion via optimal transport. Advances in Neural Information Processing Systems, 33:22045-22055, 2020.  
+Strang, G. Linear algebra and its applications. Academic Press, 1976.  
+Tatro, N., Chen, P.-Y., Das, P., Melnyk, I., Sattigeri, P., and Lai, R. Optimizing mode connectivity via neuron alignment. Advances in Neural Information Processing Systems, 33:15300-15311, 2020.  
+Tjong Kim Sang, E. F. and De Meulder, F. Introduction to the CoNLL-2003 shared task: Language-independent named entity recognition. In Proceedings of the Seventh Conference on Natural Language Learning at HLT-NAACL 2003, pp. 142-147, 2003. URL https://aclanthology.org/W03-0419.  
+Tran, V.-H., Vo, T. N., The, A. N., Huu, T. T., Nguyen-Nhat, M.-K., Tran, T., Pham, D.-T., and Nguyen, T. M. Equivariant neural functional networks for transformers. arXiv preprint arXiv:2410.04209, 2024.  
+Umeyama, S. Least-squares estimation of transformation parameters between two point patterns. IEEE Transactions on Pattern Analysis & Machine Intelligence, 13(04): 376-380, 1991.  
+Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., and Polosukhin, I. Attention is all you need. Advances in neural information processing systems, 30, 2017.  
+Wang, A., Singh, A., Michael, J., Hill, F., Levy, O., and Bowman, S. R. Glue: A multi-task benchmark and analysis platform for natural language understanding, 2019.
+
+Wang, H., Yurochkin, M., Sun, Y., Papailiopoulos, D., and Khazaeni, Y. Federated learning with matched averaging. In International Conference on Learning Representations, 2020.  
+Wei, X. and Liu, L. Are large language models good in-context learners for financial sentiment analysis? arXiv preprint arXiv:2503.04873, 2025.  
+Wortsman, M., Ilharco, G., Gadre, S. Y., Roelofs, R., Gontijo-Lopes, R., Morcos, A. S., Namkoong, H., Farhadi, A., Carmon, Y., Kornblith, S., et al. Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time. In International conference on machine learning, pp. 23965-23998, 2022.  
+Yadav, P., Tam, D., Choshen, L., Raffel, C. A., and Bansal, M. Ties-merging: Resolving interference when merging models. Advances in Neural Information Processing Systems, 36, 2023.  
+Yadav, P., Vu, T., Lai, J., Chronopoulou, A., Faruqui, M., Bansal, M., and Munkhdalai, T. What matters for model merging at scale? arXiv preprint arXiv:2410.03617, 2024.  
+Yang, E., Wang, Z., Shen, L., Liu, S., Guo, G., Wang, X., and Tao, D. Adamerging: Adaptive model merging for multi-task learning. In International Conference on Learning Representations, 2024.  
+Yun, S., Jeong, M., Kim, R., Kang, J., and Kim, H. J. Graph transformer networks. Advances in neural information processing systems, 32, 2019.  
+Yurochkin, M., Agarwal, M., Ghosh, S., Greenewald, K., Hoang, N., and Khazaeni, Y. Bayesian nonparametric federated learning of neural networks. In International conference on machine learning, pp. 7252-7261, 2019.  
+Zamir, G., Dokania, A., Zhao, B., and Yu, R. Improving learning to optimize using parameter symmetries. arXiv preprint arXiv:2504.15399, 2025.  
+Zhang, B., Chen, Z., Zheng, Z., Li, J., and Chen, H. Resolving editing-unlearning conflicts: A knowledge codebook framework for large language model updating. arXiv preprint arXiv:2502.00158, 2025.  
+Zhao, B., Dehmamy, N., Walters, R., and Yu, R. Symmetry teleportation for accelerated optimization. Advances in neural information processing systems, 35:16679-16690, 2022.  
+Zhao, B., Ganev, I., Walters, R., Yu, R., and Dehmamy, N. Symmetries, flat minima, and the conserved quantities of gradient flow. In The Eleventh International Conference on Learning Representations, 2023.
+
+Zhao, B., Gower, R. M., Walters, R., and Yu, R. Improving convergence and generalization using parameter symmetries. In International Conference on Learning Representations, 2024.  
+Zheng, Z., Dong, Y., Wang, S., Liu, H., Wang, Q., and Li, J. Kg-cf: Knowledge graph completion with context filtering under the guidance of large language models. In 2024 IEEE International Conference on Big Data (BigData), pp. 805-810, 2024.  
+Zhou, A., Yang, K., Burns, K., Cardace, A., Jiang, Y., Sokota, S., Kolter, J. Z., and Finn, C. Permutation equivariant neural functionals. Advances in neural information processing systems, 36:24966-24992, 2023a.  
+Zhou, A., Yang, K., Jiang, Y., Burns, K., Xu, W., Sokota, S., Kolter, J. Z., and Finn, C. Neural functional transformers. Advances in neural information processing systems, 36: 77485-77502, 2023b.  
+Zhou, H., Zhang, S., Peng, J., Zhang, S., Li, J., Xiong, H., and Zhang, W. Informer: Beyond efficient transformer for long sequence time-series forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pp. 11106-11115, 2021.  
+Zhou, Y., Song, L., Wang, B., and Chen, W. Metagpt: Merging large language models using model exclusive task arithmetic. In Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing, pp. 1711-1724, 2024.  
+Zhou, Z., Yang, Y., Yang, X., Yan, J., and Hu, W. Going beyond linear mode connectivity: The layerwise linear feature connectivity. Advances in Neural Information Processing Systems, 36, 2023c.  
+Zhu, Y., Wu, L., Zhang, B., Wang, S., Guo, Q., Hong, L., Simon, L., and Li, J. Understanding and modeling job marketplace with pretrained language models. In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management, pp. 5143-5150, 2024.  
+Ziyin, L., Wang, M., Li, H., and Wu, L. Parameter symmetry and noise equilibrium of stochastic gradient descent. arXiv preprint arXiv:2402.07193, 2024.  
+Ziyin, L., Xu, Y., and Chuang, I. L. Remove symmetries to control model expressivity and improve optimization. In International Conference on Learning Representations, 2025.
+
+# A. Proof
+
+Theorem 4.1. The following optimization problem has a closed-form solution.
+
+$$
+\min  _ {\boldsymbol {R} _ {1}, \boldsymbol {R} _ {2} \in \mathcal {R}} \left\| \left[ \begin{array}{l l} \boldsymbol {W} _ {Q _ {1}} ^ {\top} & \boldsymbol {W} _ {Q _ {2}} ^ {\top} \\ \boldsymbol {b} _ {Q _ {1}} & \boldsymbol {b} _ {Q _ {2}} \end{array} \right] \left[ \begin{array}{l} \boldsymbol {R} _ {1} \\ - \boldsymbol {R} _ {2} \end{array} \right] \right\| _ {F} ^ {2} + \left\| \left[ \begin{array}{l l} \boldsymbol {W} _ {K _ {1}} ^ {\top} & \boldsymbol {W} _ {K _ {2}} ^ {\top} \\ \boldsymbol {b} _ {K _ {1}} & \boldsymbol {b} _ {K _ {2}} \end{array} \right] \left[ \begin{array}{l} \boldsymbol {R} _ {1} \\ - \boldsymbol {R} _ {2} \end{array} \right] \right\| _ {F} ^ {2}. \tag {13}
+$$
+
+The solution is given by
+
+$$
+\boldsymbol {R} _ {1} = \boldsymbol {U} \boldsymbol {V} ^ {\top}, \boldsymbol {R} _ {2} = \boldsymbol {I}, \tag {14}
+$$
+
+where  $\pmb{I}$  is the identity matrix and  $\pmb{U} \pmb{\Sigma} \pmb{V}^{\top} = \pmb{W}_{Q_1} \pmb{W}_{Q_2}^{\top} + \pmb{W}_{K_1} \pmb{W}_{K_2}^{\top} + \pmb{b}_{Q_1}^{\top} \pmb{b}_{Q_2} + \pmb{b}_{K_1}^{\top} \pmb{b}_{K_2}$  is the result of eigendecomposition.
+
+Proof. We first show the process of converting the optimization problem to an Orthogonal Procrustes problem (Schönemann, 1966; Gower & Dijksterhuis, 2004). We can obtain that the optimization problem is equivalent to the following one.
+
+$$
+\min  _ {\boldsymbol {R} _ {1}, \boldsymbol {R} _ {2} \in \mathcal {R}} \left\| \boldsymbol {W} _ {Q _ {1}} ^ {\top} \boldsymbol {R} _ {1} - \boldsymbol {W} _ {Q _ {2}} ^ {\top} \boldsymbol {R} _ {2} \right\| _ {F} ^ {2} + \left\| \boldsymbol {b} _ {Q _ {1}} \boldsymbol {R} _ {1} - \boldsymbol {b} _ {Q _ {2}} \boldsymbol {R} _ {2} \right\| _ {F} ^ {2} + \left\| \boldsymbol {W} _ {K _ {1}} ^ {\top} \boldsymbol {R} _ {1} - \boldsymbol {W} _ {K _ {2}} ^ {\top} \boldsymbol {R} _ {2} \right\| _ {F} ^ {2} + \left\| \boldsymbol {b} _ {K _ {1}} \boldsymbol {R} _ {1} - \boldsymbol {b} _ {K _ {2}} \boldsymbol {R} _ {2} \right\| _ {F} ^ {2}. \tag {15}
+$$
+
+Considering that  $\pmb{R}_1, \pmb{R}_2 \in \mathcal{R}$  and  $\pmb{R}_1, \pmb{R}_2$  are nonsingular matrices, we let  $\pmb{R} = \pmb{R}_1\pmb{R}_2^{-1}$  and convert Equation (15) to the following one:
+
+$$
+\min  _ {\boldsymbol {R}, \boldsymbol {R} _ {2} \in \mathcal {R}} \left\| \left(\boldsymbol {W} _ {Q _ {1}} ^ {\top} \boldsymbol {R} - \boldsymbol {W} _ {Q _ {2}} ^ {\top}\right) \boldsymbol {R} _ {2} \right\| _ {F} ^ {2} + \left\| \left(\boldsymbol {b} _ {Q _ {1}} \boldsymbol {R} - \boldsymbol {b} _ {Q _ {2}}\right) \boldsymbol {R} _ {2} \right\| _ {F} ^ {2} + \left\| \left(\boldsymbol {W} _ {K _ {1}} ^ {\top} \boldsymbol {R} - \boldsymbol {W} _ {K _ {2}} ^ {\top}\right) \boldsymbol {R} _ {2} \right\| _ {F} ^ {2} + \left\| \left(\boldsymbol {b} _ {K _ {1}} \boldsymbol {R} - \boldsymbol {b} _ {K _ {2}}\right) \boldsymbol {R} _ {2} \right\| _ {F} ^ {2}. \tag {16}
+$$
+
+As multiplying  $R_{2}$  preserves the Frobenius norm, we can remove the  $R_{2}$  terms from the objective and obtain an Orthogonal Procrustes problem as follows:
+
+$$
+\min  _ {\boldsymbol {R} \in \mathcal {R}} \left\| \boldsymbol {W} _ {Q _ {1}} ^ {\top} \boldsymbol {R} - \boldsymbol {W} _ {Q _ {2}} ^ {\top} \right\| _ {F} ^ {2} + \| \boldsymbol {b} _ {Q _ {1}} \boldsymbol {R} - \boldsymbol {b} _ {Q _ {2}} \| _ {F} ^ {2} + \left\| \boldsymbol {W} _ {K _ {1}} ^ {\top} \boldsymbol {R} - \boldsymbol {W} _ {K _ {2}} ^ {\top} \right\| _ {F} ^ {2} + \| \boldsymbol {b} _ {K _ {1}} \boldsymbol {R} - \boldsymbol {b} _ {K _ {2}} \| _ {F} ^ {2}. \tag {17}
+$$
+
+We take a look at the first term  $\left\| W_{Q_1}^\top R - W_{Q_2}^\top \right\| _F^2$  and have:
+
+$$
+\begin{array}{l} \min  _ {\boldsymbol {R} \in \mathcal {R}} \left\| \boldsymbol {W} _ {Q _ {1}} ^ {\top} \boldsymbol {R} - \boldsymbol {W} _ {Q _ {2}} ^ {\top} \right\| _ {F} ^ {2} \\ = \min  _ {\boldsymbol {R} \in \mathcal {R}} \left\langle \boldsymbol {W} _ {Q _ {1}} ^ {\top} \boldsymbol {R} - \boldsymbol {W} _ {Q _ {2}} ^ {\top}, \boldsymbol {W} _ {Q _ {1}} ^ {\top} \boldsymbol {R} - \boldsymbol {W} _ {Q _ {2}} ^ {\top} \right\rangle_ {F} \\ = \max  _ {\boldsymbol {R} \in \mathcal {R}} \left\langle \boldsymbol {W} _ {Q _ {1}} ^ {\top} \boldsymbol {R}, \boldsymbol {W} _ {Q _ {2}} ^ {\top} \right\rangle_ {F} \tag {18} \\ = \max  _ {\boldsymbol {R} \in \mathcal {R}} \operatorname {t r} \left(\boldsymbol {R} ^ {\top} \boldsymbol {W} _ {Q _ {1}} \boldsymbol {W} _ {Q _ {2}} ^ {\top}\right) \\ = \max  _ {\boldsymbol {R} \in \mathcal {R}} \left\langle \boldsymbol {R}, \boldsymbol {W} _ {Q _ {1}} \boldsymbol {W} _ {Q _ {2}} ^ {\top} \right\rangle_ {F}. \\ \end{array}
+$$
+
+Similarly, we can convert Equation (17) to the following one:
+
+$$
+\max  _ {\boldsymbol {R} \in \mathcal {R}} \left\langle \boldsymbol {R}, \boldsymbol {W} _ {Q _ {1}} \boldsymbol {W} _ {Q _ {2}} ^ {\top} + \boldsymbol {W} _ {K _ {1}} \boldsymbol {W} _ {K _ {2}} ^ {\top} + \boldsymbol {b} _ {Q _ {1}} ^ {\top} \boldsymbol {b} _ {Q _ {2}} + \boldsymbol {b} _ {K _ {1}} ^ {\top} \boldsymbol {b} _ {K _ {2}} \right\rangle_ {F}. \tag {19}
+$$
+
+We then conduct the singular value decomposition to the matrix  $\mathbf{W}_{Q_1}\mathbf{W}_{Q_2}^\top +\mathbf{W}_{K_1}\mathbf{W}_{K_2}^\top +\mathbf{b}_{Q_1}^\top \mathbf{b}_{Q_2} + \mathbf{b}_{K_1}^\top \mathbf{b}_{K_2} = \mathbf{U}\boldsymbol {\Sigma}\mathbf{V}^\top$  and have:
+
+$$
+\begin{array}{l} \max  _ {\boldsymbol {R} \in \mathcal {R}} \left\langle \boldsymbol {R}, \boldsymbol {W} _ {Q _ {1}} \boldsymbol {W} _ {Q _ {2}} ^ {\top} + \boldsymbol {W} _ {K _ {1}} \boldsymbol {W} _ {K _ {2}} ^ {\top} + \boldsymbol {b} _ {Q _ {1}} ^ {\top} \boldsymbol {b} _ {Q _ {2}} + \boldsymbol {b} _ {K _ {1}} ^ {\top} \boldsymbol {b} _ {K _ {2}} \right\rangle_ {F} \\ = \max  _ {\boldsymbol {R} \in \mathcal {R}} \operatorname {t r} \left(\boldsymbol {V} \boldsymbol {\Sigma} \boldsymbol {U} ^ {\top} \boldsymbol {R}\right) \tag {20} \\ = \max  _ {\boldsymbol {R} \in \mathcal {R}} \operatorname {t r} \left(\boldsymbol {\Sigma} \boldsymbol {U} ^ {\top} \boldsymbol {R} \boldsymbol {V}\right) \\ = \operatorname {t r} (\boldsymbol {\Sigma}), \\ \end{array}
+$$
+
+where  $\boldsymbol{U}^{\top}\boldsymbol{R}\boldsymbol{V} = \boldsymbol{I}$ , i.e.,  $\boldsymbol{R} = \boldsymbol{U}\boldsymbol{V}^{\top}$ . Finally, we incorporate  $\boldsymbol{R} = \boldsymbol{U}\boldsymbol{V}^{\top}$  into  $\boldsymbol{R} = \boldsymbol{R}_1\boldsymbol{R}_2^{-1}$  and let  $\boldsymbol{R}_2 = \boldsymbol{I}$  for simplicity. Subsequently, we have  $\boldsymbol{R}_1 = \boldsymbol{U}\boldsymbol{V}^{\top}$  and  $\boldsymbol{R}_2 = \boldsymbol{I}$  as a closed-form solution of the optimization problem in Theorem 4.1.
+
+![](images/4785e5da74a3a0ec678c830cfa80a6f9e8c3b1978a962e83c2b7e1f0b56f87dc.jpg)
+
+From the proof, we can observe that the optimization problem in Theorem 4.1 has infinite pairs of solutions  $\{R_1, R_2\}$  regarding the value of  $R_2$ . In this paper, we let  $R_2 = I$  for simplicity, which makes model 2 an anchor model. However, the value of  $R_2$  can be further adjusted to benefit some data-dependent model fusion techniques (Singh & Jaggi, 2020; Jin et al., 2023). We leave this part to future work.
+
+# B. Comparison with Related Work
+
+Git Re-Basin (Ainsworth et al., 2023) introduces three model fusion algorithms: weight matching, activation matching, and straight-through matching, based on permutation symmetry. While these methods are effective for CNNs and MLPs, they do not easily extend to transformers due to the discrete nature of permutation symmetry. In a discrete space, the parameter matching problem is equivalent to solving a sum of bilinear assignment problems, which is NP-hard (Ainsworth et al., 2023). To address this limitation, we introduce rotation symmetry, which extends the symmetry space to a continuous domain, allowing for a closed-form solution to parameter matching. Unlike permutation symmetry, rotation symmetry provides a more flexible and efficient solution for aligning parameters in transformers.
+
+Additionally, our study emphasizes the advantage of weight matching as a plug-and-play module for model fusion, as it preserves model functionality while reducing the inner distance of end models after matching. OT-ACTS (Imfeld et al., 2024) also includes a parameter matching step, leveraging optimal transport for model fusion. However, its matching module is restricted to permutation operations, and its fusion method is limited to simple merging. In contrast, our approach generalizes parameter matching to rotation operations and integrates it with more advanced merging strategies, demonstrating the potential of parameter matching to enhance model fusion.
+
+A concurrent work by Tran et al. (2024) explores a similar form of rotation symmetry. However, their focus is on constructing functionally equivalent networks for transformers, whereas we focus on leveraging rotation symmetry for model fusion, proposing a theoretically optimal parameter matching algorithm specifically designed to improve fusion performance.
+
+# C. Experimental Details
+
+Datasets. Similar with (Jin et al., 2023), We employ emotion classification and named entity recognition (NER) as the tasks for the main experiments. The emotion classification datasets are extracted from (Bostan & Klinger, 2018). Five of them are selected as in-domain datasets (emoint, ssec, electoraltweets, grounded_emotions, affectivetext), and five others are good datasets (dailydialog, crowdflower, tec, tales-emotion, isear). For the NER task, we choose OntoNotes (Hovy et al., 2006) for model finetuning and CoNLL (Tjong Kim Sang & De Meulder, 2003) for out-of-domain evaluation. Additionally, in the ablation study and subset layers matching, we employ datasets (STS-B, SST-2, CoLA) from GLUE (Wang et al., 2019) to analyze the module performance.
+
+Baselines. We compared the performance of our method with two matching baselines:
+
+- OTFusion (Singh & Jaggi, 2020) mitigates the positional mismatch of functionally analogous neurons across end models by employing optimal transport to align neurons layer-wise, thus enabling more effective model merging.  
+- Git Re-Basin (Ainsworth et al., 2023) matches the parameters of end models through learned permutation symmetries in the parameter space, aiming to transform their parameters into a shared basin within the loss landscape and facilitate seamless merging.
+
+Meanwhile, to address RQ1, we further select six merging methods as baselines:
+
+- Simple (Wortsman et al., 2022) directly averages the weight matrices of independently pretrained end models.  
+- Fisher (Matena & Raffel, 2022) approximates each end model's posterior as a Gaussian distribution with a precision matrix derived from its Fisher information, thereby casting model merging as a Maximum Likelihood Estimation (MLE) problem over these posterior distributions.  
+- RegMean (Jin et al., 2023) formulates model merging as a regression problem, aiming to minimize the Euclidean distance between the predictions of the merged model and individual end models over a set of input samples.  
+- OT-WTS (Imfeld et al., 2024) extends OTFusion by introducing soft neuron alignment, which replaces the rigid one-to-one neuron matching with a more flexible, weighted correspondence between neurons. Subsequent model merging is then guided by these soft alignment weights.  
+- OT-ACTS and OT-ACTS-EMD (Imfeld et al., 2024) are two variants of OT-WTS that match neurons based on their
+
+activations rather than directly aligning parameter matrices. Specifically, OT-ACTS measures activation similarity using Euclidean distance, while OT-ACTS-EMD employs Earth Mover's Distance for a more fine-grained comparison.
+
+The code implementation is based on (Jin et al., 2023)² for NLP tasks and (Imfeld et al., 2024)³ for vision tasks.
+
+# D. Limitations
+
+In this study, we proposed a theoretically optimal parameter matching algorithm based on rotation symmetry. However, the optimality guarantees of our method are subject to several important prerequisites.
+
+First, our theoretical analysis in Theorem 4.1 focuses on the binary model matching case, where the optimal solution is independent of the choice of anchor model. This property, however, does not naturally extend to the multi-model setting. When matching multiple models, a straightforward extension involves selecting a single model as an anchor and aligning the others to it using pairwise rotation-based matching. Yet, in this case, the global optimality depends on the definition of the overall distance metric and may not be preserved. An alternative is iterative pairwise matching, but this introduces path dependency, i.e., the final result can vary depending on the order of merging. Extending our framework to the optimal multi-model matching problem remains an open challenge for future work.
+
+Second, the scope of our parameter matching algorithm is within the rotation symmetry, as defined in Equation (5). While this serves as a tractable and well-motivated instance of parameter space symmetry, we recognize that Transformers exhibit a richer set of structural symmetries beyond permutation, rotation, and rescaling. For example, recent work suggests that geometric properties of normalization and softmax layers may enlarge the symmetry set of previous model layers (Kunin et al., 2021; Lim et al., 2024a). Characterizing the complete symmetry group of transformers remains an open question.
+
+We also considered the possibility of extending rotation matrices to invertible matrices in terms of parameter space symmetries. Although invertible matrices expand the symmetry set, they are less suitable for practical parameter matching. Specifically, our closed-form solution in Equation (9) relies on the orthogonality of the rotation matrices. If replacing them with general invertible matrices, it becomes difficult to derive a closed-form solution for Equation (9), leading to suboptimal solution and increased complexity. Additionally, parameter matching requires computation of the inverse matrix of every  $\mathbf{R}$ , which is impractical for invertible matrices. In contrast, orthogonal matrices benefit from computationally efficient inverses (via transposition) and theoretical simplicity.
+
+Finally, we apply the rescaling symmetry after optimizing for rotation-based matching. However, our sequential approach (first optimizing rotation  $\mathbf{R}$ , then rescaling weight  $\alpha$ ) does not guarantee global optimality for the joint optimization over  $(\mathbf{R}, \alpha)$ . Deriving a globally optimal solution would require solving Equation (12) in closed form, which remains non-trivial. Hence, we consider our method as a practical approximation to the joint optimization solution which balances theoretical soundness with computational efficiency. Exploring a fully joint optimization framework for rotation and rescaling symmetries can be a promising direction for future research.

2025/Beyond the Permutation Symmetry of Transformers_ The Role of Rotation for Model Fusion/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6cd778ad85d02f0094ec6f3ff2420ee7c53d96a4a2e9dd0b9f88363021bc24cf
+size 589513

2025/Bi-perspective Splitting Defense_ Achieving Clean-Seed-Free Backdoor Security/1a2572fb-8fce-427d-a09e-f6c5c0d94ca3_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4d83993789d10523b2e3486334690ac263672c8cb5f4563bd364fc341c2cb047
+size 5754140

2025/Bi-perspective Splitting Defense_ Achieving Clean-Seed-Free Backdoor Security/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ecc000f8f3bc495f68980f0d6a437268944e196f1a0cf192bb2b9995b1476b2a
+size 1489755

2025/BiAssemble_ Learning Collaborative Affordance for Bimanual Geometric Assembly/332e0677-babf-4a85-9347-30c2a8d88e3c_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:50f3e3c73103b8b8f8178ffff2f289c8c71e7aed42275e83c0990fdcc283fe07
+size 13718826

2025/BiAssemble_ Learning Collaborative Affordance for Bimanual Geometric Assembly/full.md

@@ -0,0 +1,534 @@
+# BiAssemble: Learning Collaborative Affordance for Bimanual Geometric Assembly
+
+Yan Shen  $^{*12}$  Ruihai Wu  $^{*1}$  Yubin Ke  $^{1}$  Xinyuan Song  $^{1}$  Zeyi Li  $^{1}$  Xiaoqi Li  $^{12}$  Hongwei Fan  $^{12}$  Haoran Lu  $^{1}$  Hao Dong  $^{12}$
+
+# Abstract
+
+Shape assembly, the process of combining parts into a complete whole, is a crucial robotic skill with broad real-world applications. Among various assembly tasks, geometric assembly—where broken parts are reassembled into their original form (e.g., reconstructing a shattered bowl)—is particularly challenging. This requires the robot to recognize geometric cues for grasping, assembly, and subsequent bimanual collaborative manipulation on varied fragments. In this paper, we exploit the geometric generalization of point-level affordance, learning affordance aware of bimanual collaboration in geometric assembly with long-horizon action sequences. To address the evaluation ambiguity caused by geometry diversity of broken parts, we introduce a real-world benchmark featuring geometric variety and global reproducibility. Extensive experiments demonstrate the superiority of our approach over both previous affordance-based and imitation-based methods. Project page: https://sites.google.com/view/biassembly/.
+
+# 1. Introduction
+
+Shape assembly, the task of assembling individual parts into a complete whole, is a critical skill for robots with wide-ranging real-world applications. This task can be broadly categorized into two main branches: furniture assembly (Zhan et al., 2020; Heo et al., 2023; Lee et al., 2021) and geometric assembly (Wu et al., 2023c; Sellan et al., 2022; Lu et al., 2024c). Furniture assembly focuses on combining functional components, such as chair legs and arms,
+
+*Equal contribution  ${}^{1}$  CFCS, School of Computer Science, Peking University  ${}^{2}$  PKU-Agibot Lab. Correspondence to: Hao Dong <hao.dong@pku.edu.cn>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+into a fully constructed piece, emphasizing both the functional role of each part and the overall structural design. In contrast, geometric assembly involves reconstructing broken objects, like piecing together parts of a shattered mug, to restore their original form. While furniture assembly has been relatively well-studied—ranging from computer vision tasks that predict part poses in the assembled object (Zhan et al., 2020) to robotic systems that assemble parts in both simulation (Ankile et al., 2024; Yu et al., 2021; Wang et al., 2022a) and real-world environments (Heo et al., 2023; Suárez-Ruiz et al., 2018; Xian et al., 2017)—geometric assembly remains under-explored despite its significant potential for real-world applications (Sellán et al., 2022; Lu et al., 2024b), such as repairing broken household items, reconstructing archaeological artifacts (Papaioannou & Karabassi, 2003), assembling irregularly shaped objects in industrial tasks, aligning bone fragments in surgery (Liu et al., 2014), and reconstructing fossils in paleontology (Clarke et al., 2005).
+
+Previous works on geometric assembly primarily focused on generating physically plausible broken parts through precise physics simulations in the graphics domain (Sellán et al., 2022; 2023), and estimating the target assembled part poses based on observations in the computer vision domain (Wu et al., 2023c; Lu et al., 2024c; Qi et al., 2025). These studies only consider the geometries and ideal assembled poses of broken parts, dismissing the process of step-by-step assembling parts to the complete shape. However, different from opening a door or closing a drawer, only with the ideal part poses, it is difficult for a robot to directly and successfully manipulate broken parts to the complete shape.
+
+The challenges of the above robotic geometric shape assembly task mainly come from the exceptionally large observation and action spaces. For the observation space, the broken parts have arbitrary geometries, and the grasps on the object surface should consider not only the local geometry itself, but also whether grasping on such point can afford the subsequent bimanual assembly actions. For the action space, as illustrated in Figure 1, it requires long-horizon action trajectories. Given the contact-rich nature of the task, where collisions among the two parts and two robots will easily exist, the actions should be fine-grained and aware
+
+![](images/2e47e2a7a1c688a2a8c70865ae589f6870fe1ddee4f8508082f360c39583823d.jpg)  
+Figure 1. (A) Direct learning long-horizon action trajectories of geometric assembly may face many challenges: grasping ungraspable points, grasping points not suitable for assembly (e.g., seams of fragments), robot colliding with parts and the other robot. (B) We formulate this task into 3 steps: pick-up, alignment and assembly. For assembly, we predict the direction that will not result in part collisions. For alignment, we transformed any assembled poses to pose easy for the robot to manipulate from the initial poses without collisions. For pick-up, we learn point-level affordance aware of graspness and the following 2 steps. (C) Real-World Evaluations with affordance predictions on two mugs and the corresponding manipulation.
+
+of bimanual collaboration. Consequently, the policy must account for geometry, contact-rich assembly processes, and bimanual coordination.
+
+We propose our BiAssemble framework for this challenging task. For geometric awareness, we utilize point-level affordance, which is trained to focus on local geometry. This approach has demonstrated strong geometric generalization in diverse tasks (Wu et al., 2022; 2023b), including short-term bimanual manipulation (Zhao et al., 2022), such as pushing a box or lifting a basket. To enhance the affordance model with an understanding of subsequent long-horizon bimanual assembly actions, we draw inspiration from how humans intuitively assemble fragments: after picking up two fragments, we align them at the seam, deliberately leaving a gap (since directly placing them in the target pose often causes
+
+geometric collisions), with part poses denoted as alignment poses. We then gradually move the fragments toward each other to fit them together precisely. The alignment poses of the two fragments can be obtained by disassembling assembled parts in opposite directions. With this information, it becomes straightforward to extend the geometry-aware affordance to further be aware of whether the controller can move fragments into the alignment poses without collisions.
+
+We develop a simulation environment where robots can be controlled to assemble broken parts. This simulation environment bridges the gap between vision-based pose prediction for broken parts and the real-world robotic geometric assembly. Moreover, since broken parts exhibit varied geometries (e.g., the same bowl falling from different heights breaking into different groups of fragments), it is challeng-
+
+ing to fairly assess policy performance in real-world settings. To address this, we further introduce a real-world benchmark featuring globally available objects with reproducible broken parts, along with their corresponding 3D meshes, which can be integrated into simulation environment. This benchmark enables consistent and fair evaluation of robotic geometric assembly policies. Extensive experiments on diverse categories demonstrate the superiority of our method both quantitatively and qualitatively.
+
+# 2. Related Work
+
+# 2.1.3D Shape Assembly
+
+Shape assembly is a well-established problem in visual manipulation, with many studies focusing on constructing a complete shape from given parts, typically involving the pose prediction of each part for accurate placement (Zhan et al., 2020; Wang et al., 2022a). Further work (Heo et al., 2023; Tian et al., 2022; Jones et al., 2021; Willis et al., 2022; Tie et al., 2025) studied assembly with robotic execution, requiring robots to carry out each step, with benchmarks spanning various applications from home furniture assembly (Lee et al., 2021) to factory-based nut-and-bolt interactions (Narang et al., 2022). We categorize these tasks into two types: furniture assembly and geometric assembly. This paper focuses on geometric assembly, where pieces are irregular and lack semantic definitions, such as in the case of a broken bowl, making categorization difficult. This contrasts with furniture assembly that involves parts like nuts, bolts, and screws, each with specific functions and clear categorization.
+
+Previous work on geometric assembly (Sellán et al., 2022; Chen et al., 2022; Lu et al., 2024c), includes Wu et al. (2023c), which learns SE(3)-equivariant part representations by capturing part correlations for assembly, and Lee et al. (2024), which introduces a low-complexity, high-order feature transform layer that refines feature pair matching. However, these methods primarily focus on synthesizing parts into a cohesive object based on pose considerations without incorporating robotic execution, which is impractical in real-world scenarios where collisions may occur if the assembly process ignores actions. To tackle this challenge, we introduce a robotic bimanual geometric assembly framework that leverages two robots to collaboratively assemble pieces, enhancing stability in real-world execution.
+
+# 2.2. Bimanual Manipulation
+
+Bimanual manipulation (Chen et al., 2023; Grannen et al., 2023; Mu et al., 2021; Chitnis et al., 2020; Lee et al., 2015; Xie et al., 2020; Ren et al., 2024b; Liu et al., 2024; 2022; Li et al., 2023; Mu et al., 2024) offers several advantages, particularly in tasks requiring stable control or wide action
+
+space. Current research primarily focuses on planning and collaboration. ACT (Fu et al., 2024; Zhao et al., 2023) introduces a transformer-based encoder-decoder architecture that leverages semantic knowledge from image inputs to predict bimanual actions. PerAct2 (Grotz et al., 2024) learns features at both voxel and language levels, utilizing shared and private transformer blocks to coordinate two robotic arms based on semantic instructions. However, in tasks rich in geometric complexity, where objects have limited semantic information but intricate geometric structures, these approaches—focused on semantic understanding—may encounter generalization limits. DualAfford (Zhao et al., 2022) learns point-level collaborative visual affordance, while only for short-term tasks like pushing or rotating. To address this, we leverage the geometric generalization capability of point-level affordance, and enhance it with the awareness of subsequent long-horizon assembly actions.
+
+# 2.3. Visual Affordance for Robotic Manipulation
+
+Among various vision-based approaches for robotic manipulation (Goyal et al., 2023; Brohan et al., 2023; Ze et al., 2024; Ju et al., 2024), for objects with rich geometric information and tasks requiring geometric generalization, point-level affordance, which reflects the functionality of each point for downstream manipulation (Mo et al., 2021; Li et al., 2024a), is broadly leveraged and can easily generalize to novel shapes with similar local geometries. A series of research have leverage this representation to a broad range of robotic manipulation tasks, such as deformable object manipulation (Wu et al., 2023b; Lu et al., 2024a; Wu et al., 2024), object manipulation in complex environments (Ding et al., 2024; Li et al., 2024b; Wu et al., 2023a), object manipulation with efficient exploration (Ning et al., 2024; Wang et al., 2022b), and short-term bimanual manipulation (Zhao et al., 2022). Leveraging the strengths of affordance representation, we design a sophisticated approach that incorporates this representation into bimanual geometric assembly task requiring long-horizon fine-grained actions, enhancing generalization and enabling more effective collaboration in addressing long-horizon geometric assembly challenges.
+
+# 3. Problem Formulation
+
+The task is to use two grippers to assemble a pair of 3D fractured parts initialized in random poses on the table. A camera situated in front of the table captures a partially scanned point cloud  $O$ . Given  $O$ , current state-of-the-art methods (Scarpellini et al., 2024; Lu et al., 2024c; Chen et al., 2022) can imagine the assembled part poses and the assembled object  $S$  in any pose. Thus we assume taking imaginary assembled shape  $S$  as the input.
+
+For the policy  $\pi$ , as illustrated in Figure 1, we can simplify this long-term process into 3 key steps:
+
+![](images/c991c6da8f388b349d730c1240577d43135f42e45ec2edcec9d73d92c6262a73.jpg)  
+Figure 2. Framework Overview. With the point cloud observation and Imaginary Assembled Shape, the model predicts the disassembly direction in which the disassembled part poses can be easily reached by manipulating the raw parts under the guidance Bi-Affordance.
+
+- Pick-up: the two grippers pick up the fractured parts with actions  $(g_1^{pick}, g_2^{pick})$ ;  
+- Alignment: grippers carry parts to alignment poses with actions  $(g_{1}^{align}, g_{2}^{align})$ , positioning part seams to face each other and ensuring precise alignment for a perfect assembly;  
+Assembly: grippers move forward to complete the assembly with actions  $(g_1^{asm}, g_2^{asm})$ .
+
+Here, 1 and 2 denote the left and the right grippers, respectively. Each gripper action  $g$  is formulated as an  $SE(3)$  matrix, representing the gripper pose in 3D space.
+
+# 4. Method
+
+# 4.1. Overview
+
+Our BiAssembly framework is designed to predict collaborative affordance and gripper actions for bimanual geometric shape assembly. First, to propose the assembly direction on two aligned parts, we develop the Disassembly Predic
+
+tor to learn the feasible disassembly directions in which the opposite assembly direction will result in no collisions, based on the fracture geometry of the imaginary assembled shape in any pose (4.2). Next, we design the Transformation Predictor, to transform disaassembled parts to poses where the controller can successfully manipulate the initial parts to these alignment poses (4.3). Based on the predicted part alignment poses, we propose the BiAffordance Predictor, which not only predicts where to grasp the fractured parts, but also considers the subsequent collaborative alignment and assembly steps (4.4). Finally, we explain training strategy and loss functions (4.6).
+
+# 4.2. Disassembly Prediction Based on Shape Geometry
+
+The set of feasible disassembly directions (in which the disassembly and opposite assembly processes will not result in collisions) is an inherent attribute of a pair of fractured parts, determined by fracture geometries. Therefore, we predict the disassembly directions, from the object-centric perspective, on the imaginary assembled shape  $S$  in any
+
+pose. Additionally, we observe that when fractured parts rotate, the feasible disassembly directions will rotate correspondingly, maintaining SO(3) equivariance relative to part poses. This SO(3) equivariance property is advantageous for disentangling shape geometry from shape poses, as demonstrated in previous works (Wu et al., 2023c; Scarpellini et al., 2024). Therefore, we adopt VN-DGCNN (Wu et al., 2023c; Deng et al., 2021) to encode the imaginary assembled shape parts  $S$  and acquire the SO(3)-equivariant shape feature  $f_{s}$ .
+
+Inputting the equivariant representation  $f_{s}$ , we use the Disassembly Predictor to predict the distribution of disassembly directions. Concretely, the Disassembly Predictor is implemented as a conditional variational autoencoder (cVAE) (Sohn et al., 2015), where the cVAE encoder maps the input disassembly direction  $v$  into Gaussian noise  $z \in \mathbb{R}^{32}$ , and the cVAE decoder reconstructs the disassembly direction  $v$  from  $z$ , with  $f_{s}$  as the condition.
+
+# 4.3. Transformation Prediction For Alignment Pose
+
+Given the object-centric disassembly direction resulting in no collisions in the last-step assembly, we want to predict the alignment poses, where the robot can manipulate two parts from the initial poses to the alignment poses without collisions, and then robot can execute the assembly step. This problem can be formulated as predicting an SE(3) transformation  $M \in \mathbb{R}^{4 \times 4}$  that is applied to the combination of the imaginary assembled shape  $S$  and the disassembly direction  $v$ . To capture this, we adopt PointNet++ (Qi et al., 2017a;b) to encode the initial point cloud observation  $O$  into the global feature  $f_{O}$ . We also employ an MLP to encode disassembly direction  $v$  into feature  $f_{v}$ . The transformation predictor, which is implemented as a cVAE, takes in concatenation of  $(f_{O}, f_{v})$  to predict the SE(3) transformation  $M$ . The yellow and blue arrow in Fig. 2 illustrate the data flow in this process: by applying the transformation  $M$  to the imaginary assembly  $S$  and disassembly direction  $v$ , we obtain the transformed  $S'$  and  $v'$ . Therefore, we can get the target poses to which the objects should be moved during the alignment and assembly phases.
+
+# 4.4. BiAffordance Predictor
+
+We build the BiAffordance Predictor to propose actions in the Pick-up step, indicating where to grasp the two fractured parts that can facilitate the whole long-horizon robotic assembly task. The BiAffordance Predictor should both identify easy-to-grasp regions on fractured parts and consider the subsequent Alignment and Assembly steps. This means (1) avoiding grasping regions in the seam and (2) preventing each gripper from adopting poses that could collide with the other part or gripper during subsequent steps.
+
+Following DualAfford (Zhao et al., 2022), we disentangle the bimanual task into two conditional submodules. As
+
+presented in Fig. 2 (bottom), during inference, the BiAffordance Predictor conditionally predicts two gripper actions. The first Affordance Network generates the affordance map for the first gripper, highlighting the actionable regions for the bimanual assembly task, and we select a contact point  $p_1^*$  with high actionable value. Then, the Actor Network predicts the gripper orientation  $r_1$  for interaction at  $p_1^*$ . Based on the first action  $g_1 = (p_1^*, r_1)$ , we can then predict the second gripper action  $g_2 = (p_2^*, r_2)$  using the second Affordance Network and Actor Network.
+
+Different from DualAfford that only predicts affordance for short-term tasks, we use whether the manipulation points can satisfy the following alignment pose (by the robotic controller) and the assembly step as the training signal.
+
+To encode input information, one PointNet++ encodes the initial point cloud  $O$  obtains per-point features  $\{f_p\}$ . Another PointNet++ encodes the transformed shape  $S^{\prime}$  and derives the global feature  $f_{s^{\prime}}$ . Additionally, an MLP encodes the transformed disassembly direction  $v^{\prime}$  into  $f_{v^{\prime}}$ .
+
+For Affordance and Actor Networks' designs, the first Affordance Network is implemented as an MLP that receives the concatenated features  $(f_{p}, f_{S'}, f_{v'})$  and predicts an affordance score in the range of  $[0, 1]$  for each point  $p$ . By aggregating the affordance scores of all points, we obtain the first affordance map, and from which we select  $p_1^*$ . The first Actor Network is implemented as a cVAE that takes the concatenated features  $(f_{p_1^*}, f_{S'}, f_{v'})$  as condition, and outputs the gripper orientation  $r_1$ . The design of the second Affordance Network and Actor Network follows a similar structure, with the difference that they additionally incorporate the first gripper action's feature  $(f_{p_1^*}, f_{r_1})$  along with  $(f_{p}, f_{S'}, f_{v'})$ . More details about the BiAffordance Predictor can be found in Appendix C.
+
+# 4.5. Alignment and Assembly Actions
+
+After successfully grasping a part, we now predict the gripper alignment poses  $g_{i}^{align}$  and assembly poses  $g_{i}^{asm}$ , with  $i \in \{1,2\}$  denotes the gripper id. We assume the relative pose between the gripper and the object remains stable. For example, in the first pickup step and the third assembly step, the relative gripper-object pose remains consistent, as expressed in Equation 1:
+
+$$
+g _ {i} ^ {\text {p i c k}} \cdot q _ {i} ^ {\text {p i c k}} = g _ {i} ^ {\text {a s m}} \cdot q _ {i} ^ {\text {a s m}}; \quad g, q \in S E (3). \tag {1}
+$$
+
+Here,  $g$  and  $q$  denote gripper and object poses, respectively.
+
+Next, as we have the imaginary part shapes  $\mathcal{P}$  with pose  $q_{i}^{init}$ , we can utilize a pretrained pose estimation model (Wen et al., 2024) to predict the relative pose of  $q_{i}^{pick}$  with respect to  $q_{i}^{init}$ . Besides, by applying the predicted transformation  $M$  to  $\mathcal{P}$ , we obtain the target assembled part  $\mathcal{P}'$  and its pose as  $q_{i}^{asm} = M \cdot q_{i}^{init}$ . The gripper pose  $g_{i}^{pick}$
+
+![](images/59fa15baf318d073b3b8b1d7b9764c5f1b50762ada7112a28a3c948337ef080d.jpg)  
+Figure 3. Part A illustrates the pipeline for scanning and reconstructing real objects. Part B presents examples of fractured parts from various categories, showcasing diverse geometries.
+
+can be acquired from the robot control interface. Therefore, the gripper's final pose for assembling the parts can be calculated using Equation 2:
+
+$$
+g _ {i} ^ {a s m} = g _ {i} ^ {p i c k} \cdot q _ {i} ^ {p i c k} \cdot \left(q _ {i} ^ {i n i t}\right) ^ {- 1} \cdot M ^ {- 1}; \quad g, q \in S E (3). \tag {2}
+$$
+
+It is important to note that, as indicated in the above simplified equation, we do not need to define a canonical pose or try to obtain the values of  $q_{i}^{init}$ ; we only require the relative pose of  $q_{i}^{pick}$  to  $q_{i}^{init}$ .
+
+A similar relationship can be established between the first and the second intermediate steps, with the difference being that  $q_{i}^{align} = M \cdot q_{i}^{init} + v'$ .
+
+# 4.6. Training and Losses
+
+Disassembly Direction Loss. The Disassembly Predictor is implemented as cVAE. We apply Cosine Similarity Loss to measure the error between the reconstructed disassembly direction  $v$  and ground-truth  $v^{*}$ , and KL Divergence to measure the difference between two distributions:
+
+$$
+\mathcal {L} _ {D} = \mathcal {L} _ {C L S} (v, v ^ {*}) + D _ {K L} \left(q (z | v ^ {*}, f _ {s}) | | \mathcal {N} (0, 1)\right). \tag {3}
+$$
+
+Transformation Loss. The predicted SE(3) transformation matrix  $M$  consists of translation  $T$  and rotation  $R$ . Our model predicts the translation as a 3D-vector using L1 Loss. The rotation, represented as a SO(3) matrix, can be expressed as a 6D vector by using two 3D vectors that correspond to the directions of the two orthogonal axes. Consequently, ours model predicts the rotation as a 6D-vector and employs the geodesic loss. In summary, let  $T^{*}$  and  $R^{*}$  denote the ground-truth, and for simplicity, denote
+
+$D_{KL}\big(q(z|x,f)||\mathcal{N}(0,1)\big)$  as  $D(x,f)$ , and denote the concatenated features  $(f_s,f_v)$  as  $f_{sv}$ . The loss function is:
+
+$$
+\mathcal {L} _ {T} = \mathcal {L} _ {1} (T, T ^ {*}) + \mathcal {L} _ {g e o} (R, R ^ {*}) + D (T ^ {*}, f _ {s v}) + D (R ^ {*}, f _ {s v}).
+$$
+
+For the losses used in the Bi-Affordance Predictor, we provide detailed explanation in Appendix C.
+
+# 5. Benchmark
+
+# 5.1. Simulation Benchmark
+
+Constructing a large-scale dataset with real objects is both time-consuming and costly. To address this challenge, we utilize the Breaking Bad Dataset (Sellán et al., 2022), which models the natural destruction of objects into fragments. This dataset features multiple categories, diverse objects, and varying fracture patterns. For physics simulation, we employ the SAPIEN (Xiang et al., 2020) platform along with two two-finger Franka Panda grippers as robot actuators.
+
+We randomly select a pair of 3D fractured parts from a randomly chosen shape within a random category. The initial part poses are also randomized. Given the considerable diversity in fractured parts, collecting successful manipulation data for assembly can be quite challenging. To enhance data collection efficiency, we implement several heuristic strategies, with details in Appendix B.
+
+# 5.2. Real-World Benchmark
+
+A real-world benchmark is crucial not only for the evaluation of various methods but also for providing a standardized platform that enables researchers to reproduce and share solutions. Illustrated in Figure 3, we build the real-world benchmark by scanning with a smart phone camera. First,
+
+Table 1. Quantitative results for novel instances within training categories and for unseen categories.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="10">Novel Instances in Training Categories</td><td colspan="6">Novel Categories</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>AVG</td><td></td><td></td><td></td><td></td><td></td><td>AVG</td></tr><tr><td>ACT</td><td>2%</td><td>0%</td><td>0%</td><td>0%</td><td>1%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>0.30%</td><td>0%</td><td>1%</td><td>0%</td><td>0%</td><td>1%</td></tr><tr><td>Heuristic</td><td>5%</td><td>8%</td><td>0%</td><td>3%</td><td>2%</td><td>4%</td><td>3%</td><td>5%</td><td>10%</td><td>2%</td><td>4.20%</td><td>1%</td><td>5%</td><td>2%</td><td>0%</td><td>14%</td></tr><tr><td>SE(3)-Equiv</td><td>0%</td><td>0%</td><td>4%</td><td>0%</td><td>1%</td><td>5%</td><td>0%</td><td>7%</td><td>2%</td><td>11%</td><td>3.00%</td><td>4%</td><td>0%</td><td>2%</td><td>0%</td><td>2%</td></tr><tr><td>DualAfford</td><td>21%</td><td>17%</td><td>0%</td><td>2%</td><td>2%</td><td>4%</td><td>14%</td><td>8%</td><td>10%</td><td>6%</td><td>8.40%</td><td>5%</td><td>10%</td><td>4%</td><td>1%</td><td>16%</td></tr><tr><td>w/o SE(3)</td><td>59%</td><td>29%</td><td>13%</td><td>14%</td><td>11%</td><td>8%</td><td>15%</td><td>19%</td><td>24%</td><td>20%</td><td>21.20%</td><td>13%</td><td>24%</td><td>4%</td><td>9%</td><td>22%</td></tr><tr><td>Ours</td><td>60%</td><td>38%</td><td>13%</td><td>13%</td><td>12%</td><td>9%</td><td>26%</td><td>18%</td><td>27%</td><td>25%</td><td>24.10%</td><td>14%</td><td>31%</td><td>10%</td><td>7%</td><td>25%</td></tr><tr><td>w/ GT Target</td><td>71%</td><td>28%</td><td>4%</td><td>9%</td><td>9%</td><td>13%</td><td>27%</td><td>19%</td><td>25%</td><td>19%</td><td>22.40%</td><td>14%</td><td>33%</td><td>12%</td><td>9%</td><td>27%</td></tr></table>
+
+we put the object on an automatic turntable with 6 aruco markers around for precise camera localization, and capture a RGB video from a top-down view to a level view, lowering the height by one level for each 360-degree rotation. After capturing 4-5 levels, we uniformly sample around 300 frames, and feed them to COLMAP (Schönberger & Frahm, 2016; Schönberger et al., 2016) for estimating camera poses. Then, we use Grounded SAM 2 (Ren et al., 2024a; Ravi et al., 2024) to generate object masks and Depth Anything V2 (Yang et al., 2024) to predict monocular depths, and use SDFStudio (Yu et al., 2022; Wang et al., 2021) with depth ranking loss (Wang et al., 2023) to reconstruct object mesh. To annotate the ground-truth of scanned object assembly, we import the object slices to Blender (Community, 2018) and edit the object transformations.
+
+Our real-world benchmark encompasses a diverse range of object categories, including wine glass, plate, beer bottle, bowl, mug, and teapot. These objects have been primarily selected from well-known international brands, ensuring both durability and accessibility. To promote object diversity, our shapes vary in size, geometry, transparency, and texture, with different seam geometries.
+
+# 6. Experiments
+
+# 6.1. Simulation and Settings
+
+The simulation environment is built on the SAPIEN (Xiang et al., 2020) platform, utilizing the Franka Panda grippers as the robot actuator. We employ the EverydayColorPieces dataset from the Breaking Bad Dataset (Sellán et al., 2022), covering 15 categories, 445 shapes and 11,820 fragment pairs, with 10 categories for training and the remaining 5 for testing. Training categories are further divided into training shapes and novel instances, allowing the evaluation on generalization capabilities at both the object and category levels. More details can be found in Appendix A. For each method, we provide 7,000 positive and 7,000 negative samples. Negative samples encompass failures occurring during grasping, alignment, and assembly steps.
+
+# 6.2. Evaluation Metric, Baselines and Ablation
+
+Evaluation Metric. Our metric evaluates whether relative distance (measured in unit-length) and rotation angle (measured in degrees) of two parts are within the threshold range at the end of assembly. These thresholds ensure the success of assembly can be measured consistently and meaningfully. To evaluate each method, we prepare 100 samples in each category. For each sample, all methods are presented with the same initial observation for a fair comparison.
+
+Baselines and Ablations. We compare with four baselines and two ablated versions:
+
+(1) ACT (Zhao et al., 2023), a transformer network with action chunking that imitates successful action sequences in closed-loop manner. We enhance this method by providing depth information, object pose, and an additional target image as inputs. Besides, this baseline is trained and tested on individual categories, whereas other learning-based methods are trained on all training categories and evaluated on both novel instances and unseen categories.  
+(2) Heuristic, where we hand-engineer a set of heuristic strategies to improve manipulation success rate. These strategies are similar to the data collection heuristics (Appendix B).  
+(3) SE(3)-Equiv (Wu et al., 2023c) that learns SE(3)-equivariant part representations to estimate target assembled part poses in the computer vision domain. Though originally designed for assembled part pose predictions without considering robotic execution, we adapted it as a baseline by generating the pick-up actions of robots via heuristic strategies. Additionally, given the predicted pose  $q_{i}^{asm}$  for part  $i$ , we compute the corresponding gripper pose  $g_{i}^{asm}$  for assembly action, following Equation 1.  
+(4) DualAfford (Zhao et al., 2022) that learns collaborative visual affordance for bimanual manipulation. While it focuses on short-term manipulation, we adapt it to determine where to grasp the two fractured parts, using heuristic methods for alignment and assembly steps.
+
+![](images/e486745c2ff8c15f8e3b0095a11a1c85c0b52a07278ce9c4148858622f05e94f.jpg)  
+Figure 4. Simulation Experiment. We present qualitative results of the predicted affordance maps and robot actions from our method.
+
+(5) w/o SE(3), an ablation that replaces SO(3)-equivariant VN-DGCNN encoder with PointNet++.  
+(6) w/ GT target, where we provide additional ground-truth disassembly direction  $v$  and transformation  $M$ . Ground truth is sampled using a heuristic method that ensures at least one feasible assembly.
+
+# 6.3. Quantitative Results and Analysis
+
+Table 1 shows the success rate comparisons across different methods on both the novel instance dataset and the unseen category dataset. Our method outperforms the baselines and ablation models in most cases, demonstrating its effectiveness and geometric generalization capabilities.
+
+For ACT, though we provide additional input such as depth, object poses, and the goal image, it achieves lower scores on our task. Although ACT successfully picks up parts in approximately  $40\%$  of trials, it often fails during the alignment phase, with a misalignment of over  $100^{\circ}$  between the parts in many cases. Furthermore, ACT struggles to avoid grasping the fractured seam regions, leading to collisions during the assembly process. This is because the observation and action spaces in robotic geometric assembly are exceptionally large, making it challenging to directly learn the appropriate fine-grained actions.
+
+For heuristic, it achieves higher scores because we provide substantial ground-truth information in simulation. However, due to the significant diversity in both inter- and intracategory shape geometries, it is unrealistic to expect hand-engineered rules to generalize effectively across all shapes.
+
+For SE(3)-Equiv, the lower accuracy is primarily due to its disregard for the robotic execution process, which is a common limitation of prior visual assembly estimation methods. Specifically, these methods do not determine suitable grasping locations on the fragments, not only for successful pick-up but also for avoiding the seam region critical to subsequent assembly. They also lack the capability to align the fragments properly at the seam, significantly increasing the likelihood of collisions when the parts are brought together.
+
+For DualAfford, it performs better than the heuristic policy, demonstrating its superior ability to learn geometric-aware pick-up poses compared to heuristic sampling. However, only with designs focused on short-term manipulation, it still lacks awareness of the subsequent alignment and assembly steps.
+
+In comparison with the ablation models, our method also shows consistent improvements. Compared to the ablation w/o SE(3), we observe that utilizing the SE(3)-equivariant representation enhances the performance across most categories. Compared to the ablation w/ GT Target, our method performs better on novel instances but worse on novel categories. This suggests that, for the training categories, our method learns to predict a more accurate distribution of disassembly and transformation, surpassing those sampled from the heuristic strategy. However, on novel unseen categories, while our method still demonstrates generalization capability, the ablated version with the ground-truth target remains more effective.
+
+Appendix E presents additional experiments and analyses, including multi-fragment assembly, the use of "imperfect" imaginary assembly shape inputs, and further ablations.
+
+# 6.4. Qualitative Results and Analysis
+
+In Figure 4, we present collaborative affordance maps and robot manipulations predicted by our methods across multiple categories, including novel instances in training categories and unseen categories. Predicted affordance demonstrates an awareness of part geometry, highlighting graspable regions while avoiding areas near the table that could result in collisions between the gripper and surface. Additionally, the affordance accounts for subsequent alignment and assembly steps, avoiding seam areas that may cause collisions during the approach phase. Based on predicted affordance map, our model predicts appropriate gripper actions for assembling parts. Moreover, the results demonstrate the ability to generalize to unseen categories and shapes.
+
+![](images/bfa59aa387ccda12d988a57ee4567ce4a9f6064c300656ad4cae5a774549e9ca.jpg)  
+Figure 5. Real-World Experiment. We present the results of our model tested on real-world scans.
+
+# 6.5. Real-World Experiments
+
+We set up two Franka Panda with fractures positioned between them. An Azure Kinect camera is mounted in front of the robots, capturing partial 3D point cloud data as inputs for our models. The robots are controlled via the Robot Operating System (ROS) (Quigley et al., 2009), with control and communication managed through the frankapy library (Zhang et al., 2020). Communication with the Kinect Azure is facilitated by the pyk4a library (pyk4a, 2019).
+
+Figure 5 and the bottom row of Figure 1 present promising results by directly testing our method in real-world scenarios. Detailed manipulation processes are shown in Appendix F and videos are available in the supplementary materials. We observe that our model not only learns which regions of the fractured parts to grasp but also avoids manipulating areas near the fracture regions or too close to the table surface, reducing the likelihood of collisions during manipulation. The results from the real-world experiments demonstrate our model's capacity for generalization to real-world scenarios.
+
+# 7. Conclusion
+
+We leverage the geometric generalization capability of point-level affordance to develop a method that enables both generalization and collaboration in long-horizon geometric assembly tasks. To evaluate performance across diverse geometries, we introduced a real-world benchmark that features significant geometric variety and global reproducibility. Extensive experiments have shown that our approach outperforms previous methods, demonstrating its effectiveness in handling complex and long-horizon assembly tasks.
+
+# Impact Statement
+
+This work contributes to the development of shape assembly skills in future robots, specifically focusing on visual collaborative affordance predictions. These affordances can be visualized and analyzed prior to deployment, which may reduce potential risks and dangers. We do not anticipate any significant ethical concerns or societal consequences arising from our research.
+
+# Acknowledgements
+
+This paper is supported by National Natural Science Foundation of China (No. 62376006) and National Youth Talent Support Program (8200800081).
+
+# References
+
+Ankile, L., Simeonov, A., Shenfeld, I., and Agrawal, P. Juicer: Data-efficient imitation learning for robotic assembly. arXiv preprint arXiv:2404.03729, 2024.  
+Brohan, A., Brown, N., Carbajal, J., Chebotar, Y., Chen, X., Choromanski, K., Ding, T., Driess, D., Dubey, A., Finn, C., et al. Rt-2: Vision-language-action models transfer web knowledge to robotic control. arXiv preprint arXiv:2307.15818, 2023.  
+Chen, Y., Geng, Y., Zhong, F., Ji, J., Jiang, J., Lu, Z., Dong, H., and Yang, Y. Bi-dexhands: Towards human-level bimanual dexterous manipulation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2023.  
+Chen, Y.-C., Li, H., Turpin, D., Jacobson, A., and Garg,
+
+A. Neural shape mating: Self-supervised object assembly with adversarial shape priors. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12724-12733, 2022.  
+Chitnis, R., Tulsiani, S., Gupta, S., and Gupta, A. Efficient bimanual manipulation using learned task schemas. In 2020 IEEE International Conference on Robotics and Automation (ICRA), pp. 1149-1155. IEEE, 2020.  
+Clarke, J. A., Tambussi, C. P., Noriega, J. I., Erickson, G. M., and Ketcham, R. A. Definitive fossil evidence for the extant avian radiation in the cretaceous. Nature, 433 (7023):305-308, 2005.  
+Community, B. O. Blender - a 3D modelling and rendering package. Blender Foundation, Stichting Blender Foundation, Amsterdam, 2018. URL http://www.blender.org.  
+Deng, C., Litany, O., Duan, Y., Poulenard, A., Tagliasacchi, A., and Guibas, L. J. Vector neurons: A general framework for so (3)-equivariant networks. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 12200-12209, 2021.  
+Ding, K., Chen, B., Wu, R., Li, Y., Zhang, Z., Gao, H.-a., Li, S., Zhu, Y., Zhou, G., Dong, H., et al. Preafford: Universal affordance-based pre-grasping for diverse objects and environments. IROS, 2024.  
+Fu, Z., Zhao, T. Z., and Finn, C. Mobile aloha: Learning bimanual mobile manipulation with low-cost whole-body teleoperation. arXiv preprint arXiv:2401.02117, 2024.  
+Goyal, A., Xu, J., Guo, Y., Blukis, V., Chao, Y.-W., and Fox, D. Rvt: Robotic view transformer for 3d object manipulation. In Conference on Robot Learning, pp. 694-710. PMLR, 2023.  
+Grannen, J., Wu, Y., Vu, B., and Sadigh, D. Stabilize to act: Learning to coordinate for bimanual manipulation. In Conference on Robot Learning, pp. 563-576. PMLR, 2023.  
+Grotz, M., Shridhar, M., Asfour, T., and Fox, D. Peract2: A perceiver actor framework for bimanual manipulation tasks. arXiv preprint arXiv:2407.00278, 2024.  
+Heo, M., Lee, Y., Lee, D., and Lim, J. J. Furniturebench: Reproducible real-world benchmark for long-horizon complex manipulation. arXiv preprint arXiv:2305.12821, 2023.  
+Jones, B., Hildreth, D., Chen, D., Baran, I., Kim, V. G., and Schulz, A. Automate: A dataset and learning approach for automatic mating of cad assemblies. ACM Transactions on Graphics (TOG), 40(6):1-18, 2021.
+
+Ju, Y., Hu, K., Zhang, G., Zhang, G., Jiang, M., and Xu, H. Robo-abc: Affordance generalization beyond categories via semantic correspondence for robot manipulation. arXiv preprint arXiv:2401.07487, 2024.  
+Lee, A. X., Lu, H., Gupta, A., Levine, S., and Abbeel, P. Learning force-based manipulation of deformable objects from multiple demonstrations. In 2015 IEEE International Conference on Robotics and Automation (ICRA), pp. 177-184. IEEE, 2015.  
+Lee, N., Min, J., Lee, J., Kim, S., Lee, K., Park, J., and Cho, M. 3d geometric shape assembly via efficient point cloud matching. arXiv preprint arXiv:2407.10542, 2024.  
+Lee, Y., Hu, E. S., and Lim, J. J. Iea furniture assembly environment for long-horizon complex manipulation tasks. In 2021 iee international conference on robotics and automation (icra), pp. 6343-6349. IEEE, 2021.  
+Li, X., Zhang, M., Geng, Y., Geng, H., Long, Y., Shen, Y., Zhang, R., Liu, J., and Dong, H. Manipllm: Embodied multimodal large language model for object-centric robotic manipulation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 18061-18070, 2024a.  
+Li, Y., Pan, C., Xu, H., Wang, X., and Wu, Y. Efficient bi-manual handover and rearrangement via symmetry-aware actor-critic learning. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pp. 3867-3874. IEEE, 2023.  
+Li, Y., Wu, R., Lu, H., Ning, C., Shen, Y., Zhan, G., and Dong, H. Broadcasting support relations recursively from local dynamics for object retrieval in clutters. RSS, 2024b.  
+Liu, B., Luo, X., Huang, R., Wan, C., Zhang, B., Hu, W., and Yue, Z. Virtual plate pre-bending for the long bone fracture based on axis pre-alignment. Computerized medical imaging and graphics, 38(4):233-244, 2014.  
+Liu, J., Chen, Y., Dong, Z., Wang, S., Calinon, S., Li, M., and Chen, F. Robot cooking with stir-fry: Bimanual non-prehensile manipulation of semi-fluid objects. IEEE Robotics and Automation Letters, 7(2):5159-5166, 2022.  
+Liu, Y., Yang, H., Si, X., Liu, L., Li, Z., Zhang, Y., Liu, Y., and Yi, L. Taco: Benchmarking generalizable bimanual tool-action-object understanding. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 21740-21751, 2024.  
+Lu, H., Li, Y., Wu, R., Ning, C., Shen, Y., and Dong, H. Unigarment: A unified simulation and benchmark for garment manipulation. In ICRA Workshop on Deformable Object Manipulation, 2024a.
+
+Lu, J., Liang, Y., Han, H., Hua, J., Jiang, J., Li, X., and Huang, Q. A survey on computational solutions for reconstructing complete objects by reassembling their fractured parts. arXiv preprint arXiv:2410.14770, 2024b.  
+Lu, J., Sun, Y., and Huang, Q. Jigsaw: Learning to assemble multiple fractured objects. Advances in Neural Information Processing Systems, 36, 2024c.  
+Mo, K., Guibas, L. J., Mukadam, M., Gupta, A., and Tulsiani, S. Where2act: From pixels to actions for articulated 3d objects. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 6813-6823, 2021.  
+Mu, T., Ling, Z., Xiang, F., Yang, D., Li, X., Tao, S., Huang, Z., Jia, Z., and Su, H. Maniskill: Learning-from-demonstrations benchmark for generalizable manipulation skills. CoRR, abs/2107.14483, 2021b. URL https://arxiv.org/abs/2107.14483, 2021.  
+Mu, Y., Chen, T., Peng, S., Chen, Z., Gao, Z., Zou, Y., Lin, L., Xie, Z., and Luo, P. Robotwin: Dual-arm robot benchmark with generative digital twins (early version). arXiv preprint arXiv:2409.02920, 2024.  
+Narang, Y., Storey, K., Akinola, I., Macklin, M., Reist, P., Wawrzyniak, L., Guo, Y., Moravanszky, A., State, G., Lu, M., et al. Factory: Fast contact for robotic assembly. arXiv preprint arXiv:2205.03532, 2022.  
+Ning, C., Wu, R., Lu, H., Mo, K., and Dong, H. Where2explore: Few-shot affordance learning for unseen novel categories of articulated objects. Advances in Neural Information Processing Systems, 36, 2024.  
+Papaioannou, G. and Karabassi, E.-A. On the automatic assemblage of arbitrary broken solid artefacts. Image and Vision Computing, 21(5):401-412, 2003.  
+pyk4a. pyk4a, 2019. URL https://github.com/ etiennedub/pyk4a.  
+Qi, C. R., Su, H., Mo, K., and Guibas, L. J. Pointnet: Deep learning on point sets for 3d classification and segmentation. Proc. Computer Vision and Pattern Recognition (CVPR), IEEE, 2017a.  
+Qi, C. R., Yi, L., Su, H., and Guibas, L. J. Pointnet++: Deep hierarchical feature learning on point sets in a metric space. Advances in neural information processing systems, 30, 2017b.  
+Qi, Y., Ju, Y., Wei, T., Chu, C., Wong, L. L., and Xu, H. Two by two: Learning multi-task pairwise objects assembly for generalizable robot manipulation. CVPR 2025, 2025.  
+Quigley, M., Conley, K., Gerkey, B., Faust, J., Foote, T., Leibs, J., Wheeler, R., Ng, A. Y., et al. Ros: an open-source robot operating system. In ICRA workshop on
+
+open source software, volume 3, pp. 5. Kobe, Japan, 2009.  
+Ravi, N., Gabeur, V., Hu, Y.-T., Hu, R., Ryali, C., Ma, T., Khedr, H., Rädle, R., Rolland, C., Gustafson, L., Mintun, E., Pan, J., Alwala, K. V., Carion, N., Wu, C.-Y., Girshick, R., Dollar, P., and Feichtenhofer, C. Sam 2: Segment anything in images and videos. arXiv preprint arXiv:2408.00714, 2024. URL https://arxiv.org/abs/2408.00714.  
+Ren, T., Liu, S., Zeng, A., Lin, J., Li, K., Cao, H., Chen, J., Huang, X., Chen, Y., Yan, F., Zeng, Z., Zhang, H., Li, F., Yang, J., Li, H., Jiang, Q., and Zhang, L. Grounded sam: Assembling open-world models for diverse visual tasks, 2024a.  
+Ren, Y., Zhou, Z., Xu, Z., Yang, Y., Zhai, G., Leibold, M., Ni, F., Zhang, Z., Buss, M., and Zheng, Y. Enabling versatility and dexterity of the dual-arm manipulators: A general framework toward universal cooperative manipulation. IEEE Transactions on Robotics, 2024b.  
+Scarpellini, G., Fiorini, S., Giulari, F., Morerio, P., and Del Bue, A. Diffassemble: A unified graph-diffusion model for 2d and 3d reassembly. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2024.  
+Schönberger, J. L. and Frahm, J.-M. Structure-from-motion revisited. In Conference on Computer Vision and Pattern Recognition (CVPR), 2016.  
+Schönberger, J. L., Zheng, E., Pollefeys, M., and Frahm, J.-M. Pixelwise view selection for unstructured multiview stereo. In European Conference on Computer Vision (ECCV), 2016.  
+Sellán, S., Chen, Y.-C., Wu, Z., Garg, A., and Jacobson, A. Breaking bad: A dataset for geometric fracture and reassembly. In Thirty-sixth Conference on Neural Information Processing Systems Datasets and Benchmarks Track, 2022.  
+Sellán, S., Luong, J., Mattos Da Silva, L., Ramakrishnan, A., Yang, Y., and Jacobson, A. Breaking good: Fracture modes for realtime destruction. ACM Transactions on Graphics, 42(1):1-12, 2023.  
+Sera, I., Yamanobe, N., Ramirez-Alpizar, I. G., Wang, Z., Wan, W., and Harada, K. Assembly planning by recognizing a graphical instruction manual. In 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3138-3145. IEEE, 2021.  
+Sohn, K., Lee, H., and Yan, X. Learning structured output representation using deep conditional generative models. Advances in neural information processing systems, 28: 3483-3491, 2015.
+
+Suárez-Ruiz, F., Zhou, X., and Pham, Q.-C. Can robots assemble an ikea chair? Science Robotics, 3(17):eaat6385, 2018.  
+Tian, Y., Xu, J., Li, Y., Luo, J., Sueda, S., Li, H., Willis, K. D., and Matusik, W. Assemble them all: Physics-based planning for generalizable assembly by disassembly. ACM Transactions on Graphics (TOG), 41(6):1-11, 2022.  
+Tie, C., Sun, S., Zhu, J., Liu, Y., Guo, J., Hu, Y., Chen, H., Chen, J., Wu, R., and Shao, L. Manual2skill: Learning to read manuals and acquire robotic skills for furniture assembly using vision-language models. In Proceedings of Robotics: Science and Systems (RSS), 2025.  
+Tsesmelis, T., Palmieri, L., Khoroshiltseva, M., Islam, A., Elkin, G., Shahar, O. I., Scarpellini, G., Fiorini, S., Ohayon, Y., Alali, N., et al. Re-assembling the past: The repair dataset and benchmark for real world 2d and 3d puzzle solving. arXiv preprint arXiv:2410.24010, 2024.  
+Wan, Y., Zhou, K., Dong, H., et al. Scanet: Correcting lego assembly errors with self-correct assembly network. arXiv preprint arXiv:2403.18195, 2024.  
+Wang, G., Chen, Z., Loy, C. C., and Liu, Z. Sparsenerf: Distilling depth ranking for few-shot novel view synthesis. In IEEE/CVF International Conference on Computer Vision (ICCV), 2023.  
+Wang, P., Liu, L., Liu, Y., Theobalt, C., Komura, T., and Wang, W. Neus: Learning neural implicit surfaces by volume rendering for multi-view reconstruction. *NeurIPS*, 2021.  
+Wang, R., Zhang, Y., Mao, J., Zhang, R., Cheng, C.-Y., and Wu, J. Ikee-manual: Seeing shape assembly step by step. Advances in Neural Information Processing Systems, 35: 28428-28440, 2022a.  
+Wang, Y., Wu, R., Mo, K., Ke, J., Fan, Q., Guibas, L. J., and Dong, H. Adaafford: Learning to adapt manipulation affordance for 3d articulated objects via few-shot interactions. In European conference on computer vision, pp. 90-107. Springer, 2022b.  
+Wen, B., Yang, W., Kautz, J., and Birchfield, S. Foundationpose: Unified 6d pose estimation and tracking of novel objects. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 17868-17879, 2024.  
+Willis, K. D., Jayaraman, P. K., Chu, H., Tian, Y., Li, Y., Grandi, D., Sanghi, A., Tran, L., Lambourne, J. G., Solar-Lezama, A., et al. Joinable: Learning bottom-up assembly of parametric cad joints. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 15849-15860, 2022.
+
+Wu, R., Zhao, Y., Mo, K., Guo, Z., Wang, Y., Wu, T., Fan, Q., Chen, X., Guibas, L., and Dong, H. Vat-mart: Learning visual action trajectory proposals for manipulating 3d articulated objects. ICLR, 2022.  
+Wu, R., Cheng, K., Zhao, Y., Ning, C., Zhan, G., and Dong, H. Learning environment-aware affordance for 3d articulated object manipulation under occlusions. In Thirty-seventh Conference on Neural Information Processing Systems, 2023a. URL https://openreview.net/forum?id=Re2NHYoZ51.  
+Wu, R., Ning, C., and Dong, H. Learning foresightful dense visual affordance for deformable object manipulation. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 10947-10956, 2023b.  
+Wu, R., Tie, C., Du, Y., Zhao, Y., and Dong, H. Leveraging se (3) equivariance for learning 3d geometric shape assembly. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 14311-14320, 2023c.  
+Wu, R., Lu, H., Wang, Y., Wang, Y., and Dong, H. Unigarmmentmanip: A unified framework for category-level garment manipulation via dense visual correspondence. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 16340-16350, 2024.  
+Xian, Z., Lertkultanon, P., and Pham, Q.-C. Closed-chain manipulation of large objects by multi-arm robotic systems. IEEE Robotics and Automation Letters, 2(4):1832-1839, 2017.  
+Xiang, F., Qin, Y., Mo, K., Xia, Y., Zhu, H., Liu, F., Liu, M., Jiang, H., Yuan, Y., Wang, H., et al. Sapien: A simulated part-based interactive environment. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 11097-11107, 2020.  
+Xie, F., Chowdhury, A., De Paolis Kaluza, M., Zhao, L., Wong, L., and Yu, R. Deep imitation learning for bimanual robotic manipulation. Advances in neural information processing systems, 33:2327-2337, 2020.  
+Yang, L., Kang, B., Huang, Z., Zhao, Z., Xu, X., Feng, J., and Zhao, H. Depth anything v2. arXiv:2406.09414, 2024.  
+Yu, M., Shao, L., Chen, Z., Wu, T., Fan, Q., Mo, K., and Dong, H. Roboassembly: Learning generalizable furniture assembly policy in a novel multi-robot contact-rich simulation environment. arXiv preprint arXiv:2112.10143, 2021.  
+Yu, Z., Chen, A., Antic, B., Peng, S., Bhattacharyya, A., Niemeyer, M., Tang, S., Sattler, T., and Geiger,
+
+A. Sdfstudio: A unified framework for surface reconstruction, 2022. URL https://github.com/ autonomousvision/sdfstudio.  
+Ze, Y., Zhang, G., Zhang, K., Hu, C., Wang, M., and Xu, H. 3d diffusion policy. arXiv preprint arXiv:2403.03954, 2024.  
+Zhan, G., Fan, Q., Mo, K., Shao, L., Chen, B., Guibas, L. J., Dong, H., et al. Generative 3d part assembly via dynamic graph learning. Advances in Neural Information Processing Systems, 33:6315-6326, 2020.  
+Zhang, K., Sharma, M., Liang, J., and Kroemer, O. A modular robotic arm control stack for research: Franka-interface and frankapy. arXiv preprint arXiv:2011.02398, 2020.  
+Zhao, T. Z., Kumar, V., Levine, S., and Finn, C. Learning fine-grained bimanual manipulation with low-cost hardware. arXiv preprint arXiv:2304.13705, 2023.  
+Zhao, Y., Wu, R., Chen, Z., Zhang, Y., Fan, Q., Mo, K., and Dong, H. Dualafford: Learning collaborative visual affordance for dual-gripper manipulation. arXiv preprint arXiv:2207.01971, 2022.
+
+# Appendix
+
+# A. Details about Data Statistics
+
+![](images/692c614e8dffc3cbd8ef89fd9a6a253c03397032da7229eb02af0b8799c0bf25.jpg)  
+BreakingBad Dataset  
+Figure 6. Visualization of simulation data. We present one example shape from each object category used in our paper.
+
+In Figure 6, we present a representative example for each object category from the dataset used in our experiments.
+
+In this paragraph, we detail the data split for our experiments. We randomly select 10 out of the 15 categories for training, reserving the remaining 5 categories exclusively for testing. Within the 10 training categories,  $60\%$  of the shapes are randomly chosen for the training set, while the remaining  $40\%$  serve as a test set to assess the models' performance on novel instances within the training categories (shape-level). For the reserved 5 categories, all shapes are included in the test set to evaluate the methods' generalization capabilities on unseen categories (category-level). In summary, the training set consists of 10 categories, totaling 237 shapes and 6,403 pairs of fragments. The shape-level test set includes 10 categories, comprising 131 shapes and 3,638 pairs of fragments. The category-level test set encompasses 5 categories, containing 77 shapes and 1,779 pairs of fragments. Detailed statistics for each category can be found in Table 2.
+
+# B. Details about Data Collection in Simulation
+
+In this section, we provide detailed information about data collection in the simulation.
+
+Due to the complexity of bimanual geometric assembly tasks, which stems from the vast observation and action spaces, it is nearly impossible to directly acquire positive data by randomly manipulating the fractured parts. To address this, we apply several heuristic strategies to improve the efficiency of data collection. Specifically, our strategies focus on the following three key steps in the process:
+
+1. Sampling the grasping poses for the two grippers  
+2. Sampling the alignment poses for the two grippers  
+3. Sampling the assembly poses for the two grippers
+
+Table 2. Shape and Fracture Counts Across Categories. Numbers before the slash represent the training set, and numbers after the slash represent the testing set. The top 10 categories are the training categories, and the bottom 5 categories are the unseen categories.  
+
+<table><tr><td>Category</td><td>Shape (Train/Test)</td><td>Fracture (Train/Test)</td></tr><tr><td>BeerBottle</td><td>6 / 3</td><td>100 / 61</td></tr><tr><td>Bottle</td><td>51 / 22</td><td>1296 / 559</td></tr><tr><td>Bowl</td><td>16 / 32</td><td>446 / 801</td></tr><tr><td>Mug</td><td>32 / 15</td><td>876 / 545</td></tr><tr><td>PillBottle</td><td>7 / 3</td><td>217 / 60</td></tr><tr><td>Statue</td><td>2 / 0</td><td>57 / 35</td></tr><tr><td>Teapot</td><td>7 / 3</td><td>315 / 104</td></tr><tr><td>ToyFigure</td><td>36 / 16</td><td>1118 / 556</td></tr><tr><td>Vase</td><td>74 / 32</td><td>1842 / 872</td></tr><tr><td>WineGlass</td><td>6 / 3</td><td>136 / 45</td></tr><tr><td>Cup</td><td>0 / 31</td><td>0 / 663</td></tr><tr><td>DrinkBottle</td><td>0 / 7</td><td>0 / 230</td></tr><tr><td>DrinkingUtensil</td><td>0 / 14</td><td>0 / 343</td></tr><tr><td>Teacup</td><td>0 / 7</td><td>0 / 167</td></tr><tr><td>WineBottle</td><td>0 / 18</td><td>0 / 376</td></tr><tr><td>Total</td><td>237 / 208</td><td>6403 / 5417</td></tr></table>
+
+Each of these steps is described in detail in the following subsections.
+
+# B.1. Sampling Grasping Poses
+
+Different from furniture assembly task, where grasp points are easier to define, the objects in geometric shape assembly tasks have more diverse geometries, making it challenging to establish a consistent grasping policy. As a result, our heuristic strategy for grasping primarily focuses on the orientation of the grippers rather than specific grasp points.
+
+At initialization, the two parts are randomly placed on the table. From the simulation, we obtain the ground-truth depth map and normalization map of the two parts. The normal directions often closely align with feasible grasping directions (i.e., the z-axis of the gripper). Consequently, we randomly select a grasp point on the part, and then choose a grasping direction within a cone that deviates less than 30 degrees from the normal direction at that point. To avoid potential collisions between the grippers and the table during grasping, we discard any directions that point towards the upper hemisphere of the world coordinate system.
+
+In addition to the gripper's z-axis, the x-axis also significantly impacts grasping accuracy. Therefore, we uniformly sample a list of  $n$  x-axis candidates that are orthogonal to the gripper's z-axis. By combining each candidate x-axis with the z-axis, we determine the gripper pose. We test each of these gripper poses sequentially. If a grasp pose successfully grasps the object, we proceed to the next stage; otherwise, we reset the scene and move on to the next x-axis candidate. If all grasp pose candidates fail, we record this as negative data for the grasping step. In our implementation, we empirically set  $n = 6$ , resulting in each x-axis candidate being spaced 60 degrees apart.
+
+# B.2. Sampling Alignment Poses
+
+To sample the grippers' alignment poses, we begin by sampling feasible part poses during the alignment step. Our heuristic strategy also follows a reverse disassembly process. Specifically, we load the ground-truth assembled object into the simulation at a height of 0.5 meters above the tabletop, and allow the assembled object to take any pose rather than being restricted to a canonical pose. Next, we randomly explore feasible disassembly directions for the two parts, ensuring that these directions are collision-free. The resulting poses of the parts, after moving in their respective disassembly directions, represent the parts' alignment poses. It is important to note that we will discard alignment poses that are too distant from the parts' initial poses (for example, if the initial left part has an alignment pose to the right, while the initial right part has an alignment pose to the left).
+
+Once we have determined the parts' alignment poses, we can calculate the grippers' alignment poses using the functions described in Section 4.5.
+
+# B.3. Sampling Assembly Directions
+
+In the previous step, we identified the feasible disassembly directions for the ground-truth assembled parts. Consequently, we can obtain the assembly directions by simply inverting these disassembly directions, allowing the two grippers to assemble the parts accordingly. However, it is important to note that this assembly process may lead to failures. This is because, although the parts can be successfully aligned in an idealized scenario without grippers, the presence of grippers increases the risk of collisions. For instance, if one gripper is positioned too close to the seam area of a fractured part, it may collide with another part or the other gripper during the assembly process.
+
+# C. More Details about the BiAffordance Framework
+
+In this section, we provide more details about the BiAffordance Predictor. Following DualAfford (Zhao et al., 2022), we decompose the bimanual cooperation task into two separate yet closely interconnected submodules,  $\mathcal{M}_1$  and  $\mathcal{M}_2$ , which conditionally predict the first and second gripper actions, respectively.
+
+During inference, the first module  $\mathcal{M}_1$  predicts the first gripper action  $g_1 = (p_1^*, r_1)$ , followed by the second module  $\mathcal{M}_2$ , which predicts the second action  $g_2 = (p_2^*, r_2)$  conditioned on  $g_1$ , as described in Section 4.4 of the main paper.
+
+During training,  $\mathcal{M}_2$  still takes the first gripper action  $g_1$  as input, and then generates a complementary second action  $g_2$ . However, since  $\mathcal{M}_1$  lacks knowledge of how  $g_2$  will be predicted, it faces challenges in predicting a collaborative action  $g_1$ . To address this issue, we aim to make  $\mathcal{M}_1$  aware of the types of actions that can be easily collaborated on. We assess the quality of  $\mathcal{M}_1$ 's actions by evaluating whether  $\mathcal{M}_2$  can generate cooperative actions, which encourages  $\mathcal{M}_1$  to predict actions with high collaborative quality. Following this approach,  $\mathcal{M}_2$  guides the training of  $\mathcal{M}_1$ . Thus, we first train  $\mathcal{M}_2$  and then use the trained  $\mathcal{M}_2$  to train  $\mathcal{M}_1$ , ensuring cooperative predictions.
+
+During training, each submodule  $\mathcal{M}_i$  consists of three components: (1) an Affordance Network  $\mathcal{A}_i$ , which predicts an affordance map to indicate where interaction should occur; (2) an Actor Network  $\mathcal{U}_i$ , which predicts manipulation orientations to determine how to interact at the selected point; and (3) a Critic Network  $\mathcal{C}_i$ , which assesses the likelihood of the action's success.
+
+To explain the training process, we begin with the more straightforward second module,  $\mathcal{M}_2$ , which is also the first to be trained.
+
+The Actor Network  $\mathcal{U}_2$  in  $\mathcal{M}_2$  is implemented as a conditional Variational Autoencoder (cVAE). As detailed in Section 4.4, it takes concatenated input features  $f_{in} = (f_p,f_{S'},f_{v'})$  and the ground-truth feature of the first action  $f_{g_1} = (f_{p_1^*},f_{r_1})$  from the collected data. We apply a geodesic distance loss to measure the error between the reconstructed gripper orientation  $r_2$  and the ground-truth orientation  $\hat{r}_2$ , along with KL divergence to quantify the difference between the two distributions:
+
+$$
+\mathcal {L} _ {\mathcal {U} _ {2}} = \mathcal {L} _ {g e o} \left(r _ {2}, \hat {r} _ {2}\right) + D _ {K L} \left(q \left(z \mid \hat {r} _ {2}, f _ {i n}, f _ {g _ {1}}\right) \mid \mid \mathcal {N} (0, 1)\right). \tag {4}
+$$
+
+The Critic Network  $\mathcal{C}_2$  in  $\mathcal{M}_2$  is implemented as a multilayer perceptron (MLP) and evaluates how well the predicted second gripper action  $g_{2} = (p_{2}^{*},r_{2})$  collaborates with the first action  $g_{1}$ . Using the collected data along with the corresponding ground-truth interaction results  $r$  (where  $r = 1$  indicates a positive interaction and  $r = 0$  indicates a negative one), we train  $\mathcal{C}_2$  with the standard binary cross-entropy loss:
+
+$$
+\mathcal {L} _ {\mathcal {C} _ {2}} = r _ {j} \log \left(\mathcal {C} _ {2} \left(f _ {i n}, f _ {g _ {1}}, f _ {p _ {2} ^ {*}}, f _ {r _ {2}}\right)\right) + (1 - r _ {j}) \log \left(1 - \mathcal {C} _ {2} \left(f _ {i n}, f _ {g _ {1}}, f _ {p _ {2} ^ {*}}, f _ {r _ {2}}\right)\right). \tag {5}
+$$
+
+The Affordance Network  $\mathcal{A}_2$  in  $\mathcal{M}_2$  is implemented as a multilayer perceptron (MLP). The predicted affordance score represents the expected success rate for executing action proposals generated by the Actor Network, which can be directly evaluated by the Critic Network. To obtain the ground-truth affordance score  $\hat{a}_{p_i}$  on  $p_i$ , we use the Actor Network  $\mathcal{U}_2$  to
+
+sample  $n$  gripper orientations at the point  $p_i$  and calculate the average action scores assigned by the Critic Network  $\mathcal{C}_2$ . We apply L1 loss to measure the error between the predicted and ground-truth affordance scores at a specific point  $p_i$ :
+
+$$
+\hat {a} _ {p _ {i}} = \frac {1}{n} \sum_ {j = 1} ^ {n} \mathcal {C} _ {2} \left(f _ {i n}, f _ {g _ {1}}, f _ {p _ {i}}, \mathcal {U} _ {2} \left(f _ {i n}, f _ {g _ {1}}, f _ {p _ {i}}, z _ {j}\right)\right); \quad \mathcal {L} _ {\mathcal {A} _ {2}} = \left| \mathcal {A} _ {2} \left(f _ {i n}, f _ {g _ {1}}, f _ {p _ {i}}\right) - \hat {a} _ {p _ {i}} \right|. \tag {6}
+$$
+
+After training the expert model  $\mathcal{M}_2 = (\mathcal{A}_2, \mathcal{U}_2, \mathcal{C}_2)$ , we can utilize it to generate collaborative actions  $g_2$  for a given  $g_1$  predicted by  $\mathcal{M}_1$ . Thus, we can assess the quality of  $\mathcal{M}_1$ 's actions by evaluating whether  $\mathcal{M}_2$  can generate cooperative actions. Specifically, to evaluate the predicted  $g_1$ , we use the trained  $\mathcal{A}_2$  and  $\mathcal{U}_2$  to generate multiple second gripper action candidates  $\{g_2\}$ . We then employ  $\mathcal{C}_2$  to determine how well these second gripper candidates  $\{g_2\}$  collaborate with the proposed  $g_1$ . The average critic score reflects how easily the second gripper can cooperate with the proposed first action  $g_1$ . Consequently, this average score serves as the ground truth for the first Critic Network  $\mathcal{C}_1$ , and we apply L1 loss for supervision:
+
+$$
+\hat {c} _ {g _ {1}} = \frac {1}{n m} \sum_ {j = 1} ^ {n} \sum_ {k = 1} ^ {m} \mathcal {C} _ {2} \left(f _ {i n}, f _ {g _ {1}}, f _ {p _ {j}}, \mathcal {U} _ {2} \left(f _ {i n}, f _ {g _ {1}}, f _ {p _ {j}}, z _ {j k}\right)\right); \quad \mathcal {L} _ {\mathcal {C} _ {1}} = \left| \mathcal {C} _ {1} \left(f _ {i n}, f _ {g _ {1}}\right) - \hat {c} _ {g _ {1}} \right|. \tag {7}
+$$
+
+To train the Affordance Network  $\mathcal{A}_1$  and Actor Network  $\mathcal{U}_1$  in  $\mathcal{M}_1$ , the loss functions are similar to those used for  $\mathcal{A}_2$  and  $\mathcal{U}_2$ . Therefore, with the trained Critic Network  $\mathcal{C}_1$ , the Affordance Network  $\mathcal{A}_1$  assigns high scores to points that can be easily manipulated collaboratively by the subsequent gripper action.
+
+In this training pipeline, the two gripper modules can generate collaborative affordance maps and manipulation actions for bimanual tasks. Note that during inference, the use of the Critic Networks is optional.
+
+# D. Details about Training and Computational Costs
+
+During training, there are two main components: (1) the Disassembly Predictor and the Transformation Predictor are trained together in an end-to-end manner, and (2) all modules within the BiAssembly Predictor are also trained collectively in an end-to-end manner. These two training components can be conducted simultaneously on a single GPU. Using a single NVIDIA V100 GPU, the total training time for our model is approximately 48 hours: the combination of the Disassembly Predictor and Transformation Predictor converges in about 20 hours, while the BiAffordance Predictor converges in about 48 hours.
+
+During inference, our method utilizes only 1,600 MB of GPU memory and processes each data point in an average of 0.1 seconds.
+
+# E. More Experimental Results
+
+# E.1. Handling Multiple Broken Parts
+
+Table 3. Quantitative Results for multi-fragment assembly.  
+
+<table><tr><td>Method</td><td>R</td><td>R</td><td>R</td><td>AVG</td></tr><tr><td>Ours</td><td>24%</td><td>22%</td><td>16%</td><td>9%</td></tr></table>
+
+Our method can be extended to handle multi-fragment assembly tasks. In Table 3, we present the accuracy of our method on three-fragment assembly tasks. Additionally, Figure 7 showcases visualizations that include the predicted affordance maps and collaborative actions. Both the quantitative and qualitative results demonstrate that our proposed method can be
+
+![](images/1ba74f5534cb8ff845a93a737c3e4df9f765d682dc758a90f7e10616a2fd2f8c.jpg)  
+Figure 7. We provide the visualization of the predicted affordance maps and actions for multi-part assembly task.
+
+effectively adapted to multi-fragment assembly tasks. Below, we provide a detailed explanation of how our method can be adapted for multi-fragment assembly.
+
+The multi-fragment assembly task can be achieved by iteratively applying the two-fragment assembly process. First, at each iteration, we can identify which two fragments,  $p_i$  and  $p_j$ , should be assembled next. (If some parts have already been assembled in previous iterations, their combination is treated as a new fragment.) Specifically, based on the imaginary assembled shape  $S$ , we can calculate the minimum distance,  $\min \| p_i - p_j \|$ , between sampled points from every pair of fragments, and the pair  $(p_i, p_j)$  with the minimum distance is chosen for assembly:  $(p_i, p_j) = \arg \min_{(p_i, p_j) \in S_1 \times S_2} \| p_i - p_j \|$ . Once
+
+$p_i$  and  $p_j$  are identified on  $S$ , we then map these fragments to their corresponding parts in the observed point cloud  $O$ . This mapping is formulated as a classification task, where the similarity between parts in  $S$  and  $O$  is estimated.
+
+Finally, using the imaginary assembled shape of the selected fragments,  $S_{p_i} \cup S_{p_j}$ , and the corresponding observed point cloud  $O_{p_i} \cup O_{p_j}$ , our method predicts the actions to pick up and assemble the fragments. This process mirrors the steps of the standard two-fragment assembly method. By iteratively applying this two-fragment assembly process, the complete assembly of all fragments can be achieved.
+
+To validate the feasibility of this multi-fragment assembly process, we fine-tune our pretrained BiAssembly model on multi-category three-piece assembly data and present the results on unseen objects. Based on the accuracy reported in Table 3 and the visualizations in Figure 7, we can see that even for multi-fragment assembly task, our method can still generate reasonable results.
+
+# E.2. Utilizing the Imperfect Imaginary Assessed Shape Prediction
+
+Predicting the imaginary assembled shape from multiple fractured parts is a relatively well-studied vision problem (Sellán et al., 2022; Wu et al., 2023c; Lu et al., 2024c; Tsesmelis et al., 2024; Scarpellini et al., 2024). Previous works have demonstrated the ability to predict precise fragment poses that allow for an imaginary assembled shape, making it reasonable to assume the existence of such shapes in our framework. Additionally, in traditional furniture assembly tasks, several studies (Wang et al., 2022a; Sera et al., 2021; Wan et al., 2024) also assume the existence of an imaginary Therefore, for simplicity, we adopt the concept of the imaginary assembled shape, aligning with advancements in prior works.
+
+While this simplification is reasonable, we also conduct experiments to evaluate our BiAssembly Method on "imperfect" imaginary assembled shapes. Specifically, we first utilize the pretrained model from a prior visual assembly method (Wu et al., 2023c) to generate the assembled point cloud prediction. We then replace the input "perfect" imaginary assembled shapes with these "imperfect" predictions. Importantly, the only difference is the change in the input imaginary shape, the subsequent BiAssembly framework remains unchanged, and we do not fine-tune the model on the new predictions. The average accuracy based on imperfect imaginary shapes is  $20.80\%$  on training categories and  $17.20\%$  on novel unseen categories, which is comparable to the accuracy on perfect imaginary shapes ( $24.10\%$  on training categories and  $17.40\%$  on
+
+novel unseen categories). These results demonstrate that our method is capable of tolerating imperfect input shapes, even without any fine-tuning process, supporting the reasonableness of our assumption for simplification.
+
+# E.3. More Ablation Studies
+
+In this section, we conduct three additional ablation studies, and provide the quantitative results in Table 4 for novel objects within training categories and in Table 5 for novel unseen categories.
+
+w/o Affordance Network: During inference, we do not use the trained Affordance Network to highlight actionable regions. Instead, we randomly sample a contact point on the part. The results show a significant drop in the success rates, which decrease to  $4.60\%$  for training categories and  $2.80\%$  in unseen categories. This demonstrates that the Affordance Network plays a crucial role in filtering out non-graspable points and points that are unsuitable for the subsequent assembly process.
+
+w/o Transformation Predictor: In this ablation, we remove the Transformation Predictor during inference. This results in success rates of  $7.40\%$  on training categories and  $4.80\%$  on unseen categories, both substantially lower than our original method. These results show that the Transformation Predictor plays an essential role in predicting alignment poses, enabling the robot to manipulate parts from their initial to alignment poses without collisions.
+
+w/ heuristic disassembly direction  $v$ : In this case, we remove the Disassembly Predictor during inference. Instead, we compute the center of each part from the imaginary assembled shape  $S$  by averaging the part points, and then use the relative direction of the two parts' centers as the disassembly direction  $v$ . This ablation achieves success rates of  $19.70\%$  on training categories and  $15.20\%$  on unseen categories, both of which are lower than those achieved by our method. While this ablated version performs well on certain categories, suggesting that the calculated relative direction can approximate the relative positions of the two parts, it falls short in categories with complex geometries. In such cases, the heuristic method lacks the accuracy needed to replace the assembly direction required for our task. This highlights the critical role of the Disassembly Predictor in achieving better performance.
+
+Table 4. More ablation studies: quantitative results in novel instances within training categories.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="11">Novel Instances in Training Categories</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>AVG</td></tr><tr><td>w/o Affordance</td><td>7%</td><td>11%</td><td>0%</td><td>0%</td><td>1%</td><td>8%</td><td>1%</td><td>4%</td><td>6%</td><td>8%</td><td>4.60%</td></tr><tr><td>w/o Transformation</td><td>29%</td><td>19%</td><td>0%</td><td>0%</td><td>0%</td><td>0%</td><td>8%</td><td>4%</td><td>5%</td><td>9%</td><td>7.40%</td></tr><tr><td>w/ heuristic v</td><td>54%</td><td>28%</td><td>0%</td><td>3%</td><td>10%</td><td>5%</td><td>28%</td><td>23%</td><td>21%</td><td>25%</td><td>19.70%</td></tr><tr><td>Ours</td><td>60%</td><td>38%</td><td>13%</td><td>13%</td><td>12%</td><td>9%</td><td>26%</td><td>18%</td><td>27%</td><td>25%</td><td>24.10%</td></tr></table>
+
+Table 5. More Ablation studies: quantitative results in the novel unseen categories.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="6">Unseen Categories</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>AVG</td></tr><tr><td>w/o Affordance</td><td>2%</td><td>6%</td><td>2%</td><td>0%</td><td>4%</td><td>2.8%</td></tr><tr><td>w/o Transformation</td><td>4%</td><td>10%</td><td>1%</td><td>0%</td><td>9%</td><td>4.8%</td></tr><tr><td>w/ heuristic v</td><td>18%</td><td>22%</td><td>15%</td><td>9%</td><td>12%</td><td>15.20%</td></tr><tr><td>Ours</td><td>14%</td><td>31%</td><td>10%</td><td>7%</td><td>25%</td><td>17.4%</td></tr></table>
+
+# E.4. More Bimanual Tasks
+
+While we demonstrate the effectiveness on the challenging geometric shape assembly task, our approach is also applicable to a broader range of bimanual tasks that require coordinated manipulation. To illustrate this generalizability, we present an additional experiment on a bottle cap closing task. This task can be formulated into three sequential steps: (1) The two arms pick up the bottle and cap from the table. (2) The two arms align the cap with the bottle opening. (3) The two arms place
+
+and secure the cap onto the bottle. Our results show that, after training on a few bottle shapes, our method generalizes well to novel bottle shapes, achieving an average accuracy of  $67\%$ . Figure 8 provides visualizations of the predicted affordance maps and the manipulation process.
+
+![](images/1d18fd1c08c772c29de4bb4a36d921a4d263ecca7a34e6ec428dd8e181d35587.jpg)  
+Figure 8. We provide visualizations of the predicted affordance maps and the manipulation process for the bottle cap closing task.
+
+# F. More Visualizations
+
+In Figure 9, we present a detailed visualization of the manipulation process in real-world experiments, serving as a supplement to Figure 5 in the main paper.
+
+In Figure 10, we present additional qualitative results from our method.
+
+![](images/3a088dda7157396ec9691bda6d820f4b04030e6f6638423b8e3fb15bf72ef6d2.jpg)  
+Figure 9. Real-World Experiment. We present the results of our model tested on real-world scans. For each data, We visualized the affordance map, and the bimanual actions for the pick-up, alignment, and assembly steps. Manipulation videos can be found in our supplementary materials.
+
+![](images/962a4da083c9d9e9d50b88a21bab30b5092aded19e2624a8f400d500d16391d4.jpg)  
+Figure 10. We visualize additional qualitative results that augment Figure 4 in the main paper. In each row, from left to right, we respectively present the input observation, the predicted affordance maps for the two fractured parts, and the bimanual actions for the pick-up, alignment, and assembly steps. In the top part are novel shapes from the training categories, while in the bottom part are shapes from unseen categories.
+
+![](images/4788a1616f4690d1019605f70ef0bb6ed47014fb8341f43d83005e2c773b2fc1.jpg)  
+Initial Observation
+
+![](images/65ea26d363d713817960bce3f9575b4e678aa8cee256cca1324cd001cf0224d8.jpg)
+
+![](images/4c25048622183f1a97414560c9efee33457dec43ac47a4af5d6ef0b4f12e4dea.jpg)
+
+![](images/cf438a80ab17d8174c57e0a4d00d9d69d4858e5d8ae3a7d41a0135d8c5b827a4.jpg)  
+Pick-up
+
+![](images/b3370c7f0e48ff40c0efc2b1e7518dfbd1a553c4f189f33e358f66570f254250.jpg)  
+Initial Observation
+
+![](images/8429d5f8d93c8b9c43ce7618eca7372426b5ada509a82388c67b0b2de336acaa.jpg)  
+Pick-up  
+(c)
+
+![](images/414eda1ca25be01535ab9e9b7d8fd933729b242c6dba18a57a92ace32d5a9d8d.jpg)  
+Initial Observation  
+(d)  
+Figure 11. We visualize some failure cases, which demonstrate the challenges of the tasks and some cases that are difficult for robots to determine appropriate actions. The first row presents three cases where the fractured parts are either too large or too flat to grasp. The second row includes two cases where the graspable region corresponds exactly to the seam areas; while the objects can be grasped, collisions may occur during the assembly of the parts.
+
+![](images/b446068b255ff4fc89afb625683e2f3b88b606c19dcb610a0e564c7d437b6db5.jpg)  
+Pick-up  
+Assembly
+
+![](images/2887460acd782e954543fa45ea34b40db67feee48a2ca6c948e61edd69812ee7.jpg)  
+Initial Observation  
+Initial Observation  
+(e)
+
+![](images/b0b1cac48158ca63d99d94332c1e9605f92a35a36e5c4e1d4f954b13014d4310.jpg)  
+）  
+Assembly
+
+# G. Failure Cases
+
+We provide a detailed analysis of failure cases and illustrate the inherent difficulty of the task with scenarios that are particularly challenging for robots to figure out. Additionally, we provide insights into potential future improvements to address these complexities more effectively.
+
+# G.1. Hard to Grasp
+
+Heavy or Smooth-Surfaced Parts. Fractured parts that are heavy or have smooth surfaces often result in grasping failures. For instance, as shown in Figure 11 (a), categories such as teapots and vases, which are relatively large and feature smooth curved surfaces, exhibit notably high failure rates during grasping.
+
+Flat Parts. Flat fractured parts, particularly some shapes in categories like statues and mugs, are challenging to pick up due to the limited gripping area. For example, as shown in (b), the statue part on the left is too close to the desktop and has a very small thickness, which prevent the gripper from grasping it. Similarly, in (c), the fragment on the right is too flat, making it impossible for the gripper to grasp it. A potential solution is incorporating pre-grasp operations, such as moving the fractured part to the table edge, allowing the shape to hang off slightly and thus become graspable.
+
+# G.2. Hard to Assemble
+
+Graspable Regions Overlapping Seam Areas. When the graspable regions of a fractured part align with its seam areas, collisions during assembly become frequent. This issue is common in categories such as wineglasses, mugs, and bowls. For example, as shown in Figure 11 (d), the left gripper avoids collision-prone regions, but the right gripper must grasp the neck of the wine bottle. Similarly, in (e), while the left gripper avoids collisions, the right gripper ends up grasping the handle of a mug. A potential solution is to perform a series of pick-and-place operations to adjust the object's initial pose. This adjustment can reduce the overlap between the object's graspable regions and seam areas, thereby minimizing collisions during the assembly process.
+
+Complex Object Shapes. Objects with intricate shapes, like those in the statues category, pose challenges due to irregular edges and complex curves. Such designs increase the difficulty of alignment and manipulation, leading to higher failure rates during assembly.
+
+Relative Displacement During Operations. Relative displacement between the gripper and fractured parts often occurs due to small contact areas and insufficient support, which can cause sliding or tipping during manipulation. For example, wine bottles with narrow necks, which have unstable center of gravity, making the gripper prone to sliding during movement and leading to operational failures.

2025/BiAssemble_ Learning Collaborative Affordance for Bimanual Geometric Assembly/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:139914dc2defdfb3caf5c66f5d94807279e36255faa18500c5d5a585a52aac99
+size 1608318

2025/BiMaCoSR_ Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution/015a9533-715f-4e8b-a7b5-d0f1265ce5d2_content_list.json

@@ -0,0 +1,1681 @@
+[
+    {
+        "type": "text",
+        "text": "BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution",
+        "text_level": 1,
+        "bbox": [
+            145,
+            109,
+            831,
+            156
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kai Liu  $^{*1}$  Kaicheng Yang  $^{*1}$  Zheng Chen  $^{1}$  Zhiteng Li  $^{1}$  Yong Guo  $^{2}$  Wenbo Li  $^{3}$  Linghe Kong  $^{†1}$  Yulun Zhang  $^{†1}$",
+        "bbox": [
+            279,
+            180,
+            687,
+            213
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
+        "bbox": [
+            241,
+            213,
+            320,
+            227
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While super-resolution (SR) methods based on diffusion models (DM) have demonstrated inspiring performance, their deployment is impeded due to the heavy request of memory and computation. Recent researchers apply two kinds of methods to compress or fasten the DM. One is to compress the DM into 1-bit, aka binarization, alleviating the storage and computation pressure. The other distills the multi-step DM into only one step, significantly speeding up the inference process. Nonetheless, it remains impossible to deploy DM to resource-limited edge devices. To address this problem, we propose BiMaCoSR, which combines binarization and one-step distillation to obtain extreme compression and acceleration. To prevent the catastrophic collapse of the model caused by binarization, we propose sparse matrix branch (SMB) and low rank matrix branch (LRMB). Both auxiliary branches pass the full-precision (FP) information but in different ways. SMB absorbs the extreme values and its output is high rank, carrying abundant FP information. Whereas, the design of LRMB is inspired by LoRA and is initialized with the top  $r$  SVD components, outputting low rank representation. The computation and storage overhead of our proposed branches can be safely ignored. Comprehensive comparison experiments are conducted to exhibit BiMaCoSR outperforms current state-of-the-art binarization methods and gains competitive performance compared with FP one-step model. Moreover, we achieve excellent compression and acceleration. BiMaCoSR achieves a  $23.8 \\times$  compression ratio and a  $27.4 \\times$  speedup ratio compared to the FP counterpart. Our code and model are available at https://github.com/KaiLiu001/BiMaCoSR.",
+        "bbox": [
+            117,
+            237,
+            444,
+            794
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/baaab697a0437f0f13cceab2954ae5d48d6a6810aeaf440382239a942084e70b.jpg",
+        "image_caption": [
+            "Figure 1: Performance comparison between binarization methods on the RealSR dataset. BiMaCoSR achieves consistently leading scores on all evaluation metrics."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            535,
+            217,
+            851,
+            434
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. Introduction",
+        "text_level": 1,
+        "bbox": [
+            498,
+            487,
+            629,
+            501
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Single image super-resolution (SR) (Keys, 1981; Zibetti & Mayer, 2007; Lu et al., 2013; Dong et al., 2014; Yang et al., 2014; Dong et al., 2016) is a traditional yet challenging low-level vision problem. Serving as a fundamental research task, it has attracted long-standing and considerable attention in the computer vision community. The final object of SR is to restore a high-quality (HQ) image from its low-quality (LQ) observation, which suffers from various image quality degradations. The difficulty of SR mainly lies in two parts: (1) the unknown degradations (e.g., blur, downsampling, noise, compression, and their combinations) of LQ. (2) the multiple solutions for a given LQ input image.",
+        "bbox": [
+            495,
+            505,
+            888,
+            686
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In recent years, numerous studies have been made to tackle this challenge, utilizing convolution neural networks (CNNs) (Dong et al., 2014; 2016; Ledig et al., 2017; Zhang et al., 2018c), vision transformers (ViTs) (Zhang et al., 2018b; Liang et al., 2021; Wang et al., 2022; Chen et al., 2022; 2023), and their combinations. Though achieving inspiring results, these methods mostly fail in real-world scenarios. This failure is attributed to assuming degradation as an ideal bicubic downsampling kernel, way too different from the unknown and complex degradation in real world. Therefore, it draws researchers' increasing attention to reconstruct perceptually realistic HQ images in real-world scenarios. Thereafter, this challenging and meaningful task is called real-world super-resolution (Real-SR) (Gu et al.,",
+        "bbox": [
+            495,
+            694,
+            888,
+            905
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "*Equal contribution  ${}^{1}$  Shanghai Jiao Tong University  ${}^{2}$  Huawei Consumer Business Group  ${}^{3}$  The Chinese University of Hong Kong. Correspondence to: Linghe Kong <linghe.kong@sjtu.edu.cn>, Yulun Zhang <yulun100@gmail.com>.",
+        "bbox": [
+            84,
+            801,
+            475,
+            854
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).",
+        "bbox": [
+            84,
+            864,
+            475,
+            905
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2019; Zhang et al., 2021; Wang et al., 2021; Cai et al., 2019; Yu et al., 2024; Wu et al., 2024b; Sun et al., 2024).",
+        "bbox": [
+            84,
+            84,
+            475,
+            116
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recently, diffusion models (DMs) demonstrate remarkable performance in image-generating tasks, particularly in perceptual quality. The excellence of DM comes from its vast prior knowledge in modeling real-world objects, especially generating clear textures and mitigating artifacts and distortions. The powerful realistic texture generation ability is inherently the same as the object of Real-SR problem, leading to plentiful breakthroughs (Yang et al., 2025; Yu et al., 2024; Rombach et al., 2022). However, the inference cost is still too high to run on edge devices. Therefore, it's essential to further compress DMs to accelerate the inference, reduce storage cost, and minimize degradation.",
+        "bbox": [
+            84,
+            122,
+            475,
+            305
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Popular model compression techniques include pruning, distillation, and quantization, among which, 1-bit quantization (i.e., binarization) gains significant effectiveness. As an extreme quantization method, binarization compresses models' weights from 32-bit to only 1-bit, significantly reducing memory and computational cost. However, applying naive binarization will lead to catastrophic model collapse. Hence, additional structures are required to be designed.",
+        "bbox": [
+            84,
+            311,
+            475,
+            434
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A popular solution is adding full-precision information, i.e., skip connection branch. However, due to UNet's frequent changes in resolution and dimension, skip connection faces the mismatch challenge. To address this problem, we propose two auxiliary branches, namely low rank matrix branch (LRMB) and sparse matrix branch (SMB). Inspired by LoRA, the proposed LRMB leverages low rank decomposition to achieve dimension shift. We select the top  $r$  singular values in SVD and utilize its corresponding components to initialize the LRMB. As for SMB, we employ sparse matrix to absorb the top  $k$  absolute values in a full-precision matrix. The weights of LRMB and SMB are subtracted from the binarized matrix branch (BMB) and it concentrates on restoring textures. Three branches form the BiMaCoSR and provide excellent performance shown in Fig. 1.",
+        "bbox": [
+            84,
+            439,
+            475,
+            667
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To sum up, the contributions of our work are as follows:",
+        "bbox": [
+            84,
+            672,
+            457,
+            688
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "1. We design BiMaCoSR, a new binarized one-step diffusion model for image super-resolution. To the best of our knowledge, BiMaCoSR is the first binarized one-step diffusion model.",
+            "2. We propose LRMB, which leverages low rank decomposition and SVD initialization to carry low frequency information and decouple the effect of BMB.",
+            "3. We propose SMB, which utilizes sparse matrix compression and extreme value absorption to deliver high rank features and achieve further decoupling.",
+            "4. We conduct comprehensive comparison experiments to show the state-of-the-art performance of the proposed BiMaCoSR. Besides, extensive ablation studies are conducted to prove the robustness and efficacy."
+        ],
+        "bbox": [
+            98,
+            691,
+            475,
+            905
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. Related Work",
+        "text_level": 1,
+        "bbox": [
+            496,
+            83,
+            638,
+            99
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Image Super-Resolution. Deep learning based approaches have demonstrated striking power in the realm of SR (Dong et al., 2014; Luo et al., 2022; Wang et al., 2021; Lim et al., 2017; Chen et al., 2023). As a groundbreaking work, SRCNN (Dong et al., 2014) initiates the track of solving SR problem via deep learning based approach. Thereafter, substantial contributions have been made to explore the best SR network architecture. For example, RCAN (Zhang et al., 2018b) leverages the residual in residual structure and deepens the network to more than 400 layers. SwinIR (Liang et al., 2021) is based on vision transformer structure and utilizes spatial window self-attention to capture the overall structure information. CAT (Chen et al., 2022) combines the attention mechanism and the CNN structure to make the most of the local and the global information. However, most of these conventional image super-resolution methods can not handle the Real-SR task because of the complex degradation in the real world.",
+        "bbox": [
+            495,
+            109,
+            887,
+            381
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Diffusion Model. In recent years, the diffusion-based methods have gained remarkable performance in many computer vision tasks and SR is no exception. For instance, SR3 (Saharia et al., 2022) restores the LQ by transforming the standard normal distribution into the empirical data distribution by learning a series of iterative refinement steps. DiffBIR (Lin et al., 2024) capitalizes on two restoration stages to seek the tradeoff of fidelity and quality. SinSR (Wang et al., 2024) effectively reduces the inference step to only one step via distillation and regularization. Following SinSR, OSEDiff (Wu et al., 2024a) modifies the distillation paradigm, and novel losses are introduced to improve face restoration ability. Despite the greatly improved inference speed, the model size remains the same, and there is still room for further acceleration.",
+        "bbox": [
+            495,
+            386,
+            888,
+            613
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Binarization. As the most extreme form of quantization, binarization typically compresses the weight into only 1 bit. In binarization, all the weights are seen as  $\\pm 1$  and the multiplications between weights and activations are converted to bit operations on sign bit of activation, allowing maximum compression and acceleration. Binarization is mainly about classification tasks initially (Rastegari et al., 2016; Liu et al., 2020; Qin et al., 2020; 2022; Huang et al., 2024). Recently, researchers have begun to perform binarization on image restoration tasks. Binary Latent Diffusion (Wang et al., 2023b) trains an auto-encoder with a binary latent space and mainly focuses on the Bernoulli distribution instead of acceleration. BiDM (Zheng et al., 2024) leverages timestep-friendly binary structure and space-patched distillation to compress the diffusion model to 1 bit. BI-DiffSR (Chen et al., 2024) designs several binary friendly modules and redistributes the activation of different time step. However, the inference step remains the same. Therefore, it is necessary to further compress the model to one step.",
+        "bbox": [
+            495,
+            619,
+            888,
+            905
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution",
+        "bbox": [
+            124,
+            56,
+            848,
+            70
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/c28c1c514f0bcd8cdfd02975fa1400d41af3bd66cc8f88aba9076d9c5bf1f23a.jpg",
+        "image_caption": [
+            "Figure 2: Overview of our proposed BiMaCoSR which employs three different compressed matrix branches. (a) The structure of a convolution layer in BiMaCoSR after binarization. Two auxiliary branches, i.e., LRMB and SMB, support BiMaCoSR's excellent performance. The linear layer can be regarded as a  $1 \\times 1$  convolution layer and is processed with the same pipeline. (b) Illustration of the initialization and how the three branches solve the weakness of the other branches."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            102,
+            80,
+            867,
+            348
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3. Methodology",
+        "text_level": 1,
+        "bbox": [
+            84,
+            420,
+            220,
+            436
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe our proposed BiMaCoSR, shown in Fig. 2. First, we analyze three issues that have potential for further improvement when binarizing diffusion models. Thereafter, we describe our proposed branches, i.e., low rank matrix branch (LRMB) and sparse matrix branch (SMB), which could serve as the auxiliary branch to binarized matrix branch (BMB). Finally, we illustrate the initialization methods, shown in Fig. 3, and why our designs work.",
+        "bbox": [
+            84,
+            441,
+            475,
+            564
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1. Analysis",
+        "text_level": 1,
+        "bbox": [
+            84,
+            569,
+            179,
+            584
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Binarized blocks inevitably suffer from representation degradation due to the extreme 1-bit compression. Previous research made remarkable progress (Chen et al., 2024; Zheng et al., 2024) on binarization, yet there still remains room for improvement. We conclude three issues that deserve special attention in most diffusion models.",
+        "bbox": [
+            84,
+            589,
+            475,
+            679
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Issue I: The specially severe degradation of linear layer. Experimentally, we observe that with the same size of parameters, linear layer suffers from more severe degradation in comparison to convolution layer. This observation in SR is consistent with previous research (Le & Li, 2023). With the advancement of DiT and binarization, this issue is gradually becoming significant and an additional process to quantize or binarize the linear layer is in urgent need.",
+        "bbox": [
+            84,
+            686,
+            475,
+            808
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Issue II: The frequent dimension changes. It's necessary to leverage skip connection as an auxiliary branch in binarized network. Skip connection could carry abundant feature information and bring negligible computation and storage overhead. However, in UNet, frequent changes in dimension",
+        "bbox": [
+            84,
+            815,
+            475,
+            892
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and resolution make two forms of skip connection (i.e., addition and concatenation) not applicable. In addition, the distributions before and after binarized block are greatly different. Therefore, efficient module is critical to overcome the frequent changes and carry rich information.",
+        "bbox": [
+            496,
+            422,
+            885,
+            497
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Issue III: The inadequate usage of pre-trained FP model. The pre-trained FP model is vital to quantization methods from any perspective. However, current QAT research does not make the most of the pre-trained model, simply loading the FP parameters and directly beginning the training. It is a popular way to leverage additional modules to enhance the binarized network. The impact of the modules could vary when different initialization methods are applied.",
+        "bbox": [
+            495,
+            505,
+            885,
+            626
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We conclude that the above issues are attributed to the insufficient ability of binarization. To be specific, binarization is just one of the matrix compression methods, inherently unsuitable for multifaceted challenges. Therefore, we propose following branches to compress matrix in different ways. With the combination of various compressed matrices, the issues above can be appropriately addressed.",
+        "bbox": [
+            495,
+            633,
+            887,
+            739
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2. Low Rank Matrix Branch",
+        "text_level": 1,
+        "bbox": [
+            496,
+            748,
+            714,
+            762
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To address Issue I, i.e., the specially severe degradation of linear layer, we identify that the quantized linear layer is strongly limited by the bit-width. An FP linear layer usually serves two purposes, delivering the complete information of input and performing appropriate linear transformation. With limited bit-width, the quantized linear layer cannot play two roles at the same time. Therefore, we decouple the roles apart. Though binarized, the weight matrix is usually high rank and contributes more to high frequency, i.e., the",
+        "bbox": [
+            495,
+            768,
+            887,
+            904
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution",
+        "bbox": [
+            124,
+            56,
+            846,
+            70
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "detailed structure and texture in image. Hence, we design low rank matrix branch (LRMB) to serve as a complementary branch, transmitting low frequency information.",
+        "bbox": [
+            84,
+            85,
+            475,
+            131
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The idea of LRMB is inspired by low rank approximation in matrix theory, which is a common strategy for matrix compression. To transmit low frequency information, matrices with low rank are enough and efficient. Mathematically, given an  $m \\times n$  matrix  $\\mathbf{W}$ , it can be approximated with two low rank matrices, i.e.,  $\\mathbf{W} \\approx \\hat{\\mathbf{W}} \\coloneqq \\mathbf{BA}$ , where  $\\mathbf{B}$  and  $\\mathbf{A}$  are  $m \\times r$  and  $r \\times n$  matrices respectively, and  $r \\ll \\min(m, n)$  is a hyper-parameters denoting the rank of  $\\hat{\\mathbf{W}}$ . Therefore, the formula of LRMB is:",
+        "bbox": [
+            84,
+            137,
+            473,
+            273
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {\\mathrm {L R M B}} = \\operatorname {L R M B} \\left(\\mathbf {x} _ {\\text {i n}}\\right) := \\mathbf {x} _ {\\text {i n}} \\mathbf {B A}, \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            163,
+            277,
+            473,
+            292
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{x}_{\\mathrm{in}}\\in \\mathbb{R}^{N\\times m}$  and  $\\mathbf{x}_{\\mathrm{LRMB}}\\in \\mathbb{R}^{N\\times n}$  are the input and output of LRMB(·) respectively, and  $N$  is the number of tokens. Usually,  $\\mathbf{W}$  is a square matrix and  $m = n$",
+        "bbox": [
+            84,
+            294,
+            475,
+            340
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As for complexity, the storage overhead is:",
+        "bbox": [
+            84,
+            347,
+            372,
+            363
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nO _ {s} = (m \\times r + r \\times n) B = r B (m + n) \\ll m n B ^ {\\prime}, \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            98,
+            364,
+            473,
+            381
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $B = 32$  or 16 and  $B' = 1$  are the number of bits required by one element in LRMB branch and binarized branch, respectively. Meanwhile, the computation overhead is:",
+        "bbox": [
+            84,
+            383,
+            473,
+            441
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nO _ {c} = N \\times m \\times r + N \\times r \\times n = N r (m + n). \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            109,
+            445,
+            473,
+            460
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This means that even saved with 32 bits, the storage and computation overhead of LRMB is acceptable. In conclusion, with LRMB, Issue I and Issue II are partly solved.",
+        "bbox": [
+            84,
+            463,
+            475,
+            508
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3. Sparse Matrix Branch",
+        "text_level": 1,
+        "bbox": [
+            84,
+            518,
+            274,
+            534
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "LRMB is still not enough to replace the skip connection. This is because skip connection can be represented as a full rank identity matrix while LRMB is low rank. To compensate the missed ranks, we propose sparse matrix branch (SMB), which leverages sparse matrix compression and could efficiently deliver high rank information.",
+        "bbox": [
+            84,
+            537,
+            475,
+            628
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Sparse matrix is a matrix in which most of the elements are zero (Yan et al., 2017). A common criterion of a sparse matrix is that the number of non-zero elements is approximated to the number of rows or columns. One way to save a sparse matrix is via the coordinate format (COO), where the non-zero elements are represented by a list of triples and its component is (row, col, value).",
+        "bbox": [
+            84,
+            636,
+            475,
+            743
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Specifically, to form a sparse matrix, we select  $k$  critical values from the weight matrix and the selection method will be described in Sec. 3.4. In the forward process, the SMB process could be equivalently expressed as",
+        "bbox": [
+            84,
+            750,
+            473,
+            810
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {\\mathrm {S M B}} = \\operatorname {S M B} \\left(\\mathbf {x} _ {\\mathrm {i n}}\\right) := \\mathbf {x} _ {\\mathrm {i n}} \\mathbf {W} _ {\\text {s p a r s e}}, \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            163,
+            811,
+            473,
+            829
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{W}_{\\mathrm{sparse}}$  is a matrix with  $k$  non-zero elements. During training, the coordinates will be fixed while their values will be updated by gradient descent. With SMB, the high rank information could be delivered and the remaining parts of Issue I and Issue II could be solved.",
+        "bbox": [
+            84,
+            830,
+            475,
+            905
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/e9cb73fb1cdfaeec990d76fcf083c9423dad56686dc27e68592283df72fa3fc3.jpg",
+        "image_caption": [
+            "Figure 3: Initialization of different branch.  $W_{res}$  represents the initial quantization error. In our method,  $\\| W_{res}\\| _F^2 = 0.1855$ , while  $\\| W_{res}\\| _F^2 = 1.1275$  in direct binarization."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            506,
+            83,
+            874,
+            448
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4. Pretrained-Friendly Initialization",
+        "text_level": 1,
+        "bbox": [
+            496,
+            508,
+            764,
+            522
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We propose the following initialization method to guarantee better use of the pre-trained model, allowing better restoration performance and faster convergence.",
+        "bbox": [
+            495,
+            527,
+            888,
+            574
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The purpose of LRMB is to carry the low frequency information. Usually, compared with high frequency information, the magnitude of low frequency information is much greater. Therefore, we leverage SVD and select the top  $r$  components to initialize LRMB. To be specific, we perform SVD on  $\\mathbf{W}$ :",
+        "bbox": [
+            495,
+            580,
+            888,
+            669
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {W} = \\mathbf {U} \\boldsymbol {\\Sigma} \\mathbf {V} ^ {T} = \\mathbf {U} \\operatorname {d i a g} \\left\\{\\sigma_ {1}, \\sigma_ {2}, \\dots , \\sigma_ {n} \\right\\} \\mathbf {V} ^ {T}. \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            539,
+            671,
+            885,
+            689
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Then, we truncate the top  $r$  singular values and absorb the remaining singular value into  $\\mathbf{U}$ , forming matrices  $\\mathbf{A}$  and  $\\mathbf{B}$ . The process can be written as:",
+        "bbox": [
+            495,
+            691,
+            887,
+            736
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {W} \\approx \\underbrace {\\mathbf {U} \\operatorname {d i a g} \\left\\{\\sigma_ {1} , \\sigma_ {2} , \\dots , \\sigma_ {r} , 0 , \\dots , 0 \\right\\}} _ {B} \\underbrace {\\mathbf {V} ^ {T}} _ {A} = \\mathbf {B} \\mathbf {A}. \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            509,
+            737,
+            885,
+            771
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Thereafter, to guarantee that the binarized matrix only serves to add high frequency information, the low frequency counterpart will be subtracted. The formula is:",
+        "bbox": [
+            495,
+            780,
+            887,
+            824
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {W} _ {\\mathrm {B M B}} ^ {\\prime} := \\mathbf {W} - \\mathbf {B A}, \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            617,
+            827,
+            885,
+            844
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathbf{W}_{\\mathrm{BMB}}^{\\prime}$  is the matrix to be binarized after the initialization of LRMB. Hence, the ability of weight matrix is decoupled into two components, the low frequency part (BA) and the high frequency part  $(\\mathbf{W}_{\\mathrm{BMB}}^{\\prime})$",
+        "bbox": [
+            495,
+            845,
+            887,
+            907
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution",
+        "bbox": [
+            124,
+            56,
+            846,
+            71
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Thereafter, the initialization of SMB also matters. As described in Sec. 3.3, we select  $k$  values from the weight matrix to generate the sparse matrix. Due to the fixed coordinates of non-zero elements, random selection is apparently a good candidate. Besides, binarization often ignores the outlier values in weight matrix, which are often huge but rare (less than  $0.1\\%$ ). Though rare, these outliers play a crucial role in most models. Considering their rarity and significance, we select  $k$  elements with the greatest absolute value in  $\\mathbf{W}_{\\mathrm{BMB}}^{\\prime}$ . Subsequently, we subtract these values in  $\\mathbf{W}_{\\mathrm{BMB}}^{\\prime}$  to decouple the overlapping effect of BMB and SMB. The process can be formulated as:",
+        "bbox": [
+            84,
+            84,
+            475,
+            265
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\widetilde {w} _ {i j} = \\left\\{ \\begin{array}{l l} w _ {i j}, & \\text {i f} | w _ {i j} ^ {\\prime} | \\geq t \\\\ 0, & \\text {o t h e r w i s e} \\end{array} , \\mathbf {W} _ {\\mathrm {B M B}} := \\mathbf {W} _ {\\mathrm {B M B}} ^ {\\prime} - \\mathbf {W} _ {\\text {s p a r s e}}, \\right. \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            86,
+            268,
+            475,
+            319
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\widetilde{w}_{ij}$  and  $w_{ij}$  are the elements of  $\\mathbf{W}_{\\mathrm{sparse}}$  and  $\\mathbf{W}_{\\mathrm{BMB}}^{\\prime}$  respectively,  $t$  is the  $k$ -th largest absolute value in  $\\mathbf{W}_{\\mathrm{BMB}}^{\\prime}$ ,  $\\mathbf{W}_{\\mathrm{BMB}}$  is the final matrix to be binarized.",
+        "bbox": [
+            84,
+            319,
+            475,
+            366
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.5. Overall Structure",
+        "text_level": 1,
+        "bbox": [
+            84,
+            375,
+            243,
+            388
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "After adding LRMB and SMB and their initialization, the binarized matrix branch (BMB) is formed with  $\\mathbf{x}_{\\mathrm{BMB}}$ . To be specific, the input feature and weight matrix will be binarized with  $\\mathrm{Sign}(\\cdot)$ , which can be written as:",
+        "bbox": [
+            84,
+            391,
+            473,
+            450
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left\\{ \\begin{array}{l} \\mathbf {x} _ {\\mathrm {b}} = \\operatorname {S i g n} \\left(\\mathbf {x} _ {\\mathrm {i n}}\\right), \\\\ \\mathbf {W} _ {\\mathrm {b}} = \\operatorname {S i g n} \\left(\\mathbf {W} _ {\\mathrm {B M B}}\\right), \\end{array} \\quad \\operatorname {S i g n} (x) = \\left\\{ \\begin{array}{l l} + 1, & x \\geq 0, \\\\ - 1, & x <   0, \\end{array} \\right. \\right. \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            94,
+            454,
+            473,
+            494
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\operatorname{Sign}(\\cdot)$  is performed element-wise. Thereafter, the full-precision convolution and linear transform are replaced with efficient logical XNOR and bit-counting operations. The process can be formulated as:",
+        "bbox": [
+            84,
+            494,
+            475,
+            554
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {\\mathrm {B M B}} ^ {\\prime} = \\text {b i t - c o u n t} (\\mathrm {X N O R} (\\mathbf {x} _ {\\mathrm {b}}, \\mathbf {W} _ {\\mathrm {b}})), \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            153,
+            556,
+            473,
+            573
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, we follow the design of XNOR-Net (Rastegari et al., 2016) to compensate the precision loss, which is:",
+        "bbox": [
+            84,
+            575,
+            473,
+            604
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {\\mathrm {B M B}} = \\mathbf {x} _ {\\mathrm {B M B}} ^ {\\prime} \\odot (\\mathcal {A} \\otimes \\mathbf {k}), \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            189,
+            607,
+            473,
+            625
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathcal{A}_{H\\times W}$  is the channel-wise absolute average of input activation,  $\\mathbf{k}$  is a vector whose element is the absolute average of correspond channel,  $\\odot$  is Hadamard product, and  $(\\mathcal{A}\\otimes \\mathbf{k})_{c,i,j}:= \\mathcal{A}_{i,j}\\mathbf{k}_c$  should be calculated first.",
+        "bbox": [
+            84,
+            625,
+            475,
+            686
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We binarized every convolution layer and linear layer except the first and last convolution layers due to extremely severe degradation of quantization on head and tail. Specifically, linear layer can be seen as the  $1 \\times 1$  convolution layer and therefore processed in the same way as convolution layer.",
+        "bbox": [
+            84,
+            693,
+            475,
+            768
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For one binarized layer, the output can be written as:",
+        "bbox": [
+            84,
+            776,
+            434,
+            791
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbf {x} _ {\\text {o u t}} = \\mathbf {x} _ {\\mathrm {B M B}} + \\mathbf {x} _ {\\mathrm {L R M B}} + \\mathbf {x} _ {\\mathrm {S M B}}. \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            173,
+            794,
+            473,
+            809
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In conclusion, we propose LRMB, SMB, and their corresponding initialization methods. With these designs, our BiMaCoSR can significantly reduce the information loss caused by binarization and improve the restoration ability. Besides, the storage and computation overhead caused by LRMB and SMB can be safely ignored.",
+        "bbox": [
+            84,
+            811,
+            475,
+            902
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4. Experiments",
+        "text_level": 1,
+        "bbox": [
+            496,
+            83,
+            630,
+            99
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1. Experimental Settings",
+        "text_level": 1,
+        "bbox": [
+            496,
+            106,
+            686,
+            121
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Data. We take the models on the training set of ImageNet (Russakovsky et al., 2015) and the LR images are generated by the same pipeline of RealESRGAN (Wang et al., 2021). We evaluate the models with three benchmark datasets: RealSR (Cai et al., 2019), DRealSR (Wei et al., 2020), and DIV2K-Val (Agustsson & Timofte, 2017). RealSR and DRealSR are real world benchmarks while DIV2K-Val employs Bicubic interpolation to generate LR images. The upscale ratio of training set and test set is  $\\times 4$ .",
+        "bbox": [
+            495,
+            125,
+            885,
+            262
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Metrics. The evaluation metrics are implemented with IQA-Pytorch (Chen & Mo, 2022) and are twofold to assess the restoration and compression ability. Firstly, to thoroughly evaluate the restoration ability of our proposed BiMaCoSR, we employ the following quantitative metrics: full-reference metrics PSNR, SSIM (Wang et al., 2004), and LPIPS (Zhang et al., 2018a) and non-reference metrics DISTS (Ding et al., 2020), FID (Heusel et al., 2017), NIQE (Zhang et al., 2015), MANIQA-pipal (Yang et al., 2022), MUSIQ (Ke et al., 2021), and CLIPIQA+ (Wang et al., 2023a). PSNR and SSIM are distortion-based metrics and are calculated on the Y channel (i.e., luminance) of the YCbCr space. The rest metrics are all perceptual metrics and it is widely known that perceptual metrics are more aligned with human when rating the image quality.",
+        "bbox": [
+            495,
+            267,
+            887,
+            494
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Secondly, to demonstrate binarization's extreme compression and acceleration ability, we use the total parameters and overall operations as key metrics. Following previous work (Xia et al., 2022; Qin et al., 2023), the total parameters (Params) of the model are calculated as Params  $= \\text{Params}^b + \\text{Params}^f$ , and the overall operations (OPs) as OPs  $= \\text{OPs}^b + \\text{OPs}^f$ , where Params  $^b = \\text{Params}^f / 32$  and OPs  $^b = \\text{OPs}^f / 64$ ; the superscripts  $f$  and  $b$  denote full-precision and binarized modules, respectively. The computational complexity is tested with the input size  $3 \\times 64 \\times 64$ .",
+        "bbox": [
+            495,
+            501,
+            888,
+            652
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Implementation Details. We take SinSR (Wang et al., 2024), a one-step diffusion model, as the backbone. We initialize with SinSR's weights and use ResShift (Yue et al., 2024) as the teacher model. All the rest convolution and linear layers, except the head and tail layers, are binarized into BMB to guarantee the best compression ratio and LRMB and SMB are attached with BMB. We set the rank of LRMB as  $r = 8$  and the number of non-zero elements in SMB as  $k = 2\\max(C_1, C_2)$ , where  $C_1$  and  $C_2$  are the numbers of input and output channels, respectively.",
+        "bbox": [
+            495,
+            657,
+            887,
+            809
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Training Settings. We utilize Adam optimizer (Kingma & Ba, 2015) with  $\\beta_{1} = 0.9$  and  $\\beta_{2} = 0.99$ , and set the learning rate as  $2 \\times 10^{-5}$ . The batch size is set to 8, with 100K iterations. The input LR images are center-cropped to size  $64 \\times 64$ . All the training and testing experiments are conducted on one NVIDIA RTX A6000.",
+        "bbox": [
+            495,
+            814,
+            885,
+            904
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution",
+        "bbox": [
+            124,
+            56,
+            848,
+            71
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/3e70148ad334fc1fb86a757357dc5428d9857c4e86a8b3367b7db2d1b11adbe2.jpg",
+        "table_caption": [
+            "Table 1: Quantitative comparison with SOTA methods. We compare BiMaCoSR with full-precision models and current binarization methods. We employ 9 metrics commonly used in SR and test on three benchmarks. The best and second best value are marked with red and blue respectively. In conclusion, our proposed BiMaCoSR achieves SOTA performance."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td>Methods</td><td>Bits (W/A)</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>MUSIQ ↑</td><td>MANIQA ↑</td><td>DISTS ↓</td><td>FID ↓</td><td>NIQE ↓</td><td>CLIP-IQA+ ↑</td></tr><tr><td rowspan=\"8\">RealSR</td><td>SinSR</td><td>32/32</td><td>26.51</td><td>0.7380</td><td>0.3635</td><td>57.87</td><td>0.5139</td><td>0.2193</td><td>56.36</td><td>5.826</td><td>0.5736</td></tr><tr><td>ResShift</td><td>32/32</td><td>25.45</td><td>0.7243</td><td>0.3731</td><td>56.23</td><td>0.5005</td><td>0.2344</td><td>58.14</td><td>7.353</td><td>0.5708</td></tr><tr><td>XNOR</td><td>1/1</td><td>26.48</td><td>0.7434</td><td>0.3968</td><td>43.56</td><td>0.3732</td><td>0.2609</td><td>105.72</td><td>6.014</td><td>0.4380</td></tr><tr><td>ReActNet</td><td>1/1</td><td>26.60</td><td>0.7530</td><td>0.3834</td><td>44.18</td><td>0.3829</td><td>0.2551</td><td>109.36</td><td>6.306</td><td>0.4361</td></tr><tr><td>BBCU</td><td>1/1</td><td>26.43</td><td>0.7488</td><td>0.3902</td><td>43.70</td><td>0.3792</td><td>0.2575</td><td>108.32</td><td>6.058</td><td>0.4298</td></tr><tr><td>ReSTE</td><td>1/1</td><td>26.26</td><td>0.7408</td><td>0.4184</td><td>41.04</td><td>0.3677</td><td>0.2719</td><td>113.86</td><td>6.174</td><td>0.4083</td></tr><tr><td>BiDM</td><td>1/1</td><td>25.07</td><td>0.7036</td><td>0.5042</td><td>35.60</td><td>0.3517</td><td>0.3226</td><td>115.23</td><td>6.759</td><td>0.3935</td></tr><tr><td>BiMaCoSR</td><td>1/1</td><td>26.84</td><td>0.7698</td><td>0.3375</td><td>49.01</td><td>0.4034</td><td>0.2183</td><td>86.09</td><td>5.856</td><td>0.4800</td></tr><tr><td rowspan=\"8\">DRealSR</td><td>SinSR</td><td>32/32</td><td>27.89</td><td>0.7332</td><td>0.4499</td><td>30.81</td><td>0.4519</td><td>0.2209</td><td>16.56</td><td>5.789</td><td>0.6052</td></tr><tr><td>ResShift</td><td>32/32</td><td>26.64</td><td>0.7298</td><td>0.4478</td><td>31.09</td><td>0.4345</td><td>0.2337</td><td>18.12</td><td>6.959</td><td>0.5795</td></tr><tr><td>XNOR</td><td>1/1</td><td>29.03</td><td>0.8319</td><td>0.3712</td><td>26.19</td><td>0.3560</td><td>0.2447</td><td>29.88</td><td>6.229</td><td>0.4449</td></tr><tr><td>ReActNet</td><td>1/1</td><td>29.34</td><td>0.8431</td><td>0.3571</td><td>26.83</td><td>0.3618</td><td>0.2411</td><td>30.18</td><td>6.561</td><td>0.4380</td></tr><tr><td>BBCU</td><td>1/1</td><td>29.00</td><td>0.8385</td><td>0.3643</td><td>26.37</td><td>0.3594</td><td>0.2433</td><td>30.94</td><td>6.337</td><td>0.4383</td></tr><tr><td>ReSTE</td><td>1/1</td><td>28.91</td><td>0.8353</td><td>0.3899</td><td>25.12</td><td>0.3509</td><td>0.2641</td><td>33.64</td><td>6.459</td><td>0.4131</td></tr><tr><td>BiDM</td><td>1/1</td><td>27.40</td><td>0.7942</td><td>0.4849</td><td>23.38</td><td>0.3529</td><td>0.3118</td><td>37.83</td><td>6.753</td><td>0.4307</td></tr><tr><td>BiMaCoSR</td><td>1/1</td><td>29.33</td><td>0.8393</td><td>0.3400</td><td>29.38</td><td>0.3802</td><td>0.2278</td><td>22.31</td><td>6.150</td><td>0.4867</td></tr><tr><td rowspan=\"8\">DIV2K-Val</td><td>SinSR</td><td>32/32</td><td>27.75</td><td>0.7694</td><td>0.1903</td><td>64.62</td><td>0.5336</td><td>0.1029</td><td>6.27</td><td>4.308</td><td>0.6147</td></tr><tr><td>ResShift</td><td>32/32</td><td>27.18</td><td>0.7667</td><td>0.1775</td><td>65.04</td><td>0.5548</td><td>0.1016</td><td>7.54</td><td>5.121</td><td>0.6280</td></tr><tr><td>XNOR</td><td>1/1</td><td>26.44</td><td>0.7185</td><td>0.3727</td><td>49.10</td><td>0.3972</td><td>0.2204</td><td>55.77</td><td>5.320</td><td>0.4584</td></tr><tr><td>ReActNet</td><td>1/1</td><td>26.49</td><td>0.7260</td><td>0.3602</td><td>50.29</td><td>0.4078</td><td>0.2111</td><td>52.32</td><td>5.366</td><td>0.4726</td></tr><tr><td>BBCU</td><td>1/1</td><td>26.39</td><td>0.7221</td><td>0.3660</td><td>50.09</td><td>0.4035</td><td>0.2148</td><td>53.22</td><td>5.263</td><td>0.4653</td></tr><tr><td>ReSTE</td><td>1/1</td><td>26.07</td><td>0.7125</td><td>0.3916</td><td>46.95</td><td>0.3907</td><td>0.2295</td><td>61.52</td><td>5.399</td><td>0.4328</td></tr><tr><td>BiDM</td><td>1/1</td><td>24.29</td><td>0.6725</td><td>0.4370</td><td>40.15</td><td>0.3747</td><td>0.2916</td><td>62.28</td><td>6.090</td><td>0.4112</td></tr><tr><td>BiMaCoSR</td><td>1/1</td><td>27.35</td><td>0.7547</td><td>0.2999</td><td>53.38</td><td>0.4337</td><td>0.1806</td><td>27.99</td><td>4.987</td><td>0.5176</td></tr></table>",
+        "bbox": [
+            86,
+            133,
+            883,
+            477
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2. Comparison with State-of-the-Art Methods",
+        "text_level": 1,
+        "bbox": [
+            84,
+            488,
+            419,
+            503
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We compare our proposed BiMaCoSR with recent binarization methods, including XNOR (Rastegari et al., 2016), ReActNet (Liu et al., 2020), BBCU (Xia et al., 2022), ReSTE (Wu et al., 2023), and BiDM (Zheng et al., 2024). All binarization methods are implemented on SinSR (Wang et al., 2024) and trained with the same settings. As LRMB and SMB slightly increase the parameters, we additionally keep the first two and last two convolution layers as full-precision. We also compare BiMaCoSR with the full-precision model SinSR and ResShift (Yue et al., 2024). SinSR is the distilled version of ResShift. The comparison results are exhibited in quantitative and qualitative aspects.",
+        "bbox": [
+            84,
+            505,
+            475,
+            686
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Restoration Results. The quantitative comparison results are shown in Table 1. We can obtain the following observations. (1) The results in Table 1 demonstrate clear advantage over competing methods in both full-reference and nonreference metrics on three benchmark datasets. (2) In some situations, such as PSNR and LPIPS on RealSR, BiMa-CoSR can even surpass SinSR and ResShift. We attribute the excellence to the LRMB and SMB, which transmit the full-precision information in a parameter-efficient way. (3) Most binarization methods outperform the baseline model and ReActNet presents competing performance, especially on DRealSR. (4) The full-precision models, i.e., SinSR and ResShift, are excellent on perceptual metrics, such as DISTS and CLIP-IQA+. Whereas, the multi-step models'",
+        "bbox": [
+            84,
+            691,
+            475,
+            902
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "performance on PSNR and SSIM is on the lower side. This phenomenon means that compression on DM leads to disappearance of texture and details. However, their performance on PSNR and SSIM is only slightly affected.",
+        "bbox": [
+            495,
+            488,
+            887,
+            549
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Efficiency Comparison. Efficiency comparison results are provided in Table 2. After the distillation from multi-step to one-step, the FLOPs are significantly reduced. Furthermore, compressing the model to 1 bit makes the model tiny and fast. In comparison with SinSR, the compression ratio is  $27.45 \\times$  and the speedup ratio is  $23.81 \\times$ . Compared with current SOTA binarization methods, we keep almost the same parameters but take much less FLOPs on one-step DM, due to the calculation efficient design. To conclude, our proposed BiMaCoSR gains remarkable compression ratio and speedup ratio and achieves outstanding performance.",
+        "bbox": [
+            495,
+            553,
+            887,
+            719
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Visual Results. We present visual comparison on challenging cases in Fig. 4. One typical challenging case is the tiny and dense structures, such as hairs, grass, tiles, and faces. It is struggling with previous binarization methods to restore image details, especially in challenging cases. On the contrary, our BiMaCoSR is able to recover results with sharper edges and richer textures. For example, in 0809, BiMaCoSR reconstructs the hairs on the nose while other methods just output rough color blocks. And in 0831, BiMaCoSR restores the woman's facial structures and expression while other methods smooth the organs to the same color. In 0834, BiMaCoSR could successfully recover the tiles' texture and",
+        "bbox": [
+            495,
+            723,
+            888,
+            904
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution",
+        "bbox": [
+            124,
+            56,
+            848,
+            70
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/504da852a9ce47dce67c1cd9ce71f18c44b6438b4d36f392a9639f0eaa85d5f7.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>ResShift</td><td>SinSR</td><td>ReActNet</td><td>BBCU</td><td>ReSTE</td><td>BiDM</td><td>XNOR</td><td>Ours</td></tr><tr><td>Inference Step</td><td>15</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>FLOPs (G)</td><td>753.45</td><td>50.23</td><td>5.83</td><td>5.83</td><td>5.83</td><td>11.60</td><td>5.83</td><td>1.83</td></tr><tr><td># Total Param (M)</td><td>118.59</td><td>118.59</td><td>4.95</td><td>4.95</td><td>4.95</td><td>18.69</td><td>4.95</td><td>4.98</td></tr><tr><td>PSNR/LPIPS</td><td>25.45/0.3731</td><td>26.51/0.3635</td><td>26.60/0.3834</td><td>26.43/0.3902</td><td>26.26/0.4184</td><td>25.07/0.5042</td><td>26.48/0.3968</td><td>26.84/0.3375</td></tr></table>",
+        "bbox": [
+            86,
+            80,
+            883,
+            160
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Efficiency comparison on RealSR. BiMaCoSR takes least FLOPs while gains the best performance.",
+        "bbox": [
+            127,
+            161,
+            841,
+            176
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/b4a427ff36cfd8c96f2a6008a942d8f3546ec3b68085d4b2bf386010e7d9c8fb.jpg",
+        "image_caption": [
+            "Figure 4: Visual comparison for image SR. We compare our proposed BiMaCoSR with current competitive binarization methods and the full-precision (FP) model. The visual results illustrate that BiMaCoSR gains rich details and reasonable textures."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            84,
+            183,
+            883,
+            609
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "most binarization methods fail. What's more, the difference between BiMaCoSR and the FP model (SinSR) exists but is minimal. To conclude, our proposed BiMaCoSR surpasses other methods according to visual comparison. Additionally, we provide more visual results and corresponding analysis in the supplementary material for further comparison.",
+        "bbox": [
+            84,
+            646,
+            475,
+            737
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3. Ablation Study",
+        "text_level": 1,
+        "bbox": [
+            84,
+            746,
+            225,
+            762
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we adopt RealSR as the test set and other training settings are the same as Sec. 4.1. To test the effectiveness and robustness, we conduct five key ablation experiments, including the break down ablation, loss function, rank of low rank branch, initialization of LRMB, and initialization of SMB. What's more, detailed analysis is also provided in the following sections. These ablation studies demonstrate the robustness and efficiency of our LRMB, SMB, and their corresponding initialization methods.",
+        "bbox": [
+            84,
+            763,
+            475,
+            898
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Break Down Ablation. Table 3a shows results of the break down ablation. With only BMB, the model could be extremely compressed but the performance is somewhat on the low side. With LRMB, though the number of parameters increases to 1.83G, the performance also improves on all metrics and the FLOPs only increase to 4.98M. After adding SMB, the overhead of parameters and storage can be neglected. The performance on all perceptual metrics improves while both PSNR and SSIM drop slightly.",
+        "bbox": [
+            495,
+            646,
+            887,
+            782
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Loss Function. The loss functions of SinSR play a critical role in FP model. However, these loss functions are not suitable in the binarized situation. After removing unnecessary parts from the SinSR loss, we leave only the distillation loss and the results are shown in Table 3b. With only the distillation loss, BiMaCoSR gains improvement on both distortion metrics and perceptual metrics.",
+        "bbox": [
+            495,
+            789,
+            888,
+            896
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution",
+        "bbox": [
+            124,
+            56,
+            846,
+            70
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/1b022926a5884375f145d417e52013149bcc19509c7aada4b28eca50ac433e4c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Branch</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>MANIQA ↑</td><td>FID ↓</td><td>CLIP-IQA+ ↑</td><td>FLOPs (G)</td><td>Param (M)</td></tr><tr><td>BMB</td><td>26.41</td><td>0.7408</td><td>0.4141</td><td>0.3704</td><td>110.15</td><td>0.4325</td><td>0.78</td><td>3.69</td></tr><tr><td>+LRMB</td><td>26.95</td><td>0.7718</td><td>0.3400</td><td>0.3937</td><td>88.72</td><td>0.4663</td><td>1.83</td><td>4.98</td></tr><tr><td>+LRMB+SMB</td><td>26.84</td><td>0.7698</td><td>0.3375</td><td>0.4034</td><td>86.09</td><td>0.4800</td><td>1.83</td><td>4.98</td></tr></table>",
+        "bbox": [
+            96,
+            80,
+            883,
+            148
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/e97f1daf10ff7a158a197a8fe4591ca56048c59f19a3807aae25b328d55bab83.jpg",
+        "table_caption": [
+            "(a) Break down ablation."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Loss</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CLIP-IQA+ ↑</td></tr><tr><td>Distill loss</td><td>26.83</td><td>0.7698</td><td>0.3375</td><td>0.4800</td></tr><tr><td>SinSR loss</td><td>26.37</td><td>0.7466</td><td>0.4029</td><td>0.4273</td></tr></table>",
+        "bbox": [
+            98,
+            167,
+            478,
+            222
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/ec3b63de12511ff886a30cbeb77f1322994181e2956c13c4c01a263b37033976.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Initialization</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CLIP-IQA+ ↑</td></tr><tr><td>Zero + Random</td><td>26.88</td><td>0.7660</td><td>0.3497</td><td>0.4674</td></tr><tr><td>SVD</td><td>26.84</td><td>0.7698</td><td>0.3375</td><td>0.4800</td></tr></table>",
+        "bbox": [
+            491,
+            167,
+            885,
+            222
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/198d7dd8d4b6d5230348067ff9217fd391bf65e1d0ed89038a891b71b702852e.jpg",
+        "table_caption": [
+            "(b) Ablation study on losses."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Rank</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CLIP-IQA+↑</td><td>FLOPs (G)</td><td>Param (M)</td></tr><tr><td>4</td><td>26.29</td><td>0.7383</td><td>0.4197</td><td>0.4411</td><td>1.31</td><td>4.37</td></tr><tr><td>8</td><td>26.84</td><td>0.7698</td><td>0.3375</td><td>0.4800</td><td>1.83</td><td>4.98</td></tr><tr><td>12</td><td>26.97</td><td>0.7695</td><td>0.3400</td><td>0.4766</td><td>2.32</td><td>5.60</td></tr><tr><td>16</td><td>26.72</td><td>0.7625</td><td>0.3442</td><td>0.4971</td><td>2.83</td><td>6.21</td></tr></table>",
+        "bbox": [
+            96,
+            241,
+            529,
+            319
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/87b434b9acd4ecb2d7f0572f34d85b33abeab9b882ae9c4a5523b9edcd6bf43c.jpg",
+        "table_caption": [
+            "(c) Ablation study on LRMB initialization."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Initialization</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CLIP-IQA+ ↑</td></tr><tr><td>Zero initial</td><td>26.72</td><td>0.7628</td><td>0.3561</td><td>0.4639</td></tr><tr><td>Uni-shortcut</td><td>25.69</td><td>0.7113</td><td>0.5105</td><td>0.4008</td></tr><tr><td>Sparse skip</td><td>26.84</td><td>0.7698</td><td>0.3375</td><td>0.4800</td></tr></table>",
+        "bbox": [
+            547,
+            253,
+            880,
+            319
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(d) Ablation study on the rank of LRMB.",
+        "bbox": [
+            181,
+            324,
+            428,
+            337
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(e) Ablation study on SMB initialization.",
+        "bbox": [
+            584,
+            324,
+            828,
+            337
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Ablation studies on branches, loss, rank of LRMB and initialization of LRMB and SMB. The experiments is tested on RealSR. The comprehensive results demonstrate the robustness and efficient performance of our proposed BiMaCoSR.",
+        "bbox": [
+            83,
+            338,
+            885,
+            364
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Rank of LRMB. The rank of LRMB significantly influences the complexity, while the performance is not consistent with the increase in complexity. As shown in Table 3d, the model with rank of 16 performs best only on CLIP-IQA+, while the model with rank of 8 performs best and takes acceptable parameters. This is because the LRMB with high rank may influence high frequency generated by BMB, making the training unstable and leading to worse results. Therefore, we ultimately set the rank of LRMB to 8.",
+        "bbox": [
+            83,
+            371,
+            475,
+            507
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Initialization of LRMB. Conventional LoRA's initialization method is set the first matrix with random values and the second matrix with zero values. We compare this initialization method with our proposed SVD initialization and the result is shown in Table 3c. To conclude, our method outperforms conventional methods on SSIM, LPIPS, and CLIP-  $\\mathrm{QA + }$  .We attribute it to the decoupling of transmitting low frequency information and generating high frequency. To conclude, our proposed SVD allows better performance.",
+        "bbox": [
+            83,
+            515,
+            473,
+            651
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Initialization of SMB. We search three popular ways to initialize SMB, including (1) random position and zero initialization, (2) Uni-shortcut (Xu et al., 2022), and (3) our proposed sparse skip. The experiment results in Table 3e show that our sparse skip initialization consistently enjoys the best performance on all four metrics. This improvement compared with other two initialization methods strongly supports the effectiveness of decoupling and absorption.",
+        "bbox": [
+            83,
+            657,
+            473,
+            780
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4. Visualization Analysis.",
+        "text_level": 1,
+        "bbox": [
+            84,
+            785,
+            277,
+            800
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Sec. 3, we assume that LRMB and SMB are in charge of transmitting low frequency information while BMB generates high frequency after decoupling. To validate this assumption, we visualize the proportion of high frequency generated by three branches in first 50 Conv layers and the result is shown in Fig. 5. On average, high frequency infor",
+        "bbox": [
+            84,
+            805,
+            475,
+            896
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/fd8f96c764ce966ef4c0378e08826a27fa4c874943b9556ebbbf1a8b87381884.jpg",
+        "image_caption": [
+            "Figure 5: The proportion of high frequency information generated by three branches. The high frequency information mainly comes from BMB, which obeys our assumption."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            498,
+            369,
+            883,
+            542
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "mation is mainly generated by BMB, accounting for around  $70\\%$ . With more parameters, LRMB also provides high frequency information in some layers but overall proportion is relative low. SMB only absorbs the extreme values and indeed carries little high frequency information. Fig. 5 shows that division of work is clear and our designs in LRMB, SMB and BMB obey our assumption.",
+        "bbox": [
+            495,
+            595,
+            885,
+            700
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5. Conclusion",
+        "text_level": 1,
+        "bbox": [
+            496,
+            705,
+            614,
+            720
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose the BiMaCoSR, a binarized SR diffusion model with only one inference step. Detailedly, we first propose LRMB and SVD initialization to decouple the effect of binarized branch and deliver low frequency information. Furthermore, we propose SMB and sparse initialization to absorb the extreme values and provide high rank representations. Comprehensive comparison experiments demonstrate the SOTA restoration ability of the proposed BiMaCoSR. Extensive ablation studies exhibit the efficiency and robustness of both LRMB and SMB. In the future, we will focus on the combination of pruning and binarization on one-step diffusion models for further compression.",
+        "bbox": [
+            495,
+            724,
+            885,
+            905
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "header",
+        "text": "BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution",
+        "bbox": [
+            124,
+            56,
+            846,
+            70
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements",
+        "text_level": 1,
+        "bbox": [
+            86,
+            83,
+            253,
+            99
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work was supported by Shanghai Municipal Science and Technology Major Project (2021SHZDZX0102), the Fundamental Research Funds for the Central Universities, and the National Natural Science Foundation of China (NSFC No. W2412089).",
+        "bbox": [
+            84,
+            109,
+            475,
+            184
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Impact Statement",
+        "text_level": 1,
+        "bbox": [
+            86,
+            204,
+            240,
+            220
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.",
+        "bbox": [
+            84,
+            229,
+            475,
+            291
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
+            86,
+            310,
+            181,
+            325
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Agustsson, E. and Timofte, R. Ntire 2017 challenge on single image super-resolution: Dataset and study. In CVPRW, 2017.",
+            "Cai, J., Zeng, H., Yong, H., Cao, Z., and Zhang, L. Toward real-world single image super-resolution: A new benchmark and a new model. In CVPR, 2019.",
+            "Chen, C. and Mo, J. IQA-PyTorch: Pytorch toolbox for image quality assessment. [Online]. Available: https://github.com/chaofengc/IQA-PyTorch, 2022.",
+            "Chen, Z., Zhang, Y., Gu, J., Zhang, Y., Kong, L., and Yuan, X. Cross aggregation transformer for image restoration. In NeurIPS, 2022.",
+            "Chen, Z., Zhang, Y., Gu, J., Kong, L., Yang, X., and Yu, F. Dual aggregation transformer for image super-resolution. In CVPR, 2023.",
+            "Chen, Z., Qin, H., Guo, Y., Su, X., Yuan, X., Kong, L., and Zhang, Y. Binarized diffusion model for image superresolution. In NeurIPS, 2024.",
+            "Ding, K., Ma, K., Wang, S., and Simoncelli, E. P. Image quality assessment: Unifying structure and texture similarity. TPAMI, 2020.",
+            "Dong, C., Loy, C. C., He, K., and Tang, X. Learning a deep convolutional network for image super-resolution. In ECCV, 2014.",
+            "Dong, C., Loy, C. C., He, K., and Tang, X. Image superresolution using deep convolutional networks. TPAMI, 2016.",
+            "Gu, J., Lu, H., Zuo, W., and Dong, C. Blind super-resolution with iterative kernel correction. In CVPR, 2019.",
+            "Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., and Hochreiter, S. Gans trained by a two time-scale update rule converge to a local nash equilibrium. NIPS, 2017."
+        ],
+        "bbox": [
+            86,
+            333,
+            475,
+            905
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Huang, W., Zheng, X., Ma, X., Qin, H., Lv, C., Chen, H., Luo, J., Qi, X., Liu, X., and Magno, M. An empirical study of llama3 quantization: From llms to mllms. Visual Intelligence, 2(1):36, 2024.",
+            "Ke, J., Wang, Q., Wang, Y., Milanfar, P., and Yang, F. Musiq: Multi-scale image quality transformer. In ICCV, 2021.",
+            "Keys, R. Cubic convolution interpolation for digital image processing. IEEE Trans. Acoust., Speech, Signal Process., 1981.",
+            "Kingma, D. and Ba, J. Adam: A method for stochastic optimization. In ICLR, 2015.",
+            "Le, P.-H. C. and Li, X. Binaryvit: pushing binary vision transformers towards convolutional models. In CVPR, 2023.",
+            "Ledig, C., Theis, L., Huszár, F., Caballero, J., Cunningham, A., Acosta, A., Aitken, A. P., Tejani, A., Totz, J., Wang, Z., et al. Photo-realistic single image super-resolution using a generative adversarial network. In CVPR, 2017.",
+            "Liang, J., Cao, J., Sun, G., Zhang, K., Van Gool, L., and Timofte, R. Swinir: Image restoration using swim transformer. In ICCVW, 2021.",
+            "Lim, B., Son, S., Kim, H., Nah, S., and Lee, K. M. Enhanced deep residual networks for single image super-resolution. In CVPRW, 2017.",
+            "Lin, X., He, J., Chen, Z., Lyu, Z., Dai, B., Yu, F., Qiao, Y., Ouyang, W., and Dong, C. Diffbir: Toward blind image restoration with generative diffusion prior. In ECCV, 2024.",
+            "Liu, Z., Shen, Z., Savvides, M., and Cheng, K.-T. Reactnet: Towards precise binary neural network with generalized activation functions. In ECCV, 2020.",
+            "Lu, X., Yuan, Y., and Yan, P. Image super-resolution via double sparsity regularized manifold learning. TCSVT, 2013.",
+            "Luo, X., Qu, Y., Xie, Y., Zhang, Y., Li, C., and Fu, Y. Lattice network for lightweight image restoration. TPAMI, 2022.",
+            "Qin, H., Gong, R., Liu, X., Shen, M., Wei, Z., Yu, F., and Song, J. Forward and backward information retention for accurate binary neural networks. In CVPR, 2020.",
+            "Qin, H., Zhang, X., Gong, R., Ding, Y., Xu, Y., and Liu, X. Distribution-sensitive information retention for accurate binary neural network. IJCV, 2022.",
+            "Qin, H., Zhang, M., Ding, Y., Li, A., Cai, Z., Liu, Z., Yu, F., and Liu, X. Bibench: Benchmarking and analyzing network binarization. In ICML, 2023."
+        ],
+        "bbox": [
+            498,
+            84,
+            887,
+            905
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "header",
+        "text": "BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution",
+        "bbox": [
+            124,
+            56,
+            846,
+            71
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
+        "bbox": [
+            480,
+            922,
+            491,
+            934
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Rastegari, M., Ordonez, V., Redmon, J., and Farhadi, A. Xnor-net: Imagenet classification using binary convolutional neural networks. In ECCV, 2016.",
+            "Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. High-resolution image synthesis with latent diffusion models. In CVPR, 2022.",
+            "Russakovsky, O., Deng, J., Su, H., Krause, J., Satheesh, S., Ma, S., Huang, Z., Karpathy, A., Khosla, A., Bernstein, M., et al. Imagenet large scale visual recognition challenge. IJCV, 2015.",
+            "Saharia, C., Ho, J., Chan, W., Salimans, T., Fleet, D. J., and Norouzi, M. Image super-resolution via iterative refinement. IEEE TPAMI, 2022.",
+            "Sun, H., Li, W., Liu, J., Chen, H., Pei, R., Zou, X., Yan, Y., and Yang, Y. Coser: Bridging image and language for cognitive super-resolution. In CVPR, 2024.",
+            "Wang, J., Chan, K. C., and Loy, C. C. Exploring clip for assessing the look and feel of images. In AAAI, 2023a.",
+            "Wang, X., Xie, L., Dong, C., and Shan, Y. Real-esrgan: Training real-world blind super-resolution with pure synthetic data. In CVPR, 2021.",
+            "Wang, Y., Yang, W., Chen, X., Wang, Y., Guo, L., Chau, L.-P., Liu, Z., Qiao, Y., Kot, A. C., and Wen, B. Sinsr: diffusion-based image super-resolution in a single step. In CVPR, 2024.",
+            "Wang, Z., Bovik, A. C., Sheikh, H. R., and Simoncelli, E. P. Image quality assessment: from error visibility to structural similarity. TIP, 2004.",
+            "Wang, Z., Cun, X., Bao, J., Zhou, W., Liu, J., and Li, H. Uformer: A general u-shaped transformer for image restoration. In CVPR, 2022.",
+            "Wang, Z., Wang, J., Liu, Z., and Qiu, Q. Binary latent diffusion. In CVPR, 2023b.",
+            "Wei, P., Xie, Z., Lu, H., Zhan, Z., Ye, Q., Zuo, W., and Lin, L. Component divide-and-conquer for real-world image super-resolution. In ECCV, 2020.",
+            "Wu, R., Sun, L., Ma, Z., and Zhang, L. One-step effective diffusion network for real-world image super-resolution. arXiv, 2024a.",
+            "Wu, R., Yang, T., Sun, L., Zhang, Z., Li, S., and Zhang, L. Seesr: Towards semantics-aware real-world image super-resolution. In CVPR, 2024b.",
+            "Wu, X.-M., Zheng, D., Liu, Z., and Zheng, W.-S. Estimator meets equilibrium perspective: A rectified straight through estimator for binary neural networks training. In ICCV, 2023."
+        ],
+        "bbox": [
+            86,
+            84,
+            475,
+            904
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Xia, B., Zhang, Y., Wang, Y., Tian, Y., Yang, W., Timofte, R., and Van Gool, L. Basic binary convolution unit for binarized image restoration network. In ICLR, 2022.",
+            "Xu, Y., Chen, X., and Wang, Y. Bimlp: Compact binary architectures for vision multi-layer perceptrons. NeurIPS, 2022.",
+            "Yan, D., Wu, T., Liu, Y., and Gao, Y. An efficient sparse-dense matrix multiplication on a multicore system. In ICCT, 2017.",
+            "Yang, C.-Y., Ma, C., and Yang, M.-H. Single-image superresolution: A benchmark. In ECCV, 2014.",
+            "Yang, S., Wu, T., Shi, S., Lao, S., Gong, Y., Cao, M., Wang, J., and Yang, Y. Maniaq: Multi-dimension attention network for no-reference image quality assessment. In CVPR, 2022.",
+            "Yang, T., Wu, R., Ren, P., Xie, X., and Zhang, L. Pixel-aware stable diffusion for realistic image super-resolution and personalized stylization. In ECCV, 2025.",
+            "Yu, F., Gu, J., Li, Z., Hu, J., Kong, X., Wang, X., He, J., Qiao, Y., and Dong, C. Scaling up to excellence: Practicing model scaling for photo-realistic image restoration in the wild. In CVPR, 2024.",
+            "Yue, Z., Wang, J., and Loy, C. C. Resshift: Efficient diffusion model for image super-resolution by residual shifting. NeurIPS, 2024.",
+            "Zhang, K., Liang, J., Van Gool, L., and Timofte, R. Designing a practical degradation model for deep blind image super-resolution. In CVPR, 2021.",
+            "Zhang, L., Zhang, L., and Bovik, A. C. A feature-enriched completely blind image quality evaluator. TIP, 2015.",
+            "Zhang, R., Isola, P., Efros, A. A., Shechtman, E., and Wang, O. The unreasonable effectiveness of deep features as a perceptual metric. In CVPR, 2018a.",
+            "Zhang, Y., Li, K., Li, K., Wang, L., Zhong, B., and Fu, Y. Image super-resolution using very deep residual channel attention networks. In ECCV, 2018b.",
+            "Zhang, Y., Tian, Y., Kong, Y., Zhong, B., and Fu, Y. Residual dense network for image super-resolution. In CVPR, 2018c.",
+            "Zheng, X., Liu, X., Bian, Y., Ma, X., Zhang, Y., Wang, J., Guo, J., and Qin, H. Bidm: Pushing the limit of quantization for diffusion models. In NeurIPS, 2024.",
+            "Zibetti, M. V. W. and Mayer, J. A robust and computationally efficient simultaneous super-resolution scheme for image sequences. TCSVT, 2007."
+        ],
+        "bbox": [
+            498,
+            84,
+            885,
+            882
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "header",
+        "text": "BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution",
+        "bbox": [
+            124,
+            56,
+            846,
+            71
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "page_number",
+        "text": "10",
+        "bbox": [
+            477,
+            922,
+            495,
+            934
+        ],
+        "page_idx": 9
+    }
+]

2025/BiMaCoSR_ Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution/015a9533-715f-4e8b-a7b5-d0f1265ce5d2_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:dfe9476ca836f9752a58b59b9faab5d1ffaa68919ba98f314d11b0b8925d9404
+size 2527429

2025/BiMaCoSR_ Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution/full.md

@@ -0,0 +1,327 @@
+# BiMaCoSR: Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution
+
+Kai Liu  $^{*1}$  Kaicheng Yang  $^{*1}$  Zheng Chen  $^{1}$  Zhiteng Li  $^{1}$  Yong Guo  $^{2}$  Wenbo Li  $^{3}$  Linghe Kong  $^{†1}$  Yulun Zhang  $^{†1}$
+
+# Abstract
+
+While super-resolution (SR) methods based on diffusion models (DM) have demonstrated inspiring performance, their deployment is impeded due to the heavy request of memory and computation. Recent researchers apply two kinds of methods to compress or fasten the DM. One is to compress the DM into 1-bit, aka binarization, alleviating the storage and computation pressure. The other distills the multi-step DM into only one step, significantly speeding up the inference process. Nonetheless, it remains impossible to deploy DM to resource-limited edge devices. To address this problem, we propose BiMaCoSR, which combines binarization and one-step distillation to obtain extreme compression and acceleration. To prevent the catastrophic collapse of the model caused by binarization, we propose sparse matrix branch (SMB) and low rank matrix branch (LRMB). Both auxiliary branches pass the full-precision (FP) information but in different ways. SMB absorbs the extreme values and its output is high rank, carrying abundant FP information. Whereas, the design of LRMB is inspired by LoRA and is initialized with the top  $r$  SVD components, outputting low rank representation. The computation and storage overhead of our proposed branches can be safely ignored. Comprehensive comparison experiments are conducted to exhibit BiMaCoSR outperforms current state-of-the-art binarization methods and gains competitive performance compared with FP one-step model. Moreover, we achieve excellent compression and acceleration. BiMaCoSR achieves a  $23.8 \times$  compression ratio and a  $27.4 \times$  speedup ratio compared to the FP counterpart. Our code and model are available at https://github.com/KaiLiu001/BiMaCoSR.
+
+![](images/baaab697a0437f0f13cceab2954ae5d48d6a6810aeaf440382239a942084e70b.jpg)  
+Figure 1: Performance comparison between binarization methods on the RealSR dataset. BiMaCoSR achieves consistently leading scores on all evaluation metrics.
+
+# 1. Introduction
+
+Single image super-resolution (SR) (Keys, 1981; Zibetti & Mayer, 2007; Lu et al., 2013; Dong et al., 2014; Yang et al., 2014; Dong et al., 2016) is a traditional yet challenging low-level vision problem. Serving as a fundamental research task, it has attracted long-standing and considerable attention in the computer vision community. The final object of SR is to restore a high-quality (HQ) image from its low-quality (LQ) observation, which suffers from various image quality degradations. The difficulty of SR mainly lies in two parts: (1) the unknown degradations (e.g., blur, downsampling, noise, compression, and their combinations) of LQ. (2) the multiple solutions for a given LQ input image.
+
+In recent years, numerous studies have been made to tackle this challenge, utilizing convolution neural networks (CNNs) (Dong et al., 2014; 2016; Ledig et al., 2017; Zhang et al., 2018c), vision transformers (ViTs) (Zhang et al., 2018b; Liang et al., 2021; Wang et al., 2022; Chen et al., 2022; 2023), and their combinations. Though achieving inspiring results, these methods mostly fail in real-world scenarios. This failure is attributed to assuming degradation as an ideal bicubic downsampling kernel, way too different from the unknown and complex degradation in real world. Therefore, it draws researchers' increasing attention to reconstruct perceptually realistic HQ images in real-world scenarios. Thereafter, this challenging and meaningful task is called real-world super-resolution (Real-SR) (Gu et al.,
+
+2019; Zhang et al., 2021; Wang et al., 2021; Cai et al., 2019; Yu et al., 2024; Wu et al., 2024b; Sun et al., 2024).
+
+Recently, diffusion models (DMs) demonstrate remarkable performance in image-generating tasks, particularly in perceptual quality. The excellence of DM comes from its vast prior knowledge in modeling real-world objects, especially generating clear textures and mitigating artifacts and distortions. The powerful realistic texture generation ability is inherently the same as the object of Real-SR problem, leading to plentiful breakthroughs (Yang et al., 2025; Yu et al., 2024; Rombach et al., 2022). However, the inference cost is still too high to run on edge devices. Therefore, it's essential to further compress DMs to accelerate the inference, reduce storage cost, and minimize degradation.
+
+Popular model compression techniques include pruning, distillation, and quantization, among which, 1-bit quantization (i.e., binarization) gains significant effectiveness. As an extreme quantization method, binarization compresses models' weights from 32-bit to only 1-bit, significantly reducing memory and computational cost. However, applying naive binarization will lead to catastrophic model collapse. Hence, additional structures are required to be designed.
+
+A popular solution is adding full-precision information, i.e., skip connection branch. However, due to UNet's frequent changes in resolution and dimension, skip connection faces the mismatch challenge. To address this problem, we propose two auxiliary branches, namely low rank matrix branch (LRMB) and sparse matrix branch (SMB). Inspired by LoRA, the proposed LRMB leverages low rank decomposition to achieve dimension shift. We select the top  $r$  singular values in SVD and utilize its corresponding components to initialize the LRMB. As for SMB, we employ sparse matrix to absorb the top  $k$  absolute values in a full-precision matrix. The weights of LRMB and SMB are subtracted from the binarized matrix branch (BMB) and it concentrates on restoring textures. Three branches form the BiMaCoSR and provide excellent performance shown in Fig. 1.
+
+To sum up, the contributions of our work are as follows:
+
+1. We design BiMaCoSR, a new binarized one-step diffusion model for image super-resolution. To the best of our knowledge, BiMaCoSR is the first binarized one-step diffusion model.  
+2. We propose LRMB, which leverages low rank decomposition and SVD initialization to carry low frequency information and decouple the effect of BMB.  
+3. We propose SMB, which utilizes sparse matrix compression and extreme value absorption to deliver high rank features and achieve further decoupling.  
+4. We conduct comprehensive comparison experiments to show the state-of-the-art performance of the proposed BiMaCoSR. Besides, extensive ablation studies are conducted to prove the robustness and efficacy.
+
+# 2. Related Work
+
+Image Super-Resolution. Deep learning based approaches have demonstrated striking power in the realm of SR (Dong et al., 2014; Luo et al., 2022; Wang et al., 2021; Lim et al., 2017; Chen et al., 2023). As a groundbreaking work, SRCNN (Dong et al., 2014) initiates the track of solving SR problem via deep learning based approach. Thereafter, substantial contributions have been made to explore the best SR network architecture. For example, RCAN (Zhang et al., 2018b) leverages the residual in residual structure and deepens the network to more than 400 layers. SwinIR (Liang et al., 2021) is based on vision transformer structure and utilizes spatial window self-attention to capture the overall structure information. CAT (Chen et al., 2022) combines the attention mechanism and the CNN structure to make the most of the local and the global information. However, most of these conventional image super-resolution methods can not handle the Real-SR task because of the complex degradation in the real world.
+
+Diffusion Model. In recent years, the diffusion-based methods have gained remarkable performance in many computer vision tasks and SR is no exception. For instance, SR3 (Saharia et al., 2022) restores the LQ by transforming the standard normal distribution into the empirical data distribution by learning a series of iterative refinement steps. DiffBIR (Lin et al., 2024) capitalizes on two restoration stages to seek the tradeoff of fidelity and quality. SinSR (Wang et al., 2024) effectively reduces the inference step to only one step via distillation and regularization. Following SinSR, OSEDiff (Wu et al., 2024a) modifies the distillation paradigm, and novel losses are introduced to improve face restoration ability. Despite the greatly improved inference speed, the model size remains the same, and there is still room for further acceleration.
+
+Binarization. As the most extreme form of quantization, binarization typically compresses the weight into only 1 bit. In binarization, all the weights are seen as  $\pm 1$  and the multiplications between weights and activations are converted to bit operations on sign bit of activation, allowing maximum compression and acceleration. Binarization is mainly about classification tasks initially (Rastegari et al., 2016; Liu et al., 2020; Qin et al., 2020; 2022; Huang et al., 2024). Recently, researchers have begun to perform binarization on image restoration tasks. Binary Latent Diffusion (Wang et al., 2023b) trains an auto-encoder with a binary latent space and mainly focuses on the Bernoulli distribution instead of acceleration. BiDM (Zheng et al., 2024) leverages timestep-friendly binary structure and space-patched distillation to compress the diffusion model to 1 bit. BI-DiffSR (Chen et al., 2024) designs several binary friendly modules and redistributes the activation of different time step. However, the inference step remains the same. Therefore, it is necessary to further compress the model to one step.
+
+![](images/c28c1c514f0bcd8cdfd02975fa1400d41af3bd66cc8f88aba9076d9c5bf1f23a.jpg)  
+Figure 2: Overview of our proposed BiMaCoSR which employs three different compressed matrix branches. (a) The structure of a convolution layer in BiMaCoSR after binarization. Two auxiliary branches, i.e., LRMB and SMB, support BiMaCoSR's excellent performance. The linear layer can be regarded as a  $1 \times 1$  convolution layer and is processed with the same pipeline. (b) Illustration of the initialization and how the three branches solve the weakness of the other branches.
+
+# 3. Methodology
+
+In this section, we describe our proposed BiMaCoSR, shown in Fig. 2. First, we analyze three issues that have potential for further improvement when binarizing diffusion models. Thereafter, we describe our proposed branches, i.e., low rank matrix branch (LRMB) and sparse matrix branch (SMB), which could serve as the auxiliary branch to binarized matrix branch (BMB). Finally, we illustrate the initialization methods, shown in Fig. 3, and why our designs work.
+
+# 3.1. Analysis
+
+Binarized blocks inevitably suffer from representation degradation due to the extreme 1-bit compression. Previous research made remarkable progress (Chen et al., 2024; Zheng et al., 2024) on binarization, yet there still remains room for improvement. We conclude three issues that deserve special attention in most diffusion models.
+
+Issue I: The specially severe degradation of linear layer. Experimentally, we observe that with the same size of parameters, linear layer suffers from more severe degradation in comparison to convolution layer. This observation in SR is consistent with previous research (Le & Li, 2023). With the advancement of DiT and binarization, this issue is gradually becoming significant and an additional process to quantize or binarize the linear layer is in urgent need.
+
+Issue II: The frequent dimension changes. It's necessary to leverage skip connection as an auxiliary branch in binarized network. Skip connection could carry abundant feature information and bring negligible computation and storage overhead. However, in UNet, frequent changes in dimension
+
+and resolution make two forms of skip connection (i.e., addition and concatenation) not applicable. In addition, the distributions before and after binarized block are greatly different. Therefore, efficient module is critical to overcome the frequent changes and carry rich information.
+
+Issue III: The inadequate usage of pre-trained FP model. The pre-trained FP model is vital to quantization methods from any perspective. However, current QAT research does not make the most of the pre-trained model, simply loading the FP parameters and directly beginning the training. It is a popular way to leverage additional modules to enhance the binarized network. The impact of the modules could vary when different initialization methods are applied.
+
+We conclude that the above issues are attributed to the insufficient ability of binarization. To be specific, binarization is just one of the matrix compression methods, inherently unsuitable for multifaceted challenges. Therefore, we propose following branches to compress matrix in different ways. With the combination of various compressed matrices, the issues above can be appropriately addressed.
+
+# 3.2. Low Rank Matrix Branch
+
+To address Issue I, i.e., the specially severe degradation of linear layer, we identify that the quantized linear layer is strongly limited by the bit-width. An FP linear layer usually serves two purposes, delivering the complete information of input and performing appropriate linear transformation. With limited bit-width, the quantized linear layer cannot play two roles at the same time. Therefore, we decouple the roles apart. Though binarized, the weight matrix is usually high rank and contributes more to high frequency, i.e., the
+
+detailed structure and texture in image. Hence, we design low rank matrix branch (LRMB) to serve as a complementary branch, transmitting low frequency information.
+
+The idea of LRMB is inspired by low rank approximation in matrix theory, which is a common strategy for matrix compression. To transmit low frequency information, matrices with low rank are enough and efficient. Mathematically, given an  $m \times n$  matrix  $\mathbf{W}$ , it can be approximated with two low rank matrices, i.e.,  $\mathbf{W} \approx \hat{\mathbf{W}} \coloneqq \mathbf{BA}$ , where  $\mathbf{B}$  and  $\mathbf{A}$  are  $m \times r$  and  $r \times n$  matrices respectively, and  $r \ll \min(m, n)$  is a hyper-parameters denoting the rank of  $\hat{\mathbf{W}}$ . Therefore, the formula of LRMB is:
+
+$$
+\mathbf {x} _ {\mathrm {L R M B}} = \operatorname {L R M B} \left(\mathbf {x} _ {\text {i n}}\right) := \mathbf {x} _ {\text {i n}} \mathbf {B A}, \tag {1}
+$$
+
+where  $\mathbf{x}_{\mathrm{in}}\in \mathbb{R}^{N\times m}$  and  $\mathbf{x}_{\mathrm{LRMB}}\in \mathbb{R}^{N\times n}$  are the input and output of LRMB(·) respectively, and  $N$  is the number of tokens. Usually,  $\mathbf{W}$  is a square matrix and  $m = n$
+
+As for complexity, the storage overhead is:
+
+$$
+O _ {s} = (m \times r + r \times n) B = r B (m + n) \ll m n B ^ {\prime}, \tag {2}
+$$
+
+where  $B = 32$  or 16 and  $B' = 1$  are the number of bits required by one element in LRMB branch and binarized branch, respectively. Meanwhile, the computation overhead is:
+
+$$
+O _ {c} = N \times m \times r + N \times r \times n = N r (m + n). \tag {3}
+$$
+
+This means that even saved with 32 bits, the storage and computation overhead of LRMB is acceptable. In conclusion, with LRMB, Issue I and Issue II are partly solved.
+
+# 3.3. Sparse Matrix Branch
+
+LRMB is still not enough to replace the skip connection. This is because skip connection can be represented as a full rank identity matrix while LRMB is low rank. To compensate the missed ranks, we propose sparse matrix branch (SMB), which leverages sparse matrix compression and could efficiently deliver high rank information.
+
+Sparse matrix is a matrix in which most of the elements are zero (Yan et al., 2017). A common criterion of a sparse matrix is that the number of non-zero elements is approximated to the number of rows or columns. One way to save a sparse matrix is via the coordinate format (COO), where the non-zero elements are represented by a list of triples and its component is (row, col, value).
+
+Specifically, to form a sparse matrix, we select  $k$  critical values from the weight matrix and the selection method will be described in Sec. 3.4. In the forward process, the SMB process could be equivalently expressed as
+
+$$
+\mathbf {x} _ {\mathrm {S M B}} = \operatorname {S M B} \left(\mathbf {x} _ {\mathrm {i n}}\right) := \mathbf {x} _ {\mathrm {i n}} \mathbf {W} _ {\text {s p a r s e}}, \tag {4}
+$$
+
+where  $\mathbf{W}_{\mathrm{sparse}}$  is a matrix with  $k$  non-zero elements. During training, the coordinates will be fixed while their values will be updated by gradient descent. With SMB, the high rank information could be delivered and the remaining parts of Issue I and Issue II could be solved.
+
+![](images/e9cb73fb1cdfaeec990d76fcf083c9423dad56686dc27e68592283df72fa3fc3.jpg)  
+Figure 3: Initialization of different branch.  $W_{res}$  represents the initial quantization error. In our method,  $\| W_{res}\| _F^2 = 0.1855$ , while  $\| W_{res}\| _F^2 = 1.1275$  in direct binarization.
+
+# 3.4. Pretrained-Friendly Initialization
+
+We propose the following initialization method to guarantee better use of the pre-trained model, allowing better restoration performance and faster convergence.
+
+The purpose of LRMB is to carry the low frequency information. Usually, compared with high frequency information, the magnitude of low frequency information is much greater. Therefore, we leverage SVD and select the top  $r$  components to initialize LRMB. To be specific, we perform SVD on  $\mathbf{W}$ :
+
+$$
+\mathbf {W} = \mathbf {U} \boldsymbol {\Sigma} \mathbf {V} ^ {T} = \mathbf {U} \operatorname {d i a g} \left\{\sigma_ {1}, \sigma_ {2}, \dots , \sigma_ {n} \right\} \mathbf {V} ^ {T}. \tag {5}
+$$
+
+Then, we truncate the top  $r$  singular values and absorb the remaining singular value into  $\mathbf{U}$ , forming matrices  $\mathbf{A}$  and  $\mathbf{B}$ . The process can be written as:
+
+$$
+\mathbf {W} \approx \underbrace {\mathbf {U} \operatorname {d i a g} \left\{\sigma_ {1} , \sigma_ {2} , \dots , \sigma_ {r} , 0 , \dots , 0 \right\}} _ {B} \underbrace {\mathbf {V} ^ {T}} _ {A} = \mathbf {B} \mathbf {A}. \tag {6}
+$$
+
+Thereafter, to guarantee that the binarized matrix only serves to add high frequency information, the low frequency counterpart will be subtracted. The formula is:
+
+$$
+\mathbf {W} _ {\mathrm {B M B}} ^ {\prime} := \mathbf {W} - \mathbf {B A}, \tag {7}
+$$
+
+where  $\mathbf{W}_{\mathrm{BMB}}^{\prime}$  is the matrix to be binarized after the initialization of LRMB. Hence, the ability of weight matrix is decoupled into two components, the low frequency part (BA) and the high frequency part  $(\mathbf{W}_{\mathrm{BMB}}^{\prime})$
+
+Thereafter, the initialization of SMB also matters. As described in Sec. 3.3, we select  $k$  values from the weight matrix to generate the sparse matrix. Due to the fixed coordinates of non-zero elements, random selection is apparently a good candidate. Besides, binarization often ignores the outlier values in weight matrix, which are often huge but rare (less than  $0.1\%$ ). Though rare, these outliers play a crucial role in most models. Considering their rarity and significance, we select  $k$  elements with the greatest absolute value in  $\mathbf{W}_{\mathrm{BMB}}^{\prime}$ . Subsequently, we subtract these values in  $\mathbf{W}_{\mathrm{BMB}}^{\prime}$  to decouple the overlapping effect of BMB and SMB. The process can be formulated as:
+
+$$
+\widetilde {w} _ {i j} = \left\{ \begin{array}{l l} w _ {i j}, & \text {i f} | w _ {i j} ^ {\prime} | \geq t \\ 0, & \text {o t h e r w i s e} \end{array} , \mathbf {W} _ {\mathrm {B M B}} := \mathbf {W} _ {\mathrm {B M B}} ^ {\prime} - \mathbf {W} _ {\text {s p a r s e}}, \right. \tag {8}
+$$
+
+where  $\widetilde{w}_{ij}$  and  $w_{ij}$  are the elements of  $\mathbf{W}_{\mathrm{sparse}}$  and  $\mathbf{W}_{\mathrm{BMB}}^{\prime}$  respectively,  $t$  is the  $k$ -th largest absolute value in  $\mathbf{W}_{\mathrm{BMB}}^{\prime}$ ,  $\mathbf{W}_{\mathrm{BMB}}$  is the final matrix to be binarized.
+
+# 3.5. Overall Structure
+
+After adding LRMB and SMB and their initialization, the binarized matrix branch (BMB) is formed with  $\mathbf{x}_{\mathrm{BMB}}$ . To be specific, the input feature and weight matrix will be binarized with  $\mathrm{Sign}(\cdot)$ , which can be written as:
+
+$$
+\left\{ \begin{array}{l} \mathbf {x} _ {\mathrm {b}} = \operatorname {S i g n} \left(\mathbf {x} _ {\mathrm {i n}}\right), \\ \mathbf {W} _ {\mathrm {b}} = \operatorname {S i g n} \left(\mathbf {W} _ {\mathrm {B M B}}\right), \end{array} \quad \operatorname {S i g n} (x) = \left\{ \begin{array}{l l} + 1, & x \geq 0, \\ - 1, & x <   0, \end{array} \right. \right. \tag {9}
+$$
+
+where  $\operatorname{Sign}(\cdot)$  is performed element-wise. Thereafter, the full-precision convolution and linear transform are replaced with efficient logical XNOR and bit-counting operations. The process can be formulated as:
+
+$$
+\mathbf {x} _ {\mathrm {B M B}} ^ {\prime} = \text {b i t - c o u n t} (\mathrm {X N O R} (\mathbf {x} _ {\mathrm {b}}, \mathbf {W} _ {\mathrm {b}})), \tag {10}
+$$
+
+Finally, we follow the design of XNOR-Net (Rastegari et al., 2016) to compensate the precision loss, which is:
+
+$$
+\mathbf {x} _ {\mathrm {B M B}} = \mathbf {x} _ {\mathrm {B M B}} ^ {\prime} \odot (\mathcal {A} \otimes \mathbf {k}), \tag {11}
+$$
+
+where  $\mathcal{A}_{H\times W}$  is the channel-wise absolute average of input activation,  $\mathbf{k}$  is a vector whose element is the absolute average of correspond channel,  $\odot$  is Hadamard product, and  $(\mathcal{A}\otimes \mathbf{k})_{c,i,j}:= \mathcal{A}_{i,j}\mathbf{k}_c$  should be calculated first.
+
+We binarized every convolution layer and linear layer except the first and last convolution layers due to extremely severe degradation of quantization on head and tail. Specifically, linear layer can be seen as the  $1 \times 1$  convolution layer and therefore processed in the same way as convolution layer.
+
+For one binarized layer, the output can be written as:
+
+$$
+\mathbf {x} _ {\text {o u t}} = \mathbf {x} _ {\mathrm {B M B}} + \mathbf {x} _ {\mathrm {L R M B}} + \mathbf {x} _ {\mathrm {S M B}}. \tag {12}
+$$
+
+In conclusion, we propose LRMB, SMB, and their corresponding initialization methods. With these designs, our BiMaCoSR can significantly reduce the information loss caused by binarization and improve the restoration ability. Besides, the storage and computation overhead caused by LRMB and SMB can be safely ignored.
+
+# 4. Experiments
+
+# 4.1. Experimental Settings
+
+Data. We take the models on the training set of ImageNet (Russakovsky et al., 2015) and the LR images are generated by the same pipeline of RealESRGAN (Wang et al., 2021). We evaluate the models with three benchmark datasets: RealSR (Cai et al., 2019), DRealSR (Wei et al., 2020), and DIV2K-Val (Agustsson & Timofte, 2017). RealSR and DRealSR are real world benchmarks while DIV2K-Val employs Bicubic interpolation to generate LR images. The upscale ratio of training set and test set is  $\times 4$ .
+
+Evaluation Metrics. The evaluation metrics are implemented with IQA-Pytorch (Chen & Mo, 2022) and are twofold to assess the restoration and compression ability. Firstly, to thoroughly evaluate the restoration ability of our proposed BiMaCoSR, we employ the following quantitative metrics: full-reference metrics PSNR, SSIM (Wang et al., 2004), and LPIPS (Zhang et al., 2018a) and non-reference metrics DISTS (Ding et al., 2020), FID (Heusel et al., 2017), NIQE (Zhang et al., 2015), MANIQA-pipal (Yang et al., 2022), MUSIQ (Ke et al., 2021), and CLIPIQA+ (Wang et al., 2023a). PSNR and SSIM are distortion-based metrics and are calculated on the Y channel (i.e., luminance) of the YCbCr space. The rest metrics are all perceptual metrics and it is widely known that perceptual metrics are more aligned with human when rating the image quality.
+
+Secondly, to demonstrate binarization's extreme compression and acceleration ability, we use the total parameters and overall operations as key metrics. Following previous work (Xia et al., 2022; Qin et al., 2023), the total parameters (Params) of the model are calculated as Params  $= \text{Params}^b + \text{Params}^f$ , and the overall operations (OPs) as OPs  $= \text{OPs}^b + \text{OPs}^f$ , where Params  $^b = \text{Params}^f / 32$  and OPs  $^b = \text{OPs}^f / 64$ ; the superscripts  $f$  and  $b$  denote full-precision and binarized modules, respectively. The computational complexity is tested with the input size  $3 \times 64 \times 64$ .
+
+Implementation Details. We take SinSR (Wang et al., 2024), a one-step diffusion model, as the backbone. We initialize with SinSR's weights and use ResShift (Yue et al., 2024) as the teacher model. All the rest convolution and linear layers, except the head and tail layers, are binarized into BMB to guarantee the best compression ratio and LRMB and SMB are attached with BMB. We set the rank of LRMB as  $r = 8$  and the number of non-zero elements in SMB as  $k = 2\max(C_1, C_2)$ , where  $C_1$  and  $C_2$  are the numbers of input and output channels, respectively.
+
+Training Settings. We utilize Adam optimizer (Kingma & Ba, 2015) with  $\beta_{1} = 0.9$  and  $\beta_{2} = 0.99$ , and set the learning rate as  $2 \times 10^{-5}$ . The batch size is set to 8, with 100K iterations. The input LR images are center-cropped to size  $64 \times 64$ . All the training and testing experiments are conducted on one NVIDIA RTX A6000.
+
+Table 1: Quantitative comparison with SOTA methods. We compare BiMaCoSR with full-precision models and current binarization methods. We employ 9 metrics commonly used in SR and test on three benchmarks. The best and second best value are marked with red and blue respectively. In conclusion, our proposed BiMaCoSR achieves SOTA performance.  
+
+<table><tr><td>Datasets</td><td>Methods</td><td>Bits (W/A)</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>MUSIQ ↑</td><td>MANIQA ↑</td><td>DISTS ↓</td><td>FID ↓</td><td>NIQE ↓</td><td>CLIP-IQA+ ↑</td></tr><tr><td rowspan="8">RealSR</td><td>SinSR</td><td>32/32</td><td>26.51</td><td>0.7380</td><td>0.3635</td><td>57.87</td><td>0.5139</td><td>0.2193</td><td>56.36</td><td>5.826</td><td>0.5736</td></tr><tr><td>ResShift</td><td>32/32</td><td>25.45</td><td>0.7243</td><td>0.3731</td><td>56.23</td><td>0.5005</td><td>0.2344</td><td>58.14</td><td>7.353</td><td>0.5708</td></tr><tr><td>XNOR</td><td>1/1</td><td>26.48</td><td>0.7434</td><td>0.3968</td><td>43.56</td><td>0.3732</td><td>0.2609</td><td>105.72</td><td>6.014</td><td>0.4380</td></tr><tr><td>ReActNet</td><td>1/1</td><td>26.60</td><td>0.7530</td><td>0.3834</td><td>44.18</td><td>0.3829</td><td>0.2551</td><td>109.36</td><td>6.306</td><td>0.4361</td></tr><tr><td>BBCU</td><td>1/1</td><td>26.43</td><td>0.7488</td><td>0.3902</td><td>43.70</td><td>0.3792</td><td>0.2575</td><td>108.32</td><td>6.058</td><td>0.4298</td></tr><tr><td>ReSTE</td><td>1/1</td><td>26.26</td><td>0.7408</td><td>0.4184</td><td>41.04</td><td>0.3677</td><td>0.2719</td><td>113.86</td><td>6.174</td><td>0.4083</td></tr><tr><td>BiDM</td><td>1/1</td><td>25.07</td><td>0.7036</td><td>0.5042</td><td>35.60</td><td>0.3517</td><td>0.3226</td><td>115.23</td><td>6.759</td><td>0.3935</td></tr><tr><td>BiMaCoSR</td><td>1/1</td><td>26.84</td><td>0.7698</td><td>0.3375</td><td>49.01</td><td>0.4034</td><td>0.2183</td><td>86.09</td><td>5.856</td><td>0.4800</td></tr><tr><td rowspan="8">DRealSR</td><td>SinSR</td><td>32/32</td><td>27.89</td><td>0.7332</td><td>0.4499</td><td>30.81</td><td>0.4519</td><td>0.2209</td><td>16.56</td><td>5.789</td><td>0.6052</td></tr><tr><td>ResShift</td><td>32/32</td><td>26.64</td><td>0.7298</td><td>0.4478</td><td>31.09</td><td>0.4345</td><td>0.2337</td><td>18.12</td><td>6.959</td><td>0.5795</td></tr><tr><td>XNOR</td><td>1/1</td><td>29.03</td><td>0.8319</td><td>0.3712</td><td>26.19</td><td>0.3560</td><td>0.2447</td><td>29.88</td><td>6.229</td><td>0.4449</td></tr><tr><td>ReActNet</td><td>1/1</td><td>29.34</td><td>0.8431</td><td>0.3571</td><td>26.83</td><td>0.3618</td><td>0.2411</td><td>30.18</td><td>6.561</td><td>0.4380</td></tr><tr><td>BBCU</td><td>1/1</td><td>29.00</td><td>0.8385</td><td>0.3643</td><td>26.37</td><td>0.3594</td><td>0.2433</td><td>30.94</td><td>6.337</td><td>0.4383</td></tr><tr><td>ReSTE</td><td>1/1</td><td>28.91</td><td>0.8353</td><td>0.3899</td><td>25.12</td><td>0.3509</td><td>0.2641</td><td>33.64</td><td>6.459</td><td>0.4131</td></tr><tr><td>BiDM</td><td>1/1</td><td>27.40</td><td>0.7942</td><td>0.4849</td><td>23.38</td><td>0.3529</td><td>0.3118</td><td>37.83</td><td>6.753</td><td>0.4307</td></tr><tr><td>BiMaCoSR</td><td>1/1</td><td>29.33</td><td>0.8393</td><td>0.3400</td><td>29.38</td><td>0.3802</td><td>0.2278</td><td>22.31</td><td>6.150</td><td>0.4867</td></tr><tr><td rowspan="8">DIV2K-Val</td><td>SinSR</td><td>32/32</td><td>27.75</td><td>0.7694</td><td>0.1903</td><td>64.62</td><td>0.5336</td><td>0.1029</td><td>6.27</td><td>4.308</td><td>0.6147</td></tr><tr><td>ResShift</td><td>32/32</td><td>27.18</td><td>0.7667</td><td>0.1775</td><td>65.04</td><td>0.5548</td><td>0.1016</td><td>7.54</td><td>5.121</td><td>0.6280</td></tr><tr><td>XNOR</td><td>1/1</td><td>26.44</td><td>0.7185</td><td>0.3727</td><td>49.10</td><td>0.3972</td><td>0.2204</td><td>55.77</td><td>5.320</td><td>0.4584</td></tr><tr><td>ReActNet</td><td>1/1</td><td>26.49</td><td>0.7260</td><td>0.3602</td><td>50.29</td><td>0.4078</td><td>0.2111</td><td>52.32</td><td>5.366</td><td>0.4726</td></tr><tr><td>BBCU</td><td>1/1</td><td>26.39</td><td>0.7221</td><td>0.3660</td><td>50.09</td><td>0.4035</td><td>0.2148</td><td>53.22</td><td>5.263</td><td>0.4653</td></tr><tr><td>ReSTE</td><td>1/1</td><td>26.07</td><td>0.7125</td><td>0.3916</td><td>46.95</td><td>0.3907</td><td>0.2295</td><td>61.52</td><td>5.399</td><td>0.4328</td></tr><tr><td>BiDM</td><td>1/1</td><td>24.29</td><td>0.6725</td><td>0.4370</td><td>40.15</td><td>0.3747</td><td>0.2916</td><td>62.28</td><td>6.090</td><td>0.4112</td></tr><tr><td>BiMaCoSR</td><td>1/1</td><td>27.35</td><td>0.7547</td><td>0.2999</td><td>53.38</td><td>0.4337</td><td>0.1806</td><td>27.99</td><td>4.987</td><td>0.5176</td></tr></table>
+
+# 4.2. Comparison with State-of-the-Art Methods
+
+We compare our proposed BiMaCoSR with recent binarization methods, including XNOR (Rastegari et al., 2016), ReActNet (Liu et al., 2020), BBCU (Xia et al., 2022), ReSTE (Wu et al., 2023), and BiDM (Zheng et al., 2024). All binarization methods are implemented on SinSR (Wang et al., 2024) and trained with the same settings. As LRMB and SMB slightly increase the parameters, we additionally keep the first two and last two convolution layers as full-precision. We also compare BiMaCoSR with the full-precision model SinSR and ResShift (Yue et al., 2024). SinSR is the distilled version of ResShift. The comparison results are exhibited in quantitative and qualitative aspects.
+
+Restoration Results. The quantitative comparison results are shown in Table 1. We can obtain the following observations. (1) The results in Table 1 demonstrate clear advantage over competing methods in both full-reference and nonreference metrics on three benchmark datasets. (2) In some situations, such as PSNR and LPIPS on RealSR, BiMa-CoSR can even surpass SinSR and ResShift. We attribute the excellence to the LRMB and SMB, which transmit the full-precision information in a parameter-efficient way. (3) Most binarization methods outperform the baseline model and ReActNet presents competing performance, especially on DRealSR. (4) The full-precision models, i.e., SinSR and ResShift, are excellent on perceptual metrics, such as DISTS and CLIP-IQA+. Whereas, the multi-step models'
+
+performance on PSNR and SSIM is on the lower side. This phenomenon means that compression on DM leads to disappearance of texture and details. However, their performance on PSNR and SSIM is only slightly affected.
+
+Efficiency Comparison. Efficiency comparison results are provided in Table 2. After the distillation from multi-step to one-step, the FLOPs are significantly reduced. Furthermore, compressing the model to 1 bit makes the model tiny and fast. In comparison with SinSR, the compression ratio is  $27.45 \times$  and the speedup ratio is  $23.81 \times$ . Compared with current SOTA binarization methods, we keep almost the same parameters but take much less FLOPs on one-step DM, due to the calculation efficient design. To conclude, our proposed BiMaCoSR gains remarkable compression ratio and speedup ratio and achieves outstanding performance.
+
+Visual Results. We present visual comparison on challenging cases in Fig. 4. One typical challenging case is the tiny and dense structures, such as hairs, grass, tiles, and faces. It is struggling with previous binarization methods to restore image details, especially in challenging cases. On the contrary, our BiMaCoSR is able to recover results with sharper edges and richer textures. For example, in 0809, BiMaCoSR reconstructs the hairs on the nose while other methods just output rough color blocks. And in 0831, BiMaCoSR restores the woman's facial structures and expression while other methods smooth the organs to the same color. In 0834, BiMaCoSR could successfully recover the tiles' texture and
+
+<table><tr><td></td><td>ResShift</td><td>SinSR</td><td>ReActNet</td><td>BBCU</td><td>ReSTE</td><td>BiDM</td><td>XNOR</td><td>Ours</td></tr><tr><td>Inference Step</td><td>15</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>FLOPs (G)</td><td>753.45</td><td>50.23</td><td>5.83</td><td>5.83</td><td>5.83</td><td>11.60</td><td>5.83</td><td>1.83</td></tr><tr><td># Total Param (M)</td><td>118.59</td><td>118.59</td><td>4.95</td><td>4.95</td><td>4.95</td><td>18.69</td><td>4.95</td><td>4.98</td></tr><tr><td>PSNR/LPIPS</td><td>25.45/0.3731</td><td>26.51/0.3635</td><td>26.60/0.3834</td><td>26.43/0.3902</td><td>26.26/0.4184</td><td>25.07/0.5042</td><td>26.48/0.3968</td><td>26.84/0.3375</td></tr></table>
+
+Table 2: Efficiency comparison on RealSR. BiMaCoSR takes least FLOPs while gains the best performance.
+
+![](images/b4a427ff36cfd8c96f2a6008a942d8f3546ec3b68085d4b2bf386010e7d9c8fb.jpg)  
+Figure 4: Visual comparison for image SR. We compare our proposed BiMaCoSR with current competitive binarization methods and the full-precision (FP) model. The visual results illustrate that BiMaCoSR gains rich details and reasonable textures.
+
+most binarization methods fail. What's more, the difference between BiMaCoSR and the FP model (SinSR) exists but is minimal. To conclude, our proposed BiMaCoSR surpasses other methods according to visual comparison. Additionally, we provide more visual results and corresponding analysis in the supplementary material for further comparison.
+
+# 4.3. Ablation Study
+
+In this section, we adopt RealSR as the test set and other training settings are the same as Sec. 4.1. To test the effectiveness and robustness, we conduct five key ablation experiments, including the break down ablation, loss function, rank of low rank branch, initialization of LRMB, and initialization of SMB. What's more, detailed analysis is also provided in the following sections. These ablation studies demonstrate the robustness and efficiency of our LRMB, SMB, and their corresponding initialization methods.
+
+Break Down Ablation. Table 3a shows results of the break down ablation. With only BMB, the model could be extremely compressed but the performance is somewhat on the low side. With LRMB, though the number of parameters increases to 1.83G, the performance also improves on all metrics and the FLOPs only increase to 4.98M. After adding SMB, the overhead of parameters and storage can be neglected. The performance on all perceptual metrics improves while both PSNR and SSIM drop slightly.
+
+Loss Function. The loss functions of SinSR play a critical role in FP model. However, these loss functions are not suitable in the binarized situation. After removing unnecessary parts from the SinSR loss, we leave only the distillation loss and the results are shown in Table 3b. With only the distillation loss, BiMaCoSR gains improvement on both distortion metrics and perceptual metrics.
+
+<table><tr><td>Branch</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>MANIQA ↑</td><td>FID ↓</td><td>CLIP-IQA+ ↑</td><td>FLOPs (G)</td><td>Param (M)</td></tr><tr><td>BMB</td><td>26.41</td><td>0.7408</td><td>0.4141</td><td>0.3704</td><td>110.15</td><td>0.4325</td><td>0.78</td><td>3.69</td></tr><tr><td>+LRMB</td><td>26.95</td><td>0.7718</td><td>0.3400</td><td>0.3937</td><td>88.72</td><td>0.4663</td><td>1.83</td><td>4.98</td></tr><tr><td>+LRMB+SMB</td><td>26.84</td><td>0.7698</td><td>0.3375</td><td>0.4034</td><td>86.09</td><td>0.4800</td><td>1.83</td><td>4.98</td></tr></table>
+
+(a) Break down ablation.  
+
+<table><tr><td>Loss</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CLIP-IQA+ ↑</td></tr><tr><td>Distill loss</td><td>26.83</td><td>0.7698</td><td>0.3375</td><td>0.4800</td></tr><tr><td>SinSR loss</td><td>26.37</td><td>0.7466</td><td>0.4029</td><td>0.4273</td></tr></table>
+
+<table><tr><td>Initialization</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CLIP-IQA+ ↑</td></tr><tr><td>Zero + Random</td><td>26.88</td><td>0.7660</td><td>0.3497</td><td>0.4674</td></tr><tr><td>SVD</td><td>26.84</td><td>0.7698</td><td>0.3375</td><td>0.4800</td></tr></table>
+
+(b) Ablation study on losses.  
+
+<table><tr><td>Rank</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CLIP-IQA+↑</td><td>FLOPs (G)</td><td>Param (M)</td></tr><tr><td>4</td><td>26.29</td><td>0.7383</td><td>0.4197</td><td>0.4411</td><td>1.31</td><td>4.37</td></tr><tr><td>8</td><td>26.84</td><td>0.7698</td><td>0.3375</td><td>0.4800</td><td>1.83</td><td>4.98</td></tr><tr><td>12</td><td>26.97</td><td>0.7695</td><td>0.3400</td><td>0.4766</td><td>2.32</td><td>5.60</td></tr><tr><td>16</td><td>26.72</td><td>0.7625</td><td>0.3442</td><td>0.4971</td><td>2.83</td><td>6.21</td></tr></table>
+
+(c) Ablation study on LRMB initialization.  
+
+<table><tr><td>Initialization</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>CLIP-IQA+ ↑</td></tr><tr><td>Zero initial</td><td>26.72</td><td>0.7628</td><td>0.3561</td><td>0.4639</td></tr><tr><td>Uni-shortcut</td><td>25.69</td><td>0.7113</td><td>0.5105</td><td>0.4008</td></tr><tr><td>Sparse skip</td><td>26.84</td><td>0.7698</td><td>0.3375</td><td>0.4800</td></tr></table>
+
+(d) Ablation study on the rank of LRMB.
+
+(e) Ablation study on SMB initialization.
+
+Table 3: Ablation studies on branches, loss, rank of LRMB and initialization of LRMB and SMB. The experiments is tested on RealSR. The comprehensive results demonstrate the robustness and efficient performance of our proposed BiMaCoSR.
+
+Rank of LRMB. The rank of LRMB significantly influences the complexity, while the performance is not consistent with the increase in complexity. As shown in Table 3d, the model with rank of 16 performs best only on CLIP-IQA+, while the model with rank of 8 performs best and takes acceptable parameters. This is because the LRMB with high rank may influence high frequency generated by BMB, making the training unstable and leading to worse results. Therefore, we ultimately set the rank of LRMB to 8.
+
+Initialization of LRMB. Conventional LoRA's initialization method is set the first matrix with random values and the second matrix with zero values. We compare this initialization method with our proposed SVD initialization and the result is shown in Table 3c. To conclude, our method outperforms conventional methods on SSIM, LPIPS, and CLIP-  $\mathrm{QA + }$  .We attribute it to the decoupling of transmitting low frequency information and generating high frequency. To conclude, our proposed SVD allows better performance.
+
+Initialization of SMB. We search three popular ways to initialize SMB, including (1) random position and zero initialization, (2) Uni-shortcut (Xu et al., 2022), and (3) our proposed sparse skip. The experiment results in Table 3e show that our sparse skip initialization consistently enjoys the best performance on all four metrics. This improvement compared with other two initialization methods strongly supports the effectiveness of decoupling and absorption.
+
+# 4.4. Visualization Analysis.
+
+In Sec. 3, we assume that LRMB and SMB are in charge of transmitting low frequency information while BMB generates high frequency after decoupling. To validate this assumption, we visualize the proportion of high frequency generated by three branches in first 50 Conv layers and the result is shown in Fig. 5. On average, high frequency infor
+
+![](images/fd8f96c764ce966ef4c0378e08826a27fa4c874943b9556ebbbf1a8b87381884.jpg)  
+Figure 5: The proportion of high frequency information generated by three branches. The high frequency information mainly comes from BMB, which obeys our assumption.
+
+mation is mainly generated by BMB, accounting for around  $70\%$ . With more parameters, LRMB also provides high frequency information in some layers but overall proportion is relative low. SMB only absorbs the extreme values and indeed carries little high frequency information. Fig. 5 shows that division of work is clear and our designs in LRMB, SMB and BMB obey our assumption.
+
+# 5. Conclusion
+
+In this paper, we propose the BiMaCoSR, a binarized SR diffusion model with only one inference step. Detailedly, we first propose LRMB and SVD initialization to decouple the effect of binarized branch and deliver low frequency information. Furthermore, we propose SMB and sparse initialization to absorb the extreme values and provide high rank representations. Comprehensive comparison experiments demonstrate the SOTA restoration ability of the proposed BiMaCoSR. Extensive ablation studies exhibit the efficiency and robustness of both LRMB and SMB. In the future, we will focus on the combination of pruning and binarization on one-step diffusion models for further compression.
+
+# Acknowledgements
+
+This work was supported by Shanghai Municipal Science and Technology Major Project (2021SHZDZX0102), the Fundamental Research Funds for the Central Universities, and the National Natural Science Foundation of China (NSFC No. W2412089).
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
+# References
+
+Agustsson, E. and Timofte, R. Ntire 2017 challenge on single image super-resolution: Dataset and study. In CVPRW, 2017.  
+Cai, J., Zeng, H., Yong, H., Cao, Z., and Zhang, L. Toward real-world single image super-resolution: A new benchmark and a new model. In CVPR, 2019.  
+Chen, C. and Mo, J. IQA-PyTorch: Pytorch toolbox for image quality assessment. [Online]. Available: https://github.com/chaofengc/IQA-PyTorch, 2022.  
+Chen, Z., Zhang, Y., Gu, J., Zhang, Y., Kong, L., and Yuan, X. Cross aggregation transformer for image restoration. In NeurIPS, 2022.  
+Chen, Z., Zhang, Y., Gu, J., Kong, L., Yang, X., and Yu, F. Dual aggregation transformer for image super-resolution. In CVPR, 2023.  
+Chen, Z., Qin, H., Guo, Y., Su, X., Yuan, X., Kong, L., and Zhang, Y. Binarized diffusion model for image superresolution. In NeurIPS, 2024.  
+Ding, K., Ma, K., Wang, S., and Simoncelli, E. P. Image quality assessment: Unifying structure and texture similarity. TPAMI, 2020.  
+Dong, C., Loy, C. C., He, K., and Tang, X. Learning a deep convolutional network for image super-resolution. In ECCV, 2014.  
+Dong, C., Loy, C. C., He, K., and Tang, X. Image superresolution using deep convolutional networks. TPAMI, 2016.  
+Gu, J., Lu, H., Zuo, W., and Dong, C. Blind super-resolution with iterative kernel correction. In CVPR, 2019.  
+Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., and Hochreiter, S. Gans trained by a two time-scale update rule converge to a local nash equilibrium. NIPS, 2017.
+
+Huang, W., Zheng, X., Ma, X., Qin, H., Lv, C., Chen, H., Luo, J., Qi, X., Liu, X., and Magno, M. An empirical study of llama3 quantization: From llms to mllms. Visual Intelligence, 2(1):36, 2024.  
+Ke, J., Wang, Q., Wang, Y., Milanfar, P., and Yang, F. Musiq: Multi-scale image quality transformer. In ICCV, 2021.  
+Keys, R. Cubic convolution interpolation for digital image processing. IEEE Trans. Acoust., Speech, Signal Process., 1981.  
+Kingma, D. and Ba, J. Adam: A method for stochastic optimization. In ICLR, 2015.  
+Le, P.-H. C. and Li, X. Binaryvit: pushing binary vision transformers towards convolutional models. In CVPR, 2023.  
+Ledig, C., Theis, L., Huszár, F., Caballero, J., Cunningham, A., Acosta, A., Aitken, A. P., Tejani, A., Totz, J., Wang, Z., et al. Photo-realistic single image super-resolution using a generative adversarial network. In CVPR, 2017.  
+Liang, J., Cao, J., Sun, G., Zhang, K., Van Gool, L., and Timofte, R. Swinir: Image restoration using swim transformer. In ICCVW, 2021.  
+Lim, B., Son, S., Kim, H., Nah, S., and Lee, K. M. Enhanced deep residual networks for single image super-resolution. In CVPRW, 2017.  
+Lin, X., He, J., Chen, Z., Lyu, Z., Dai, B., Yu, F., Qiao, Y., Ouyang, W., and Dong, C. Diffbir: Toward blind image restoration with generative diffusion prior. In ECCV, 2024.  
+Liu, Z., Shen, Z., Savvides, M., and Cheng, K.-T. Reactnet: Towards precise binary neural network with generalized activation functions. In ECCV, 2020.  
+Lu, X., Yuan, Y., and Yan, P. Image super-resolution via double sparsity regularized manifold learning. TCSVT, 2013.  
+Luo, X., Qu, Y., Xie, Y., Zhang, Y., Li, C., and Fu, Y. Lattice network for lightweight image restoration. TPAMI, 2022.  
+Qin, H., Gong, R., Liu, X., Shen, M., Wei, Z., Yu, F., and Song, J. Forward and backward information retention for accurate binary neural networks. In CVPR, 2020.  
+Qin, H., Zhang, X., Gong, R., Ding, Y., Xu, Y., and Liu, X. Distribution-sensitive information retention for accurate binary neural network. IJCV, 2022.  
+Qin, H., Zhang, M., Ding, Y., Li, A., Cai, Z., Liu, Z., Yu, F., and Liu, X. Bibench: Benchmarking and analyzing network binarization. In ICML, 2023.
+
+Rastegari, M., Ordonez, V., Redmon, J., and Farhadi, A. Xnor-net: Imagenet classification using binary convolutional neural networks. In ECCV, 2016.  
+Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. High-resolution image synthesis with latent diffusion models. In CVPR, 2022.  
+Russakovsky, O., Deng, J., Su, H., Krause, J., Satheesh, S., Ma, S., Huang, Z., Karpathy, A., Khosla, A., Bernstein, M., et al. Imagenet large scale visual recognition challenge. IJCV, 2015.  
+Saharia, C., Ho, J., Chan, W., Salimans, T., Fleet, D. J., and Norouzi, M. Image super-resolution via iterative refinement. IEEE TPAMI, 2022.  
+Sun, H., Li, W., Liu, J., Chen, H., Pei, R., Zou, X., Yan, Y., and Yang, Y. Coser: Bridging image and language for cognitive super-resolution. In CVPR, 2024.  
+Wang, J., Chan, K. C., and Loy, C. C. Exploring clip for assessing the look and feel of images. In AAAI, 2023a.  
+Wang, X., Xie, L., Dong, C., and Shan, Y. Real-esrgan: Training real-world blind super-resolution with pure synthetic data. In CVPR, 2021.  
+Wang, Y., Yang, W., Chen, X., Wang, Y., Guo, L., Chau, L.-P., Liu, Z., Qiao, Y., Kot, A. C., and Wen, B. Sinsr: diffusion-based image super-resolution in a single step. In CVPR, 2024.  
+Wang, Z., Bovik, A. C., Sheikh, H. R., and Simoncelli, E. P. Image quality assessment: from error visibility to structural similarity. TIP, 2004.  
+Wang, Z., Cun, X., Bao, J., Zhou, W., Liu, J., and Li, H. Uformer: A general u-shaped transformer for image restoration. In CVPR, 2022.  
+Wang, Z., Wang, J., Liu, Z., and Qiu, Q. Binary latent diffusion. In CVPR, 2023b.  
+Wei, P., Xie, Z., Lu, H., Zhan, Z., Ye, Q., Zuo, W., and Lin, L. Component divide-and-conquer for real-world image super-resolution. In ECCV, 2020.  
+Wu, R., Sun, L., Ma, Z., and Zhang, L. One-step effective diffusion network for real-world image super-resolution. arXiv, 2024a.  
+Wu, R., Yang, T., Sun, L., Zhang, Z., Li, S., and Zhang, L. Seesr: Towards semantics-aware real-world image super-resolution. In CVPR, 2024b.  
+Wu, X.-M., Zheng, D., Liu, Z., and Zheng, W.-S. Estimator meets equilibrium perspective: A rectified straight through estimator for binary neural networks training. In ICCV, 2023.
+
+Xia, B., Zhang, Y., Wang, Y., Tian, Y., Yang, W., Timofte, R., and Van Gool, L. Basic binary convolution unit for binarized image restoration network. In ICLR, 2022.  
+Xu, Y., Chen, X., and Wang, Y. Bimlp: Compact binary architectures for vision multi-layer perceptrons. NeurIPS, 2022.  
+Yan, D., Wu, T., Liu, Y., and Gao, Y. An efficient sparse-dense matrix multiplication on a multicore system. In ICCT, 2017.  
+Yang, C.-Y., Ma, C., and Yang, M.-H. Single-image superresolution: A benchmark. In ECCV, 2014.  
+Yang, S., Wu, T., Shi, S., Lao, S., Gong, Y., Cao, M., Wang, J., and Yang, Y. Maniaq: Multi-dimension attention network for no-reference image quality assessment. In CVPR, 2022.  
+Yang, T., Wu, R., Ren, P., Xie, X., and Zhang, L. Pixel-aware stable diffusion for realistic image super-resolution and personalized stylization. In ECCV, 2025.  
+Yu, F., Gu, J., Li, Z., Hu, J., Kong, X., Wang, X., He, J., Qiao, Y., and Dong, C. Scaling up to excellence: Practicing model scaling for photo-realistic image restoration in the wild. In CVPR, 2024.  
+Yue, Z., Wang, J., and Loy, C. C. Resshift: Efficient diffusion model for image super-resolution by residual shifting. NeurIPS, 2024.  
+Zhang, K., Liang, J., Van Gool, L., and Timofte, R. Designing a practical degradation model for deep blind image super-resolution. In CVPR, 2021.  
+Zhang, L., Zhang, L., and Bovik, A. C. A feature-enriched completely blind image quality evaluator. TIP, 2015.  
+Zhang, R., Isola, P., Efros, A. A., Shechtman, E., and Wang, O. The unreasonable effectiveness of deep features as a perceptual metric. In CVPR, 2018a.  
+Zhang, Y., Li, K., Li, K., Wang, L., Zhong, B., and Fu, Y. Image super-resolution using very deep residual channel attention networks. In ECCV, 2018b.  
+Zhang, Y., Tian, Y., Kong, Y., Zhong, B., and Fu, Y. Residual dense network for image super-resolution. In CVPR, 2018c.  
+Zheng, X., Liu, X., Bian, Y., Ma, X., Zhang, Y., Wang, J., Guo, J., and Qin, H. Bidm: Pushing the limit of quantization for diffusion models. In NeurIPS, 2024.  
+Zibetti, M. V. W. and Mayer, J. A robust and computationally efficient simultaneous super-resolution scheme for image sequences. TCSVT, 2007.

2025/BiMaCoSR_ Binary One-Step Diffusion Model Leveraging Flexible Matrix Compression for Real Super-Resolution/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:83254f62b6ab498063a95b6705fa0c445e64899d7af33b25c34a39519f578dd0
+size 834718