Add Batch 6498f893-9294-4c23-aa03-782f3a88d552 data

.gitattributes

@@ -2391,3 +2391,67 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
 2025/PiCO_[[:space:]]Peer[[:space:]]Review[[:space:]]in[[:space:]]LLMs[[:space:]]based[[:space:]]on[[:space:]]Consistency[[:space:]]Optimization/572753a6-4a65-4739-804f-07dc9df60307_origin.pdf filter=lfs diff=lfs merge=lfs -text
 2025/PivotMesh_[[:space:]]Generic[[:space:]]3D[[:space:]]Mesh[[:space:]]Generation[[:space:]]via[[:space:]]Pivot[[:space:]]Vertices[[:space:]]Guidance/f52d48fe-5467-45d9-9497-c5ba888d0a0c_origin.pdf filter=lfs diff=lfs merge=lfs -text
 2025/PixWizard_[[:space:]]Versatile[[:space:]]Image-to-Image[[:space:]]Visual[[:space:]]Assistant[[:space:]]with[[:space:]]Open-Language[[:space:]]Instructions/2e663f5d-8d04-484c-8761-0d6bb1e99f16_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Planning[[:space:]]Anything[[:space:]]with[[:space:]]Rigor_[[:space:]]General-Purpose[[:space:]]Zero-Shot[[:space:]]Planning[[:space:]]with[[:space:]]LLM-based[[:space:]]Formalized[[:space:]]Programming/08557070-92b7-4701-8d22-314405276414_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Plastic[[:space:]]Learning[[:space:]]with[[:space:]]Deep[[:space:]]Fourier[[:space:]]Features/9a76bf86-ef76-4d49-8136-051df093c801_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/PnP-Flow_[[:space:]]Plug-and-Play[[:space:]]Image[[:space:]]Restoration[[:space:]]with[[:space:]]Flow[[:space:]]Matching/4058e62a-583f-44ae-9829-581a0bcb8635_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Point[[:space:]]Cluster_[[:space:]]A[[:space:]]Compact[[:space:]]Message[[:space:]]Unit[[:space:]]for[[:space:]]Communication-Efficient[[:space:]]Collaborative[[:space:]]Perception/ed2c2ba3-1518-434b-a59b-d326eb00ba86_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Point-SAM_[[:space:]]Promptable[[:space:]]3D[[:space:]]Segmentation[[:space:]]Model[[:space:]]for[[:space:]]Point[[:space:]]Clouds/b6e8bccc-f4f8-4762-844b-752b678c4fe0_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Point-based[[:space:]]Instance[[:space:]]Completion[[:space:]]with[[:space:]]Scene[[:space:]]Constraints/62e2f7f2-4d65-4c81-bc65-22dc127b6093_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/PointOBB-v2_[[:space:]]Towards[[:space:]]Simpler,[[:space:]]Faster,[[:space:]]and[[:space:]]Stronger[[:space:]]Single[[:space:]]Point[[:space:]]Supervised[[:space:]]Oriented[[:space:]]Object[[:space:]]Detection/860cef88-93f3-4f16-92f7-c5ac52feb051_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Poisson-Dirac[[:space:]]Neural[[:space:]]Networks[[:space:]]for[[:space:]]Modeling[[:space:]]Coupled[[:space:]]Dynamical[[:space:]]Systems[[:space:]]across[[:space:]]Domains/655be178-57e1-494e-92a1-8899e0bc0c03_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/PolaFormer_[[:space:]]Polarity-aware[[:space:]]Linear[[:space:]]Attention[[:space:]]for[[:space:]]Vision[[:space:]]Transformers/09aee513-49a8-410e-b6cc-2478eeb77178_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Policy[[:space:]]Decorator_[[:space:]]Model-Agnostic[[:space:]]Online[[:space:]]Refinement[[:space:]]for[[:space:]]Large[[:space:]]Policy[[:space:]]Model/616a7de7-7913-4ae9-a12a-7a7ce6ba3cab_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Policy[[:space:]]Design[[:space:]]in[[:space:]]Long-run[[:space:]]Welfare[[:space:]]Dynamics/7a8290c7-51bd-4e78-a746-4787f1485883_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Policy[[:space:]]Optimization[[:space:]]under[[:space:]]Imperfect[[:space:]]Human[[:space:]]Interactions[[:space:]]with[[:space:]]Agent-Gated[[:space:]]Shared[[:space:]]Autonomy/6f2570e7-7dad-479a-a3c1-303b168bd714_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/PolyNet_[[:space:]]Learning[[:space:]]Diverse[[:space:]]Solution[[:space:]]Strategies[[:space:]]for[[:space:]]Neural[[:space:]]Combinatorial[[:space:]]Optimization/6a2fff0f-f7db-4f51-a7c5-35c9b7621839_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/PolyPythias_[[:space:]]Stability[[:space:]]and[[:space:]]Outliers[[:space:]]across[[:space:]]Fifty[[:space:]]Language[[:space:]]Model[[:space:]]Pre-Training[[:space:]]Runs/20f835f4-9e1f-4f86-94a4-c979da5a77d8_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/PolyhedronNet_[[:space:]]Representation[[:space:]]Learning[[:space:]]for[[:space:]]Polyhedra[[:space:]]with[[:space:]]Surface-attributed[[:space:]]Graph/cfb85d4b-a294-4f57-806c-a919614f83f8_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Polynomial[[:space:]]Composition[[:space:]]Activations_[[:space:]]Unleashing[[:space:]]the[[:space:]]Dynamics[[:space:]]of[[:space:]]Large[[:space:]]Language[[:space:]]Models/fb67c7b4-5817-4834-a4a8-de899e73336c_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Polyrating_[[:space:]]A[[:space:]]Cost-Effective[[:space:]]and[[:space:]]Bias-Aware[[:space:]]Rating[[:space:]]System[[:space:]]for[[:space:]]LLM[[:space:]]Evaluation/95082ffc-2a79-4e69-b58a-02e628a9752c_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/PooDLe🐩_[[:space:]]Pooled[[:space:]]and[[:space:]]dense[[:space:]]self-supervised[[:space:]]learning[[:space:]]from[[:space:]]naturalistic[[:space:]]videos/b4e025bc-0ac4-4a30-8fb9-e51a1308826b_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Port-Hamiltonian[[:space:]]Architectural[[:space:]]Bias[[:space:]]for[[:space:]]Long-Range[[:space:]]Propagation[[:space:]]in[[:space:]]Deep[[:space:]]Graph[[:space:]]Networks/cf0787af-d4d0-46f0-8720-1f17a546d5d1_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/PortLLM_[[:space:]]Personalizing[[:space:]]Evolving[[:space:]]Large[[:space:]]Language[[:space:]]Models[[:space:]]with[[:space:]]Training-Free[[:space:]]and[[:space:]]Portable[[:space:]]Model[[:space:]]Patches/dae2a044-a376-4b3d-a4a1-621ac8043535_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Positive-Unlabeled[[:space:]]Diffusion[[:space:]]Models[[:space:]]for[[:space:]]Preventing[[:space:]]Sensitive[[:space:]]Data[[:space:]]Generation/7fbb5334-ce9d-43b0-8630-7f80e5f5b5f5_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Post-hoc[[:space:]]Reward[[:space:]]Calibration_[[:space:]]A[[:space:]]Case[[:space:]]Study[[:space:]]on[[:space:]]Length[[:space:]]Bias/94b1b340-f53d-4420-958a-9dcf861d3e80_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/PostCast_[[:space:]]Generalizable[[:space:]]Postprocessing[[:space:]]for[[:space:]]Precipitation[[:space:]]Nowcasting[[:space:]]via[[:space:]]Unsupervised[[:space:]]Blurriness[[:space:]]Modeling/4c378bd0-7de5-4c43-a6de-70d69abd0e72_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/PostEdit_[[:space:]]Posterior[[:space:]]Sampling[[:space:]]for[[:space:]]Efficient[[:space:]]Zero-Shot[[:space:]]Image[[:space:]]Editing/8a7af24f-dc7f-401f-9626-40d7b4438f83_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Posterior-Mean[[:space:]]Rectified[[:space:]]Flow_[[:space:]]Towards[[:space:]]Minimum[[:space:]]MSE[[:space:]]Photo-Realistic[[:space:]]Image[[:space:]]Restoration/e39a2209-87e3-4ec7-a952-6eab22e3a4cb_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Preble_[[:space:]]Efficient[[:space:]]Distributed[[:space:]]Prompt[[:space:]]Scheduling[[:space:]]for[[:space:]]LLM[[:space:]]Serving/8054ab91-428e-452f-9376-63400d0d3b25_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Precedence-Constrained[[:space:]]Winter[[:space:]]Value[[:space:]]for[[:space:]]Effective[[:space:]]Graph[[:space:]]Data[[:space:]]Valuation/967f6e8a-ffd2-4dc7-b330-7f04c385c8d8_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Precise[[:space:]]Localization[[:space:]]of[[:space:]]Memories_[[:space:]]A[[:space:]]Fine-grained[[:space:]]Neuron-level[[:space:]]Knowledge[[:space:]]Editing[[:space:]]Technique[[:space:]]for[[:space:]]LLMs/e84bbfe4-c52b-43bb-a8ec-8b25aa019405_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Precise[[:space:]]Parameter[[:space:]]Localization[[:space:]]for[[:space:]]Textual[[:space:]]Generation[[:space:]]in[[:space:]]Diffusion[[:space:]]Models/b2c933e9-ff78-4980-ad5f-0e27435fceb0_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Predicate[[:space:]]Hierarchies[[:space:]]Improve[[:space:]]Few-Shot[[:space:]]State[[:space:]]Classification/7c71c7c6-00c1-4468-bab3-812f9e4fd630_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Predicting[[:space:]]the[[:space:]]Energy[[:space:]]Landscape[[:space:]]of[[:space:]]Stochastic[[:space:]]Dynamical[[:space:]]System[[:space:]]via[[:space:]]Physics-informed[[:space:]]Self-supervised[[:space:]]Learning/a7b796e6-a43e-483b-9485-f3d006014a89_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Prediction[[:space:]]Risk[[:space:]]and[[:space:]]Estimation[[:space:]]Risk[[:space:]]of[[:space:]]the[[:space:]]Ridgeless[[:space:]]Least[[:space:]]Squares[[:space:]]Estimator[[:space:]]under[[:space:]]General[[:space:]]Assumptions[[:space:]]on[[:space:]]Regression[[:space:]]Errors/1b24eeb8-d785-4730-bf3d-a10aeb4a8cc4_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Predictive[[:space:]]Uncertainty[[:space:]]Quantification[[:space:]]for[[:space:]]Bird's[[:space:]]Eye[[:space:]]View[[:space:]]Segmentation_[[:space:]]A[[:space:]]Benchmark[[:space:]]and[[:space:]]Novel[[:space:]]Loss[[:space:]]Function/8f9c4f95-bc77-45da-af45-2f3ee8b5d6d8_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Preference[[:space:]]Diffusion[[:space:]]for[[:space:]]Recommendation/e12258e6-42ec-444c-ac76-afcc85a7a368_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Preference[[:space:]]Elicitation[[:space:]]for[[:space:]]Offline[[:space:]]Reinforcement[[:space:]]Learning/bc9c25fd-bd57-4e9e-9841-162acf7c7d0c_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Preserving[[:space:]]Deep[[:space:]]Representations[[:space:]]in[[:space:]]One-Shot[[:space:]]Pruning_[[:space:]]A[[:space:]]Hessian-Free[[:space:]]Second-Order[[:space:]]Optimization[[:space:]]Framework/7253570e-3268-4b87-9a60-9c9186f1b866_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Preserving[[:space:]]Diversity[[:space:]]in[[:space:]]Supervised[[:space:]]Fine-Tuning[[:space:]]of[[:space:]]Large[[:space:]]Language[[:space:]]Models/5ac59568-73c0-4c45-a6dd-68fda2c55966_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Prevalence[[:space:]]of[[:space:]]Negative[[:space:]]Transfer[[:space:]]in[[:space:]]Continual[[:space:]]Reinforcement[[:space:]]Learning_[[:space:]]Analyses[[:space:]]and[[:space:]]a[[:space:]]Simple[[:space:]]Baseline/7f3b1810-3f1c-4410-9e51-08fa840ec788_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Privacy[[:space:]]Auditing[[:space:]]of[[:space:]]Large[[:space:]]Language[[:space:]]Models/228faee7-b9de-4400-808e-d5e50c11994c_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Privacy-Aware[[:space:]]Lifelong[[:space:]]Learning/3e403618-e0b3-4da1-b604-05c487673a7c_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Privacy-Preserving[[:space:]]Personalized[[:space:]]Federated[[:space:]]Prompt[[:space:]]Learning[[:space:]]for[[:space:]]Multimodal[[:space:]]Large[[:space:]]Language[[:space:]]Models/e5ec5bc1-be1d-4772-bbdb-580e6b409d3a_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Private[[:space:]]Mechanism[[:space:]]Design[[:space:]]via[[:space:]]Quantile[[:space:]]Estimation/6b1c48ed-bcdf-4ec4-83c7-2ca64603aa7b_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Privately[[:space:]]Counting[[:space:]]Partially[[:space:]]Ordered[[:space:]]Data/b859ef91-f09f-4a8e-a9a0-8a6b434beaae_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/ProAdvPrompter_[[:space:]]A[[:space:]]Two-Stage[[:space:]]Journey[[:space:]]to[[:space:]]Effective[[:space:]]Adversarial[[:space:]]Prompting[[:space:]]for[[:space:]]LLMs/55106d97-5bef-4b45-b4dd-839e3efeb83e_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Proactive[[:space:]]Agent_[[:space:]]Shifting[[:space:]]LLM[[:space:]]Agents[[:space:]]from[[:space:]]Reactive[[:space:]]Responses[[:space:]]to[[:space:]]Active[[:space:]]Assistance/9ad6820e-9163-4689-83f1-0ce5732e47d3_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Proactive[[:space:]]Privacy[[:space:]]Amnesia[[:space:]]for[[:space:]]Large[[:space:]]Language[[:space:]]Models_[[:space:]]Safeguarding[[:space:]]PII[[:space:]]with[[:space:]]Negligible[[:space:]]Impact[[:space:]]on[[:space:]]Model[[:space:]]Utility/5f52d3c2-d38c-4e62-9bbe-2e47a1f8b4ec_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Probabilistic[[:space:]]Conformal[[:space:]]Prediction[[:space:]]with[[:space:]]Approximate[[:space:]]Conditional[[:space:]]Validity/56f4b466-8b1e-4fc6-86dd-51ab3604f5ab_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Probabilistic[[:space:]]Language-Image[[:space:]]Pre-Training/133217d3-fa30-4ed6-bca8-41926de82bb0_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Probe[[:space:]]Pruning_[[:space:]]Accelerating[[:space:]]LLMs[[:space:]]through[[:space:]]Dynamic[[:space:]]Pruning[[:space:]]via[[:space:]]Model-Probing/42bd689c-26c8-4499-a0df-f6ad347691a4_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Probe[[:space:]]before[[:space:]]You[[:space:]]Talk_[[:space:]]Towards[[:space:]]Black-box[[:space:]]Defense[[:space:]]against[[:space:]]Backdoor[[:space:]]Unalignment[[:space:]]for[[:space:]]Large[[:space:]]Language[[:space:]]Models/c169bd3c-c7f0-469b-8b38-f42d7eb81a95_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Probing[[:space:]]the[[:space:]]Latent[[:space:]]Hierarchical[[:space:]]Structure[[:space:]]of[[:space:]]Data[[:space:]]via[[:space:]]Diffusion[[:space:]]Models/767cf076-bb26-43cb-ae3e-0ea255d775b7_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Procedural[[:space:]]Knowledge[[:space:]]in[[:space:]]Pretraining[[:space:]]Drives[[:space:]]Reasoning[[:space:]]in[[:space:]]Large[[:space:]]Language[[:space:]]Models/6192cd86-dc2c-4f17-8d28-5fe7a4264c73_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Procedural[[:space:]]Synthesis[[:space:]]of[[:space:]]Synthesizable[[:space:]]Molecules/004681cb-7f9f-4182-baa5-fa0f063aacb0_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Process[[:space:]]Reward[[:space:]]Model[[:space:]]with[[:space:]]Q-value[[:space:]]Rankings/0627156d-3069-4766-8474-f68972dcdd12_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Progress[[:space:]]or[[:space:]]Regress_[[:space:]]Self-Improvement[[:space:]]Reversal[[:space:]]in[[:space:]]Post-training/2dacc1bf-77dd-46c3-ba76-18070d2e0441_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Progressive[[:space:]]Mixed-Precision[[:space:]]Decoding[[:space:]]for[[:space:]]Efficient[[:space:]]LLM[[:space:]]Inference/66ab7e2e-1879-491d-825a-c2d85eb33285_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Progressive[[:space:]]Parameter[[:space:]]Efficient[[:space:]]Transfer[[:space:]]Learning[[:space:]]for[[:space:]]Semantic[[:space:]]Segmentation/d132c3ce-fc2e-45e5-bcfb-2547ce071e89_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Progressive[[:space:]]Token[[:space:]]Length[[:space:]]Scaling[[:space:]]in[[:space:]]Transformer[[:space:]]Encoders[[:space:]]for[[:space:]]Efficient[[:space:]]Universal[[:space:]]Segmentation/c9db2d7c-452d-453c-bd62-3b2a958eb1e8_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Projection[[:space:]]Head[[:space:]]is[[:space:]]Secretly[[:space:]]an[[:space:]]Information[[:space:]]Bottleneck/ead0a0bd-d12b-4058-b61c-cf9fda9c5296_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Prompt[[:space:]]as[[:space:]]Knowledge[[:space:]]Bank_[[:space:]]Boost[[:space:]]Vision-language[[:space:]]model[[:space:]]via[[:space:]]Structural[[:space:]]Representation[[:space:]]for[[:space:]]zero-shot[[:space:]]medical[[:space:]]detection/2e078ed1-9418-48d5-a48a-c9e70badf727_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Prompting[[:space:]]Fairness_[[:space:]]Integrating[[:space:]]Causality[[:space:]]to[[:space:]]Debias[[:space:]]Large[[:space:]]Language[[:space:]]Models/33fc262a-ef52-44b2-8f8f-a3a173a98a84_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Promptriever_[[:space:]]Instruction-Trained[[:space:]]Retrievers[[:space:]]Can[[:space:]]Be[[:space:]]Prompted[[:space:]]Like[[:space:]]Language[[:space:]]Models/cf793ec0-08ad-4fb0-8be4-db5f1cd152c4_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/ProtPainter_[[:space:]]Draw[[:space:]]or[[:space:]]Drag[[:space:]]Protein[[:space:]]via[[:space:]]Topology-guided[[:space:]]Diffusion/3bac7b3e-cca5-4138-87e9-bd7265d97a81_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Protecting[[:space:]]against[[:space:]]simultaneous[[:space:]]data[[:space:]]poisoning[[:space:]]attacks/e79fa11a-8472-4cec-87f8-91e0514daac8_origin.pdf filter=lfs diff=lfs merge=lfs -text

2025/Planning Anything with Rigor_ General-Purpose Zero-Shot Planning with LLM-based Formalized Programming/08557070-92b7-4701-8d22-314405276414_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:0be8d87099c06c8ff1f112198989304e1aaad7f08cd2dd51f98011d0f8f222ed
+size 1827169

2025/Planning Anything with Rigor_ General-Purpose Zero-Shot Planning with LLM-based Formalized Programming/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:de182cd95f5e06d5c4ca5ff7371bb011d51d0a3b3341f6d3ed60d900d085e7f2
+size 3413492

2025/Plastic Learning with Deep Fourier Features/9a76bf86-ef76-4d49-8136-051df093c801_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:afc4d6a97d6f1148d6ed826caafd5945a9b1e30405457757a6f496d0de0ec913
+size 1915801

2025/Plastic Learning with Deep Fourier Features/full.md

@@ -0,0 +1,599 @@
+# PLASTIC LEARNING WITH DEEP FOURIER FEATURES
+
+Alex Lewandowski $^{1,2}$
+
+Dale Schuurmans $^{1,2,3,4}$
+
+Marlos C. Machado<sup>1,2,4</sup>
+
+<sup>1</sup>Department of Computing Science, University of Alberta, <sup>2</sup>Amii,
+
+3Google DeepMind, 4Canada CIFAR AI Chair
+
+# ABSTRACT
+
+Deep neural networks can struggle to learn continually in the face of nonstationarity, a phenomenon known as loss of plasticity. In this paper, we identify underlying principles that lead to plastic algorithms. We provide theoretical results showing that linear function approximation, as well as a special case of deep linear networks, do not suffer from loss of plasticity. We then propose deep Fourier features, which are the concatenation of a sine and cosine in every layer, and we show that this combination provides a dynamic balance between the trainability obtained through linearity and the effectiveness obtained through the nonlinearity of neural networks. Deep networks composed entirely of deep Fourier features are highly trainable and sustain their trainability over the course of learning. Our empirical results show that continual learning performance can be improved by replacing ReLU activations with deep Fourier features combined with regularization. These results hold for different continual learning scenarios (e.g., label noise, class incremental learning, pixel permutations) on all major supervised learning datasets used for continual learning research, such as CIFAR10, CIFAR100, and tiny-ImageNet.
+
+# 1 INTRODUCTION
+
+Continual learning is a problem setting that moves past some of the rigid assumptions found in supervised, semi-supervised, and unsupervised learning (Ring, 1994; Thrun, 1998). In particular, the continual learning setting involves learning from data sampled from a changing, non-stationary distribution rather than from a fixed distribution. A performant continual learning algorithm faces a trade-off due to its limited capacity: it should avoid forgetting what was previously learned while also being able to adapt to new incoming data, an ability known as plasticity (Parisi et al., 2019). Current approaches that use neural networks for continual learning are not yet capable of making this trade-off due to catastrophic forgetting (Kirkpatrick et al., 2017) and loss of plasticity (Dohare et al., 2021; Lyle et al., 2023; Dohare et al., 2024). The training of neural networks is in fact an active research area in the theory literature for supervised learning (Jacot et al., 2018; Yang et al., 2023; Kunin et al., 2024), which suggests there is much left to be understood in training neural networks continually. Compared to the relatively well-understood problem setting of supervised learning, even the formalization of the continual learning problem is an active research area (Kumar et al., 2023a; Abel et al., 2024; Liu et al., 2023). With these uncertainties surrounding current practice, we take a step back to better understand the inductive biases used to build algorithms for continual learning.
+
+One fundamental capability expected from a continual learning algorithm is its sustained ability to update its predictions on new data. Recent work has identified the phenomenon of loss of plasticity in neural networks in which stochastic gradient-based training becomes less effective when faced with data from a changing, non-stationary distribution (Dohare et al., 2024). Several methods have been proposed to address the loss of plasticity in neural networks, with their success demonstrated empirically across both supervised and reinforcement learning (Ash and Adams, 2020; Lyle et al., 2022; Lee et al., 2024). Empirically, works have identified that the plasticity of neural networks is sensitive to different components of the training process, such as the activation function (Abbas et al., 2023). However, little is known about what is required for learning with sustained plasticity.
+
+The goal of this paper is to identify a basic continual learning algorithm that does not lose plasticity in both theory and practice, rather than mitigating the loss of plasticity in existing neural network architectures. Our focus is on loss of plasticity rather than catastrophic forgetting, because plasticity is needed to sustain continual learning from new data. In particular, we investigate the effect of the nonlinearity of neural networks on the loss of plasticity. While loss of plasticity is a well-documented phenomenon in neural networks, previous empirical observations suggest that linear function
+
+![](images/bb17c51b5b5764783a84d63b0fa8f6a73c6f190d7ac04c16b003b26aa109969f.jpg)  
+Figure 1: A neural network with deep Fourier features in every layer approximately embeds a deep linear network. A single layer using deep Fourier features linearly combines the inputs,  $x$ , to compute the pre-activations,  $z$ , and each pre-activation is mapped to both a  $\cos$  unit and a  $\sin$  unit (Left). For each pre-activation, either the  $\sin$  unit (Middle) or the  $\cos$  unit (Right) is well-approximated by a linear function.
+
+approximation is capable of learning continually without suffering from loss of plasticity (Dohare et al., 2021; 2024). In this paper, we prove that linear function approximation does not suffer from loss of plasticity and can sustain their learning ability on a sequence of tasks. We then extend our analysis to a special case of deep linear networks, which provide an interesting intermediate case between deep nonlinear networks and linear function approximation. This is because deep linear networks are linear in representation but nonlinear in gradient dynamics (Saxe et al., 2014). We provide theoretical and empirical evidence that general deep linear networks also do not suffer from loss of plasticity. The plasticity of deep linear networks is surprising compared to the loss of plasticity of deep nonlinear networks. This finding suggests that loss of plasticity is not necessarily caused by the nonlinear learning dynamics, but a combination of nonlinear learning dynamics and nonlinear representations.
+
+Given this seemingly natural advantage of linearity for continual learning, as well as its inherent limitation to learning only linear representations, we explore how nonlinear networks can better emulate the dynamics of deep linear networks to sustain plasticity. We hypothesize that, to effectively learn continually, the neural network must balance between introducing too much linearity and suffering from loss of deep representations and introducing too much nonlinearity and suffering from loss of plasticity. In fact, we show that previous work partially satisfies this hypothesis, such as the concatenated ReLU (Shang et al., 2016), leaky-ReLU activations (Xu et al., 2015), and residual connections (He et al., 2016), but they fail at striking this balance. Our results build on previous work that identified issues of unit saturation (Abbas et al., 2023) and unit linearization (Lyle et al., 2024) as issues in continually training neural networks with common activation functions. In particular, we generalize both to the problem of low unit sign entropy, which indicates a lack of diversity in the activations as measured by the entropy of the sign of the hidden units. We show that linear networks have high unit sign entropy, meaning that the sign of a hidden unit on different inputs is positive on approximately half the inputs. In contrast, deep nonlinear networks with most activation functions tend to have low unit sign entropy, which indicates saturation or linearization.
+
+Periodic activation functions (Parascandolo et al., 2017), like the sinusoid function (sin), are a notable exception for having high unit sign entropy despite still suffering from loss of plasticity. Thus, in addition to unit sign entropy, we demonstrate that the network's activation function should be well-approximated by a linear function. We propose deep Fourier features as a means of approximating linearity dynamically, with every pre-activation being connected to two units, one of which will always be well-approximated by a linear function. In particular, deep Fourier features concatenate a sine and a cosine activation in each hidden layer. The resulting network is nonlinear while also approximately embedding a deep linear network using all of its parameters. Deep Fourier features differ from previous approaches that use Fourier features only in the input layer (Tancik et al., 2020; Li and Pathak, 2021; Yang et al., 2022) or that use fixed Fourier feature basis (Rahimi and Recht, 2007; Konidaris et al., 2011). We demonstrate that networks using these shallow Fourier features still exhibit a loss of plasticity. Only by using deep Fourier features in every layer is the network capable of sustaining and improving trainability in a continual learning setting. Using tiny-ImageNet (Le and Yang, 2015), CIFAR10, and CIFAR100 (Krizhevsky, 2009), we show that deep Fourier features can be used as a drop-in replacement for improving trainability in commonly used neural network architectures. Furthermore, the trainability of deep Fourier features enables training with a much larger regularization strength, leading to superior generalization performance.
+
+# 2 PROBLEM SETTING
+
+We define a deep network,  $f_{\theta}$  with a parameter set,  $\theta = \{\mathbf{W}_l,\mathbf{b}_l\}_{l = 1}^L$ , as a sequence of layers, in which each layer applies a linear transformation followed by an element-wise activation function,  $\phi$  in each hidden layer. The output of the network,  $f_{\theta}(x)\coloneqq h_L(x)$ , is defined recursively by  $h_l = [h_{l,1},\dots ,h_{l,w}] = [\phi (z_{l,1}),\dots ,\phi (z_{l,w})] = \phi (z_l)$ , and,  $z_{l} = \mathbf{W}_{l}h_{l - 1} + \mathbf{b}_{l}$  where  $w$  is the width of the network, and  $h_0 = x$ . We refer to a particular element of the hidden layer's output  $h_{l,i}$  as a unit. The deep network is a deep linear network when the activation function is the identity,  $\phi (z) = z$ . Linear function approximation is equivalent to a linear network with  $L = 1$ .
+
+The problem setting that we consider is continual supervised learning, where the learner does not have information about the task boundaries. Instead, at each iteration the learner has access to a minibatch of observation-target pairs of size  $M$ ,  $\{x_i, y_i\}_{i=1}^M$ . This minibatch is used to update the parameters  $\theta$  of a neural network  $f_\theta$  using a variant of stochastic gradient descent. The learning problem is continual because the distribution from which the data is sampled,  $p(x, y)$ , is changing. For simplicity, we assume this non-stationarity changes the distribution over the input-target pairs every  $T$  iterations. The data is sampled from a single distribution for  $T$  steps, and we refer to this particular temporary stationary problem as a task,  $\tau$ . The distribution over observations and targets that defines a task  $\tau$  is denoted by  $p_\tau$ .
+
+Loss of plasticity can refer to two related phenomena: loss of generalization (Ash and Adams, 2020; Dohare et al., 2024) or loss of trainability (Dohare et al., 2021; Lyle et al., 2023). We focus our theoretical analysis on the problem of loss of trainability, in which we evaluate the neural network at the end of each task using samples from the most recent task distribution,  $p_{\tau}$ , as is commonly done in previous work (Lyle et al., 2023). Loss of trainability refers to the problem where the neural network is unable to sustain its initial performance on the first task to later tasks. Specifically, we denote the optimisation objective by  $J_{\tau}(\theta) = \mathbb{E}_{(x,y)\sim p_{\tau}}[\ell (f_{\theta}(x),y)]$ , for some loss function  $\ell$ , and task-specific data distribution  $p_{\tau}$ . We use  $t$  to denote the iteration count of the learning algorithm, and thus the current task number can be written as  $\tau(t) = \lfloor t / T \rfloor$ .
+
+# 3 TRAINABILITY AND LINEARITY
+
+In this section, we show that, unlike nonlinear networks, linear networks do not suffer from loss of trainability. That is, if the number of iterations in each task is sufficiently large, a linear network sustain trainability on every task in the sequence. We then show theoretically that a special case of deep linear networks also does not suffer from loss of trainability, and we empirically validate the theoretical findings in more general settings. These results provide a theoretical basis for previous work that uses a linear baseline in loss of plasticity experiments.
+
+# 3.1 TRAINABILITY OF LINEAR FUNCTION APPROXIMATION
+
+We first prove that loss of trainability does not occur with linear function approximation,  $f_{\theta}(x) = \mathbf{W}_l x + \mathbf{b}_l$ . We prove this by showing that linear function approximators can sustain learning on a sequence of tasks, with a large enough number of iterations per task. In particular, the performance of the solution found on the  $\tau$ -th task can be upper bounded on a quantity that is independent of the solution found on the first  $\tau - 1$  tasks. Linear function approximation avoids loss of trainability because the optimisation problem on each task is convex (Agrawal et al., 2021; Boyd and Vandenberghe, 2004), with a unique global optimum,  $\theta_{\tau}^{\star}$ . We now state the theorem, which we prove in Appendix B
+
+Theorem 1. Let  $\theta^{(\tau T)}$  denote the linear weights learned at the end of the  $\tau$ -th task, with the corresponding unique global minimum for task  $\tau$  being denoted by  $\theta_{\tau}^{\star}$ . Assuming the objective function is  $\mu$ -strongly convex, the suboptimality gap for gradient descent on the  $\tau$ -th task is
+
+$$
+J _ {\tau} (\theta^ {(\tau T)}) - J _ {\tau} (\theta_ {\tau} ^ {\star}) <   \frac {2 D (1 - \alpha \mu) ^ {T}}{\alpha T (1 - (1 - \alpha \mu) ^ {T})},
+$$
+
+where each task lasts for  $T$  iteration,  $D$  is the assumed bound on the parameters at the global minimum for every task, and  $\alpha$  is the step-size.
+
+Intuitively, this theorem states that if the problem is bounded and effectively strongly convex due to a finite number of iterations, then the optimisation dynamics are well-behaved for every task in the bounded set. In particular, this means that the error on each task can be upper bounded by a quantity independent of the initialization found on previous tasks. Thus, given enough iterations, linear function approximation can learn continually without loss of trainability.
+
+# 3.2 TRAINABILITY OF DEEP LINEAR NETWORKS
+
+We now provide evidence that, similar to linear function approximation, deep linear networks also do not suffer from loss of trainability. Unlike deep nonlinear networks, deep linear networks use linear activation functions in their hidden layers (Bernacchia et al., 2018; Ziyin et al., 2022). This means that a deep linear network can only represent linear functions. At the same time, its gradient update dynamics are nonlinear and non-convex, similar to deep nonlinear neural networks (Saxe et al., 2014). Our central claim here is that deep linear networks under gradient descent dynamics avoid parameter configurations that would lead to loss of trainability.
+
+To simplify notation, without loss of generality, we combine the weights and biases into a single parameter for each layer in the deep linear network,  $\theta = \{\theta_{1},\dots ,\theta_{L}\}$ , and  $f_{\theta}(x) = \theta_L\theta_{L - 1}\dots \theta_1x$ . We denote the product of weight matrices, or simply product matrix, as  $\bar{\theta} = \theta_{L}\theta_{L - 1}\dots \theta_{1}$ , which allows us to write the deep linear network in terms of the product matrix:  $f_{\theta}(x) = \bar{\theta} x$ . The problem setup we use for the deep linear analysis follows previous work (Huh, 2020), and we provide additional technical details for optimisation dynamics of deep linear networks in Appendix A.3.
+
+We now provide evidence to suggest that, despite deep linear networks being nonlinear in their gradient dynamics, they do not suffer from loss of trainability. We prove this for a special case of deep diagonal linear networks, and provide empirical evidence to support this claim in general deep linear networks.
+
+Theorem 2. Let  $f_{\theta}(x) = \theta_L\theta_{L - 1}\dots \theta_1x$  be a deep diagonal linear network where  $\theta_l = \operatorname{Diag}(\theta_{l,1},\ldots ,\theta_{l,d})$ . Then, a deep diagonal linear network converges on a sequence of tasks under the same conditions for convergence in a single task (i.e., the conditions in Arora et al., 2019).
+
+Theorem 2 states that a deep diagonal linear network, a special case of general deep linear networks, can converge to a solution on each task within a sequence of tasks. The proof, provided in Appendix B, shows that the minimum singular value of the product matrix stays greater than zero,  $\sigma_{min}(\theta) > 0$ . Hence, deep diagonal linear networks do not suffer from loss of trainability. This result provides further evidence suggesting that linearity might be an effective inductive bias for learning continually.
+
+While the analysis considers a special case of deep linear networks, namely deep diagonal networks, we note that this is a common setting for the analysis of deep linear networks more generally (Nacson et al., 2022; Even et al., 2023). In particular, the analysis is motivated by the fact that, under certain conditions, the evolution of the deep linear network parameters can be analyzed through the independent singular mode dynamics (Braun et al., 2022), which simplifies the analysis of deep linear networks to deep diagonal linear networks.
+
+# 3.3 EMPIRICAL EVIDENCE FOR TRAINABILITY OF GENERAL DEEP LINEAR NETWORKS
+
+In the previous section, we proved that a special case of deep linear networks do not suffer from loss of trainability. We now provide additional empirical evidence that general deep linear networks do not suffer from loss of trainability. To do so, we use a linearly separable subset of the MNIST dataset (LeCun et al., 1998), in which the labels of each image are randomized every 100 epochs. For this experiment, the data is linearly separable so that even a linear baseline can fit the data if given enough iterations. While MNIST is a simple classification problem, memorizing random labels highlights the difficulties associated with maintaining trainability (see Lyle et al., 2023; Kumar et al., 2023b). We emphasize that the goal here is merely to validate that linear networks remain trainable in continual learning. We also provide results with traditional nonlinear neural networks on the same problem, showing that they suffer from loss of trainability in this simple problem. Later in Section 5, we extend our investigation of loss of trainability to larger-scale benchmarks.
+
+![](images/638470bd759aa69a21b8db10865ba6d1c9f1b2b97c9de2d2236a6ad60403579b.jpg)  
+Figure 2: Trainability on a linearly separable task. The higher opacity corresponds to deeper networks, ranging from  $\{1, 2, 4, 8, 16\}$ . Deep linear networks sustain trainability on new tasks, with some additional depth improving trainability. Nonlinear networks, using ReLU, suffer from loss of trainability at any depth even on this simple sequence of linearly separable problems.
+
+In Figure 2, we see that deep linear networks ranging from a depth of 1 to 16 can sustain trainability. Using a multi-layer perceptron with ReLU activations, deep nonlinear networks quickly reach a much higher accuracy on the first few tasks. However, due to loss of trainability, deep nonlinear networks of any depth eventually perform worse than the corresponding deep linear network. With additional epochs, the linear networks could achieve perfect accuracy on this task because it is linear separable. The number of epochs is comparatively low to showcase that, with some additional layers, a deep linear network is able to improve its trainability as new tasks are encountered.
+
+# 4 COMBINING LINEARITY AND NONLINEARITY
+
+In the previous section, we provided empirical and theoretical evidence that linearity provides an effective inductive bias for learning continually by avoiding loss of trainability. However, linear methods are generally not as performant as deep nonlinear networks, meaning that their sustained performance can be inadequate on complex tasks. Even deep linear networks have only linear representational power, despite their nonlinear gradient dynamics. We now seek to answer the following question:
+
+# How can the sustained trainability of linear methods be combined with the expressive power of learned nonlinear representations?
+
+To answer this question, we first seek to better understand the effects of replacing linear activation functions with nonlinear ones in deep networks for continual learning. We observe that deep linear networks have diversity in their hidden units, which can be induced in nonlinear activation functions by adding linearity through a weighted linear component, an idea we refer to as  $\alpha$ -linearization. To dynamically balance linearity and nonlinearity, we propose to use deep Fourier features for every layer in a network. We prove that such a network approximately embeds a deep linear network, a property we refer to as adaptive linearity. We demonstrate that this adaptively-linear network is plastic, maintaining trainability even on non-linearly-separable problems.
+
+# 4.1 ADDING LINEARITY TO NONLINEAR ACTIVATION FUNCTIONS
+
+Deep nonlinear networks can learn expressive representations because of their nonlinear activation functions, but these nonlinearities can also lead to issues with trainability. Although several components of common network architectures incorporate linearity, the way in which linearity is used does not avoid loss of trainability. One example is the piecewise linearity of the ReLU activation function (Shang et al., 2016),  $\operatorname{ReLU}(x) = \max(0, x)$ , which is said to be saturated if  $\operatorname{ReLU}(x) = 0$  for most inputs  $x$ , preventing gradient propagation. While saturation is generally not a problem for learning on a single distribution, it has been noted as problematic in learning from changing distributions, for example, in reinforcement learning (Abbas et al., 2023).
+
+A potential solution to saturation is to use a non-saturating activation function. Two noteworthy examples of non-saturating activation functions include a periodic activation like  $\sin(x)$  (Parascandolo et al., 2017) and leaky-ReLU $_{\alpha}(x) = \alpha x + (1 - \alpha) \mathrm{ReLU}(x)$  (Xu et al., 2015), both of which are zero on a set of measure zero. Surprisingly, using leaky-ReLU leads to a related issue, "unit linearization" (Lyle et al., 2024), in which the activation is only positive (or negative) for most inputs  $x$ . Unlike saturated units, linearized units can provide non-zero gradients but render that unit effectively linear, limiting the expressive power of the learned representation. While unit linearization seems to suggest that loss of trainability can occur due to linearity, it is important to note that a "linearized unit" is not the same as a linear unit. This is because a linearized unit provides mostly positive (or negative) outputs, whereas a linear unit can output both positive and negative values.
+
+We generalize the idea behind unit saturation and unit linearization to unit sign entropy, which is applicable to activation functions beyond saturating and piecewise linear functions, such as periodic activation functions. Intuitively, it measures the diversity of the activations of a hidden layer.
+
+Definition 1 (Unit Sign Entropy). The entropy,  $\mathbb{H}$ , of the unit's sign,  $\operatorname{sgn}(h(x))$ , on a distribution of inputs to the network,  $p(x)$ , is given by  $\mathbb{H}\left(\operatorname{sgn}(h(x))\right) = \mathbb{E}_{p(x)}[\operatorname{sgn}(h(x))]$ .
+
+The maximum value of unit sign entropy is 1, which occurs when the unit is positive on half the inputs. Conversely, a low sign entropy is associated with the aforementioned issues of saturation and linearization. For example, a low sign entropy for a deep network using ReLU activations means that the unit is almost always positive ( $P(\operatorname{sgn}(h(x)) = 1) = 1$ , meaning it is linearized) or negative ( $P(\operatorname{sgn}(h(x)) = 1) = 0$ , meaning it is saturated).
+
+With unit sign entropy, we investigate how the leak parameter for the leaky-ReLU activation function influences training as pure linearity ( $\alpha = 1$ ) is traded-off for pure nonlinearity ( $\alpha = 0$ ). The idea of mixing a linearity and nonlinearity can also be generalized to an arbitrary activation function, which we refer to as the  $\alpha$ -linearization of an activation function.
+
+Definition 2 ( $\alpha$ -linearization). The  $\alpha$ -linearization of an activation function  $\phi$ , is denoted by  $\phi_{\alpha}(x) = \alpha x + (1 - \alpha)\phi(x)$ .
+
+A natural hypothesis is that, as  $\alpha$  increases from 0 to 1, and the network becomes more linear, loss of trainability is mitigated. We emphasize that the  $\alpha$ -linearization is primarily to gain insights from empirical investigation and it is not a solution to loss of trainability. This is because any benefits of  $\alpha$ -linearization depend on tuning  $\alpha$ , and even optimal tuning can lead to overly linear representations and slow training compared to nonlinear networks.
+
+![](images/75838a27cad30fc948349a7aca1fbd05fd7d482c432da1d84adca5e0ed08daa2.jpg)  
+Figure 3: Trainability on a linearly separable task with  $\alpha$ -linearization Darker opacity lines correspond to higher values of  $\alpha$ . Unit sign entropy increases as  $\alpha$  increases (inset), leading to sustained trainability for  $\alpha$ -relu.
+
+Empirical Evidence for  $\alpha$ -linear Plasticity To understand the trainability issues introduced by nonlinearity, we present a case-study using sin and ReLU with different values of the linearization parameter,  $\alpha$ . The same experiment setup is used from Section 3.3. Referring to the results in Figure 3, we see that both ReLU and sin activation functions are able to sustain trainability for larger values of  $\alpha$ . This verifies the hypothesis: a larger  $\alpha$  pro
+
+vides more linearity to the network, allowing it to sustain trainability. Despite sustaining trainability, a larger  $\alpha$  can lead to overly linear representations, evidenced by worse performance and slower training speed on the first few tasks compared to nonlinear networks ( $\alpha = 0$ ). For  $\alpha$ -ReLU, we also verify the hypothesis that the unit sign entropy increases for larger values of  $\alpha$  (inset plot). The fact that the periodic sin activation function has a high unit sign entropy despite losing trainability is particularly interesting, which we explore in the next section.
+
+# 4.2 ADAPTIVE-LINEARITY BY CONCATENATING SINUSOID ACTIVATION FUNCTIONS
+
+Using the insight that linearity promotes unit sign entropy, we explore an alternative approach to sustain trainability. In particular, we found that linearity can sustain trainability but requires tuning  $\alpha$ , and even optimal tuning can lead to slow learning from overly linear representations. Our approach is motivated by concatenated ReLU activations (Shang et al., 2016; Abbas et al., 2023),  $\mathrm{CReLU}(z) = [\mathrm{ReLU}(z), \mathrm{ReLU}(-z)]$ , which avoids the problems from saturated units, but does not avoid the problem of low unit sign entropy. In particular, we propose using a pair of activations functions such that one activation function is always approximately linear, with a bounded error.
+
+One way to dynamically balance the linearities and nonlinearities of a network is using periodic activation functions. This is because, due to their periodicity, the properties of the activation function can re-occur as the magnitude of the preactivations grows rather than staying constant, linear, or saturating. But, as we saw in Figure 3, a single periodic activation function like sin is not enough. Instead, we propose to use deep Fourier features, meaning that every layer in the network uses Fourier features. This is a notable departure from previous work which considers only shallow Fourier features in the first layer (Rahimi and Recht, 2007; Tancik et al., 2020). In particular, each unit is a concatenation of a sinusoid basis of two elements,  $\operatorname{Fourier}(z) = [\sin(z), \cos(z)]$ . Each pre-activation is mapped to both  $\sin(z)$  and  $\cos(z)$ , which requires that a layer with deep Fourier features have half the output width to accommodate the concatenation.<sup>1</sup>
+
+The advantage of this approach is that a network with deep Fourier features maintains approximate linearity in all of its parameters. Moreover, deep Fourier features are closed under differentiation, meaning that the activations and their gradients provide a basis for representing periodic functions.
+
+Proposition 1. For any  $z$ , there exists a linear function,  $L_{z}(x) = a(z)x + b(z)$ , such that either:  $|\sin(x) - L_{z}(x)| \leq c$ , or  $|\cos(x) - L_{z}(x)| \leq c$ , for  $c = \sqrt{2}\pi^{2}/2^{8}$  and all  $x \in [z - \pi/4, z + \pi/4]$ .
+
+An intuitive description of this is provided in Figure 1. The advantage of using two sinusoids over just a single sinusoid is that whenever  $\cos (z)$  is near a critical point,  $d / dz\cos (z)\approx 0$ , we have that  $\sin (z)\approx z$ , meaning that  $d / dz\sin (z)\approx 1$  (and vice-versa). The argument follows from an analysis of the Taylor series remainder, showing that the Taylor series of half the units in a deep Fourier layer can be approximated by a linear function, with a small error of  $c = \sqrt{2}\pi^2 /2^8\approx 0.05$ . While we found that two sinusoids is sufficient, the approximation error can be further improved by concatenating additional sinusoids, at the expense of reducing the effective width of the layer.
+
+Because each pre-activation is connected to a unit that is approximately linear, we can conclude that a deep network comprised of deep Fourier features approximately embeds a deep linear network.
+
+Corollary 1. A network parameterized by  $\theta$ , with deep Fourier features, approximately embeds a deep linear network parameterized by  $\theta$  with a bounded error.
+
+Notice that piecewise linear activations also embed a deep linear network, but these embedded deep linear networks do not use the same parameter set. For example, the deep linear network embedded by a ReLU network does not depend on any of the parameters used to compute a ReLU unit that is zero. Although the leaky-ReLU function involves every parameter, the deep linear network vanishes because the leak parameter is small,  $\alpha < 1$ , and hence the embedded deep linear network is multiplied by a small constant,  $\alpha^{-L}$ , where  $L$  is the depth of the network.
+
+![](images/428c80e44abf0379cc4e6c80ef5ccd726e104686f7ab0a2d4ec4fda3a2df6fa5.jpg)  
+Figure 4: Trainability on a non linearly-separable task. Deep Fourier features improve and sustain their trainability when other networks cannot.
+
+Empirical Evidence for Nonlinear Plasticity We now consider a similar experimental setup from Sections 3.3 and 4.1, except we make the problem non linearly-separable by considering random label assignments on the entire dataset. Each task is more difficult because
+
+it involves memorizing more labels, and the effect of the non-stationarity is also stronger due to randomization of more datapoints. As a result, the deep linear network can no longer fit a single task well. Referring to Figure 4, the  $\alpha$ -linear activation functions can sustain and even improve their trainability, albeit very slowly. See also unit sign entropy in Figure 9, Appendix D.1. In contrast, using deep Fourier features within the network enables the network to easily memorize all the labels for 100 tasks. Deep Fourier features surpass the trainability of the other nonlinear baselines at initialization, CReLU and shallow Fourier features followed by ReLU. This is surprising, because deep nonlinear networks at initialization are often a gold-standard for trainability.
+
+# 5 EXPERIMENTS
+
+Our experiments demonstrate the benefits of the adaptive linearity provided by deep Fourier features. While trainability was the primary focus behind our theoretical results and empirical case studies, we show that these findings generalize to other problems in continual learning. In particular, we demonstrate that networks composed of deep Fourier features are capable of being strongly regularized leading to improved generalization performance on diminishing levels of label noise, and in class-incremental learning. The main results we present are on all of the major continual supervised learning settings considered in the plasticity literature. They build on the standard ResNet-18 architecture, widely used in practice (He et al., 2016).
+
+Datasets and Non-stationarities Our experiments use the common image classification datasets for continual learning, namely tiny-ImageNet (Le and Yang, 2015), CIFAR10, and CIFAR100 (Krizhevsky, 2009). We augment these datasets with commonly used non-stationarities to create continual learning problems, with the non-stationarity creating a sequence of tasks from the dataset. Specifically, we follow recent work that introduced the diminishing label noise problem (Lee et al., 2024), which is inspired by the warm-starting problem: We start with half the data being corrupted by label noise and reduce the noise to clean labels over 10 tasks. Additionally, for the datasets with a larger number of classes, tiny-ImageNet and CIFAR100, we also consider the class-incremental setting: the first task involves only five classes, and five new classes are added to the existing pool of classes at the beginning of each task (Van de Ven et al., 2022). Other results and more details on datasets and non-stationarities considered can be found in Appendix C.
+
+![](images/cd93eff006f24b1a623a838c93a8b519e2ec1e8ad257698b8052c8ab0a916021.jpg)  
+Figure 5: Training a ResNet-18 continually with diminishing label noise. Deep Fourier features are particularly performant on complex tasks like tiny-ImageNet. Despite networks with deep Fourier features having approximately half the number of parameters, they surpass the baselines in CIFAR100 and are on-par with spectral regularization on CIFAR10.
+
+![](images/63f3747004105c1fb541037836eaf651b053d89e092e5825eb7eae3bd37a5e42.jpg)
+
+![](images/4e01164b03fe27387e1c90317871e8358ef6979d6b408f59feba80a747ea7cc5.jpg)
+
+Architecture and Baselines We compare a ResNet-18 using only deep Fourier features against a standard ResNet-18 with ReLU activations. The network with deep Fourier features has fewer parameters because it uses a concatenation of two different activation functions, halving the effective width compared to the network with ReLU activations. This provides an advantage to the nonlinear baseline. We also include all prominent baselines that have previously been proposed to mitigate loss of plasticity in the field: L2 regularization towards zero, L2 regularization towards the initialization (Kumar et al., 2023b), spectral regularization (Lewandowski et al., 2024), Concatenated ReLU (Shang et al., 2016; Abbas et al., 2023), Dormant Neuron Recycling (ReDO, Sokar et al., 2023), Shrink and Perturb (Ash and Adams, 2020), and Streaming Elastic Weight Consolidation (S-EWC, Kirkpatrick et al., 2017; Elsayed and Mahmood, 2024).
+
+# 5.1 MAIN RESULTS
+
+Our results demonstrate that deep Fourier features, combined with regularization, are effective at continual learning. In these set of experiments, we consider the problem of sustaining test accuracy on a sequence of tasks. In addition to requiring trainability, methods must also sustain generalization.
+
+**Diminishing Label Noise** In Figure 5, we can clearly see the benefits of deep Fourier features in the diminishing label noise setting. At the end of training on ten tasks with diminishing levels of label noise, the network with deep Fourier features was always among the methods with the highest test accuracy on the uncorrupted test set. On the first of ten tasks, deep Fourier features could occasionally overfit to the corrupted labels leading to initially low test accuracy. However, as the label noise diminished on future tasks, the network with deep Fourier features was able to continue to learn to correct its previous poorly-generalizing predictions. In contrast, the improvements achieved by the other methods that we considered was oftentimes marginal compared to the baseline ReLU network. Two exceptions are: (i) networks with CReLU activations, which underperformed relative to the baseline network, and (ii) Shrink and Perturb, which was the best-performing baseline method for diminishing label noise. Interestingly, the performance benefit of deep Fourier features is most prominent on more complex datasets, like tiny-ImageNet. In Appendix D.2, we provide an ablation of the architecture, where we use a Wide Residual Network (Zagoruyko and Komodakis, 2016) and vary the width scale.
+
+Class-Incremental Learning Deep Fourier features are also effective in the class-incremental setting, where later tasks involve training on a larger subset of the classes, following the experiment described in (Dohare et al., 2024). The network is evaluated at the end of each task on the entire test set. As the network is trained on later tasks, its test set performance increases
+
+![](images/96a058b907e35839efe538b72fbb8f43bdd7748e6162839ab346abd31c48a4ec.jpg)  
+Figure 6: Class incremental learning results on tiny-Imagenet (Left) and CIFAR-100 (Right). On both datasets, deep Fourier features substantially improve over most baselines.
+
+![](images/69c6453369a3341a3f41c7c04f981244d40c3bd431a82505b23d4263c0374d00.jpg)
+
+because it has access to a larger subset of the training data. In Figure 6, we see that Deep Fourier features largely outperform the baselines in this setting, particularly on tiny-ImageNet in which the first forty tasks involve training on a growing subset of the dataset and the last forty "tasks" involve training to convergence on the full dataset. We use quotation marks to characterize the last forty tasks because they are, in fact, a single task, as the data distribution stops changing after the first forty tasks. We call them "tasks" because of the number of iterations in which they are trained. Not only are deep Fourier features quicker to learn on earlier continual learning tasks, but they are also able to improve their generalization performance by subsequently training on the full dataset. On CIFAR100, the difference between methods is not as prominent, but we can see that deep Fourier features are still among the top-performing methods. The large performance difference on tiny-ImageNet can be attributed to the fact that it is a harder problem compared to CIFAR10 and CIFAR100, with higher resolution images, more classes and more datapoints.
+
+# 5.2 SENSITIVITY ANALYSIS
+
+In the previous sections, we found that deep Fourier features used in combination with spectral regularization leads to strong generalization performance. However, the theoretical analysis and case-studies that we presented earlier concerned trainability. We now present a sensitivity result to understand the relationship between trainability and generalization. Using a ResNet-18 with different activation functions, we varied the regularization strength between no regularization (left) and high regularization (right). In Figure 5.2, we can see that deep Fourier features indeed have a high degree of trainability, sustaining higher trainability at every level of regularization strength. However, without any regularization, deep Fourier features have a tendency to overfit. Over-fitting is a known issue for shallow
+
+![](images/c1f766f256f6340eaa15d1a99335df3a941c346b5e160594b1712a52f7344c90.jpg)  
+Figure 7: Sensitivity analysis on tiny-ImageNet. Networks with deep Fourier features are highly trainable, but have a tendency to overfit without regularization, leading to high training accuracy but low test accuracy. Due to deep Fourier features being highly trainable, they are able to train with much higher regularization strengths leading to ultimately better generalization.
+
+![](images/07e3b7e17a816e37d2df76a96c5407939ab30ec41541df7c3a85b0528cbc091a.jpg)
+
+Fourier features (e.g., when using Fourier features only for the input layer, Mavor-Parker et al., 2024), and this can be attributed to their spectral bias of learning high-frequency features (Tancik et al., 2020). However, deep Fourier features are able to use their high trainability to learn effectively even when highly regularized. Thus, while high trainability does not always lead to high generalization, the trainability provided by deep Fourier features can be used in combination with regularization to improve continual learning performance. Hyperparameter sensitivity results are presented on other datasets in Appendix D.6. We also provide an in-depth sensitivity study on smaller-scale MLPs in Appendix D.8.
+
+# 6 CONCLUSION
+
+In this paper, we proved that linear function approximation and a special case of deep linearity are effective inductive biases for learning continually without loss of trainability. This surprising finding for deep linear networks suggests that nonlinearity of representations, rather than nonlinearity of gradient dynamics, contributes to loss of plasticity. We then investigated the issues that arise from using nonlinear activation functions, namely the problem of low unit sign entropy, which indicates a lack of diversity in the activations as measured by the entropy of the sign of the hidden units. Motivated by the effectiveness of linearity in sustaining trainability, we proposed deep Fourier features to approximately embed a deep linear network inside a deep nonlinear network. We found that deep Fourier features dynamically balance the trainability afforded by linearity and the effectiveness of nonlinearity, thus providing an effective inductive bias for learning continually. Experimentally, we demonstrated that networks with deep Fourier features provided benefits for continual learning across every dataset we considered. We found that networks with deep Fourier features were effective plastic learners because their trainability enabled training with a much larger regularization strength, leading to superior generalization performance.
+
+# ACKNOWLEDGMENTS
+
+The research is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada CIFAR AI Chair Program, Alberta Innovates, and the Digital Research Alliance of Canada.
+
+# REFERENCES
+
+Abbas, Z., Zhao, R., Modayil, J., White, A., and Machado, M. C. (2023). Loss of plasticity in continual deep reinforcement learning. In Conference on Lifelong Learning Agents.  
+Abel, D., Barreto, A., Van Roy, B., Precup, D., van Hasselt, H. P., and Singh, S. (2024). A definition of continual reinforcement learning. Advances in Neural Information Processing Systems.  
+Agrawal, A., Barratt, S., and Boyd, S. (2021). Learning convex optimization models. Journal of Automatica Sinica, 8(8):1355-1364.  
+Arora, S., Cohen, N., Golowich, N., and Hu, W. (2019). A convergence analysis of gradient descent for deep linear neural networks. In International Conference on Learning Representations.  
+Arora, S., Cohen, N., and Hazan, E. (2018). On the optimization of deep networks: Implicit acceleration by overparameterization. In International Conference on Machine Learning.  
+Ash, J. T. and Adams, R. P. (2020). On Warm-Starting Neural Network Training. In Advances in Neural Information Processing Systems.  
+Ba, J. L., Kiros, J. R., and Hinton, G. E. (2016). Layer normalization. CoRR, abs/1607.06450v1.  
+Bah, B., Rauhut, H., Terstiege, U., and Westdickenberg, M. (2022). Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizers. Information and Inference: A Journal of the IMA.  
+Bernacchia, A., Lengyel, M., and Hennequin, G. (2018). Exact natural gradient in deep linear networks and its application to the nonlinear case. Advances in Neural Information Processing Systems.  
+Boyd, S. P. and Vandenberghe, L. (2004). Convex optimization. Cambridge university press.  
+Braun, L., Domíne, C., Fitzgerald, J., and Saxe, A. (2022). Exact learning dynamics of deep linear networks with prior knowledge. Advances in Neural Information Processing Systems.  
+Chou, H.-H., Gieshoff, C., Maly, J., and Rauhut, H. (2024). Gradient descent for deep matrix factorization: Dynamics and implicit bias towards low rank. Applied and Computational Harmonic Analysis.  
+Cohen, G., Afshar, S., Tapson, J., and Van Schaik, A. (2017). Emmist: Extending mnist to handwritten letters. In International Joint Conference on Neural Networks (IJCNN).  
+Dohare, S., Hernandez-Garcia, J. F., Lan, Q., Rahman, P., Mahmood, A. R., and Sutton, R. S. (2024). Loss of plasticity in deep continual learning. Nature, 632(8026):768-774.  
+Dohare, S., Sutton, R. S., and Mahmood, A. R. (2021). Continual backprop: Stochastic gradient descent with persistent randomness. CoRR, abs/2108.06325v3.  
+Elsayed, M. and Mahmood, A. R. (2024). Addressing loss of plasticity and catastrophic forgetting in continual learning. In International Conference on Learning Representations.  
+Even, M., Pesme, S., Gunasekar, S., and Flammarion, N. (2023). (s)GD over diagonal linear networks: Implicit bias, large step sizes and edge of stability. In Advances in Neural Information Processing Systems.  
+Garrigos, G. and Gower, R. M. (2023). Handbook of Convergence Theorems for (Stochastic) Gradient Methods. CoRR, abs/2301.11235v3.
+
+Glorot, X. and Bengio, Y. (2010). Understanding the difficulty of training deep feedforward neural networks. In International Conference on Artificial Intelligence and Statistics.  
+He, K., Zhang, X., Ren, S., and Sun, J. (2015). Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In International Conference on Computer Vision.  
+He, K., Zhang, X., Ren, S., and Sun, J. (2016). Deep residual learning for image recognition. In Conference on Computer Vision and Pattern Recognition.  
+Huh, D. (2020). Curvature-corrected learning dynamics in deep neural networks. In International Conference on Machine Learning.  
+Ioffe, S. and Szegedy, C. (2015). Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International Conference on Machine Learning.  
+Jacot, A., Gabriel, F., and Hongler, C. (2018). Neural tangent kernel: Convergence and generalization in neural networks. Advances in Neural Information Processing Systems.  
+Kingma, D. P. and Ba, J. (2015). Adam: A Method for Stochastic Optimization. In International Conference on Learning Representations.  
+Kirkpatrick, J., Pascanu, R., Rabinowitz, N., Veness, J., Desjardins, G., Rusu, A. A., Milan, K., Quan, J., Ramalho, T., Grabska-Barwinska, A., et al. (2017). Overcoming catastrophic forgetting in neural networks. Proceedings of the National Academy of Sciences, 114(13):3521-3526.  
+Kleinman, M., Achille, A., and Soatto, S. (2024). Critical learning periods emerge even in deep linear networks. In International Conference on Learning Representations.  
+Konidaris, G., Osentoski, S., and Thomas, P. (2011). Value function approximation in reinforcement learning using the fourier basis. In AAAI Conference on Artificial Intelligence.  
+Krizhevsky, A. (2009). Learning multiple layers of features from tiny images. Technical report, University of Toronto.  
+Kumar, S., Marklund, H., Rao, A., Zhu, Y., Jeon, H. J., Liu, Y., and Van Roy, B. (2023a). Continual Learning as Computationally Constrained Reinforcement Learning. CoRR, abs/2307.04345.  
+Kumar, S., Marklund, H., and Roy, B. V. (2023b). Maintaining plasticity via regenerative regularization. CoRR, abs/2308.11958v1.  
+Kunin, D., Raventós, A., Domíne, C., Chen, F., Klindt, D., Saxe, A., and Ganguli, S. (2024). Get rich quick: exact solutions reveal how unbalanced initializations promote rapid feature learning. CoRR, abs/2406.06158v1.  
+Le, Y. and Yang, X. (2015). Tiny imagenet visual recognition challenge.  
+LeCun, Y., Cortes, C., and Burges, C. (1998). MNIST handwritten digit database. ATT Labs [Online]. Available: http://yann.lecun.com/exdb/mnist.  
+Lee, H., Cho, H., Kim, H., Kim, D., Min, D., Choo, J., and Lyle, C. (2024). Slow and steady wins the race: Maintaining plasticity with hare and tortoise networks. In International Conference on Machine Learning.  
+Lee, J., Xiao, L., Schoenholz, S., Bahri, Y., Novak, R., Sohl-Dickstein, J., and Pennington, J. (2019). Wide neural networks of any depth evolve as linear models under gradient descent. Advances in Neural Information Processing Systems.  
+Lewandowski, A., Kumar, S., Schuurmans, D., György, A., and Machado, M. C. (2024). Learning Continually by Spectral Regularization. CoRR, abs/2406.06811v1.  
+Li, A. C. and Pathak, D. (2021). Functional regularization for reinforcement learning via learned fourier features. In Advances in Neural Information Processing Systems.  
+Liu, Y., Kuang, X., and Roy, B. V. (2023). A Definition of Non-Stationary Bandits. CoRR, abs/2302.12202v2.
+
+Lyle, C., Rowland, M., and Dabney, W. (2022). Understanding and preventing capacity loss in reinforcement learning. In International Conference on Learning Representations.  
+Lyle, C., Zheng, Z., Khetarpal, K., van Hasselt, H., Pascanu, R., Martens, J., and Dabney, W. (2024). Disentangling the Causes of Plasticity Loss in Neural Networks. CoRR, abs/2402.18762v1.  
+Lyle, C., Zheng, Z., Nikishin, E., Avila Pires, B., Pascanu, R., and Dabney, W. (2023). Understanding plasticity in neural networks. In International Conference on Machine Learning.  
+Mavor-Parker, A. N., Sargent, M. J., Barry, C., Griffin, L., and Lyle, C. (2024). Frequency and Generalisation of Periodic Activation Functions in Reinforcement Learning. CoRR, abs/2407.06756v1.  
+Nacson, M. S., Ravichandran, K., Srebro, N., and Soudry, D. (2022). Implicit bias of the step size in linear diagonal neural networks. In International Conference on Machine Learning.  
+Parascandolo, G., Huttunen, H., and Virtanen, T. (2017). Taming the waves: sine as activation function in deep neural networks.  
+Parisi, G. I., Kemker, R., Part, J. L., Kanan, C., and Wermter, S. (2019). Continual lifelong learning with neural networks: A review. Neural networks, 113:54-71.  
+Rahimi, A. and Recht, B. (2007). Random features for large-scale kernel machines. Advances in Neural Information Processing Systems.  
+Ring, M. B. (1994). Continual learning in reinforcement environments. The University of Texas at Austin.  
+Saxe, A., McClelland, J., and Ganguli, S. (2014). Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. In International Conference on Learning Representations.  
+Shang, W., Sohn, K., Almeida, D., and Lee, H. (2016). Understanding and improving convolutional neural networks via concatenated rectified linear units. In International Conference on Machine Learning.  
+Sokar, G., Agarwal, R., Castro, P. S., and Evci, U. (2023). The dormant neuron phenomenon in deep reinforcement learning. In International Conference on Machine Learning.  
+Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J., and Ng, R. (2020). Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Processing Systems.  
+Thrun, S. (1998). Lifelong learning algorithms. In Learning to Learn, pages 181-209. Springer.  
+Van de Ven, G. M., Tuytelaars, T., and Tolias, A. S. (2022). Three types of incremental learning. Nature Machine Intelligence, 4(12):1185-1197.  
+Xiao, H., Rasul, K., and Vollgraf, R. (2017). Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. CoRR, abs/1708.07747.  
+Xu, B., Wang, N., Chen, T., and Li, M. (2015). Empirical Evaluation of Rectified Activations in Convolutional Network. CoRR, abs/1505.00853v2.  
+Yang, G., Ajay, A., and Agrawal, P. (2022). Overcoming the spectral bias of neural value approximation. In International Conference on Learning Representations.  
+Yang, G., Simon, J. B., and Bernstein, J. (2023). A Spectral Condition for Feature Learning. CoRR, abs/2310.17813v2.  
+Zagoruyko, S. and Komodakis, N. (2016). Wide Residual Networks. CoRR, abs/1605.07146v4.  
+Ziyin, L., Li, B., and Meng, X. (2022). Exact solutions of a deep linear network. Advances in Neural Information Processing Systems.
+
+# A ADDITIONAL DETAILS
+
+# A.1 ASSUMPTIONS FOR TRAINABILITY OF LINEAR FUNCTION APPROXIMATION
+
+We assume that the parameters at the global optimum for every task are bounded:  $\| \theta_{\tau} \|_2 < D$ . This is true for regression problems if the observations and targets are bounded. In classification tasks, the global optimum can be at infinity because activation functions such as the sigmoid and the softmax are maximized at infinity. In this case, we constrain the parameter set,  $\{\theta : \| \theta \|_2 < D\}$ , and project the optimum onto this set.
+
+In addition to convexity, we assume that the objective function is  $\mu$ -strongly convex,  $\nabla_{\theta}^{2}J_{\tau}(\theta) \succ \mu\mathbf{I}$ , where  $\nabla_{\theta}^{2}J_{\tau}(\theta)$  denotes the Hessian. Note that neither squared nor cross-entropy loss are  $\mu$ -strongly in general. However, this assumption is satisfied in continual learning problems with a finite number of iterations. For regression, denote the observations for task  $\tau$  as  $X_{\tau} \in \text{mathbb{R}}^{d \times N}$  where  $N$  is the sample size. Then the Hessian is the outer products of the data matrix,  $\nabla_{\theta}^{2}J_{\tau}^{\text{reg}}(\theta) = X_{\tau}X_{\tau}^{\top} \in \mathbb{R}^{d \times d}$ . Thus, the squared loss is strongly-convex if the data is full rank. This is satisfied in high dimensional image classification problems, which is what we consider.
+
+For classification, the Hessian involves an additional diagonal matrix of the predictions for each datapoint,
+
+$$
+\nabla_ {\theta} ^ {2} J _ {\tau} ^ {c l a s s} (\theta) = X _ {\tau} X _ {\tau} ^ {\top} \in \mathbb {R} ^ {d \times d},
+$$
+
+where  $D = \mathrm{Diag}(p_1, \ldots, p_N)$  and  $p_i = 2\sigma(f_\theta(x_i))(1 - \sigma(f_\theta(x_i)))$ . If the prediction becomes sufficiently confident,  $\sigma(f_\theta(x_i)) = 1$ , then there can be rank deficiency in the Hessian. However, because each task is only budgeted a finite number of iterations this bounds the predictions away from 1.
+
+# A.2 RELATED WORK REGARDING TRAINABILITY OF DEEP LINEAR NETWORKS
+
+Some authors have suggested deep linear networks suffer from a related issue, namely that critical learning periods also occur for deep linear networks (Kleinman et al., 2024). Unlike the focus on loss of trainability in this work where the entire network is trained, these critical learning periods are due to winner-take-all dynamics due to manufactured defects in one half of the linear network, for which the other half compensates.
+
+Finally, we note that some previous work have found that gradient dynamics have a low rank bias for deep linear networks (Chou et al., 2024). One important assumption that these works make is that the neural network weights are initialized identically across layers,  $\theta_{j} = \alpha \theta_{1}$ . Our analysis assumes that the initialization uses small random values, such as those used in practice with common neural network initialization schemes (Glorot and Bengio, 2010; He et al., 2015).
+
+# A.3 DETAILS FOR DEEP LINEAR SETUP
+
+The gradient of the loss function with respect to the parameters of a deep linear network can be written in terms of the gradient with respect to the product matrix  $\bar{\theta}$  (Bah et al., 2022):
+
+$$
+\nabla_ {\theta_ {j}} J (\theta) = \theta_ {j + 1} ^ {\top} \theta_ {j + 2} ^ {\top} \dots \theta_ {L} ^ {\top} \nabla_ {\bar {\theta}} J (\bar {\theta}) \theta_ {1} ^ {\top} \theta_ {2} ^ {\top} \dots \theta_ {j - 1} ^ {\top}, \tag {1}
+$$
+
+where the term  $\nabla_{\bar{\theta}}J(\bar{\theta})$  is the gradient of the loss with respect to the product matrix, treating it as if it was linear function approximation. The gradient is nonlinear because of the coupling between the gradient of the parameter at one layer and the value of the parameters of the other layers. Nevertheless, the gradient dynamics of the individual parameters can be combined to yield the dynamics of the product matrix (Arora et al., 2018),
+
+$$
+\bar {\nabla} _ {\theta} J (\theta) = P _ {\bar {\theta}} \nabla_ {\bar {\theta}} J (\bar {\theta}). \tag {2}
+$$
+
+The dynamics involve a preconditioner,  $P_{\bar{\theta}}$ , that accelerates optimisation (Arora et al., 2018), which we empirically demonstrate in Section 3.3. On the left-hand side of the equation, we use  $\bar{\nabla}_{\theta}J(\theta)$  to denote the combined dynamics of the gradients for each layer on the dynamics of the product matrix. This means that the effective gradient dynamics of the deep network is related to the dynamics of
+
+linear function approximation with a precondition. While the dynamics are nonlinear and non-convex, the overall dynamics are remarkably similar to that of linear function approximation, which is convex.
+
+To simplify notation, without loss of generality, we consider a deep linear network without the bias terms,  $\theta = \{\theta_{1},\dots ,\theta_{L}\}$ , and  $f_{\theta}(x) = \theta_L\theta_{L - 1}\dots \theta_1x$ . We denote the product of weight matrices, or simply product matrix, as  $\bar{\theta} = \theta_L\theta_{L - 1}\dots \theta_1$ , which allows us to write the deep linear network in terms of the product matrix:  $f_{\theta}(x) = \bar{\theta} x$ . The problem setup we use for the deep linear analysis follows previous work (Huh, 2020), and we provide additional details in Appendix A.3. We consider the squared error,  $J_{\tau}(\theta) = \mathbb{E}_{(x,y)\sim p_{\tau}}\left[\| y - \bar{\theta} x\|\right]_2^2$ . and we assume that the observations are whitened to simplify the analysis,
+
+$\Sigma_{x} = \mathbb{E}\left[x x^{\top}\right] = \mathbf{I},$  focusing on the case where the targets  $y$  are changing during continual learning. Then we can write the squared error as
+
+$$
+J (\theta) = \mathrm {T r} \left[ \Delta_ {\tau} \Delta_ {\tau} ^ {\top} \right],
+$$
+
+where  $\Delta_{\tau} = \theta_{\tau}^{\star} - \bar{\theta}$  is the distance to the optimal linear predictor,  $\theta_{\tau}^{\star} = \Sigma_{yx,\tau} = \mathbb{E}_{x,y\sim p_{\tau}}[yx^{\top}]\Sigma_{x}$ .
+
+The convergence of gradient descent for general deep linear networks requires an assumption on the deficiency margin, which is used to ensure that the solution found by a deep linear network, in terms of the product matrix, is full rank (Arora et al., 2019). That is, the deep linear network converges if the minimum singular value of the product matrix stays positive,  $\sigma_{min}(\bar{\theta}) > 0$ .
+
+We now show that a diagonal linear network maintains a positive minimum singular value under continual learning. This is a simplified setting for analysis, where we assume that the weight matrices are diagonal and thus the input, hidden, and output dimension are all equal. Let  $f_{\theta}(x)$  be a diagonal linear network, defined by a set of diagonal weight matrices,  $\theta_l = \mathrm{Diag}(\theta_{l,1},\dots,\theta_{l,d})$ . The output of the diagonal linear network is the product of the diagonal matrices,  $f_{\theta}(x) = \theta_L\theta_{L - 1}\dots \theta_1x$ . Then the product matrix is also a diagonal matrix, whose diagonals are the products of the parameters of each layer,  $\bar{\theta} = \operatorname{Diag}\left(\prod_{l = 1}^{L}\theta_{l,1},\ldots ,\prod_{l = 1}^{L}\theta_{l,d}\right)\coloneqq \operatorname{Diag}\left(\bar{\theta}_1,\dots,\bar{\theta}_d\right)$ . The minimum singular value of a diagonal matrix is the minimum of its absolute values,  $\sigma_{min}(\bar{\theta}) = \min_i|\bar{\theta}_i|$ . Thus, we must show that the minimum absolute value of the product matrix is never zero.
+
+Lemma 1. Consider a deep diagonal linear network,  $f_{\theta}(x) = \theta_{L}\theta_{L - 1}\dots \theta_{1}x$  and  $\theta_l = \text{Diag}(\theta_{l,1},\ldots ,\theta_{l,d})$ . Then, under gradient descent dynamics,  $\theta_{l,i}^{(t)} = \theta_{l',i}^{(t)}$  iff  $\theta_{l,i}^{(0)} = \theta_{l',i}^{(0)}$  for  $l^{\prime}\neq l$ .
+
+The proof of this proposition, and the next, can be found in Appendix B. This first proposition states that two parameters that are initialized to different values, such as by a random initialization, will never have the same value under gradient descent. Conversely, if the parameters are initialized identically, then they will stay the same value under gradient descent. This means that, in particular, two parameters will never be simultaneously zero.
+
+Lemma 2. Denote a deep diagonal linear network as  $f_{\theta}(x) = \text{Diag}(\bar{\theta}_1, \ldots, \bar{\theta}_d)x$  where  $\bar{\theta}_i = \prod_{l=1}^{L} \theta_{l,i}$ . Then, under gradient descent dynamics,  $\bar{\theta}_i^{(t)} = \bar{\theta}_i^{(t+1)} = 0$  iff two (or more) components are zero,  $\theta_{l,i}^{(t)} = \theta_{l',i}^{(t)} = 0$ , for  $l' \neq l$ .
+
+While the analysis considers a special case of deep linear networks, namely deep diagonal networks, we note that this is a common setting for the analysis of deep linear networks more generally (Nacson et al., 2022; Even et al., 2023). In particular, the analysis is motivated by the fact that, under certain conditions, the evolution of the deep linear network parameters can be analyzed through the independent singular mode dynamics (Saxe et al., 2014), which simplify the analysis of deep linear networks to deep diagonal linear networks. The target function being learned,  $y^{\star}(x) = \theta^{\star}x$ , is represented in terms of the singular-value decomposition,  $\theta^{\star} = U^{\star}S^{\star}V^{\star} = \sum_{j=1}^{r} s_i u_i v_i^\top$ . We also assume that the neural network has a fixed hidden dimension, so that  $\theta_1 \in \mathbb{R}^{d \times d_{in}}$ ,  $\theta_L \in \mathbb{R}^{d_{out} \times d}$ ,  $\theta_{<l<L} \in \mathbb{R}^{d \times d}$ ; and we apply the singular value decomposition to the function approximator's parameters,  $\theta_l = U_l S_l V_l \in \mathbb{R}^{d_{out} \times d_h}$ . To simplify the product of weight matrices, we assume  $V_{i+1} = U_i$ ,  $V_1 = V^{\star}$ , and  $U_L = U^{\star}$ . The simplifying result is that the squared error loss can be expressed entirely in terms of the singular values,  $\|y^{\star}x - \prod_{i=L}^{1} \theta_i x\|^2 \propto \|S^{\star} - \prod_{i=L}^{1} S_l\|^2$ , which is equivalent to our analysis of the deep diagonal network, as the matrix of singular values is a diagonal matrix. These decoupled learning dynamics are closely approximated by networks with small random weights and they persist under gradient flows (Huh, 2020).
+
+# A.4 PSEUDOCODE FOR DEEP FOURIER FEATURE LAYER
+
+<table><tr><td colspan="2">Algorithm 1 Deep Fourier Feature Layer</td></tr><tr><td colspan="2">1: function DEEPFOURIERFEATURES(x, W, b)</td></tr><tr><td>2: z← Wx + b</td><td>▷ Calculate pre-activation</td></tr><tr><td>3: a1← sin(z)</td><td>▷ Apply sine activation</td></tr><tr><td>4: a2← cos(z)</td><td>▷ Apply cosine activation</td></tr><tr><td>5: output← [a1; a2]</td><td>▷ Concatenate activations</td></tr><tr><td>6: return output</td><td></td></tr><tr><td>7: end function</td><td></td></tr></table>
+
+# B PROOFS
+
+Proof of Theorem 1. We first present the result for two tasks and we then generalize it to an arbitrary number of tasks. Let the linear weights learned on the first task be  $\theta^{(T)}$ , with the corresponding unique global minimum denoted by  $\theta_1^{\star}$ . The solution found on the first task is used as an initialization on the second task, which will end at  $\theta^{(2T)}$ , with the corresponding unique global minimum denoted by  $\theta_2^{\star}$ . We start from the known suboptimality gap for gradient descent on the second task (Garrigos and Gower, 2023):
+
+$$
+J _ {2} \left(\theta^ {(2 T)}\right) - J _ {2} \left(\theta_ {2} ^ {\star}\right) <   \frac {\left\| \theta_ {2} ^ {\star} - \theta^ {(T)} \right\| ^ {2}}{\alpha T}. \tag {3}
+$$
+
+We upper bound the distance from the initialization on the second task,  $\theta^{(T)}$ , to the optimum,  $\theta_{2}^{\star}$ , by
+
+$$
+\left\| \theta_ {2} ^ {\star} - \theta^ {(T)} \right\| ^ {2} <   \left\| \theta_ {2} ^ {\star} - \theta_ {1} ^ {\star} \right\| ^ {2} + \left\| \theta_ {1} ^ {\star} - \theta^ {(T)} \right\| ^ {2} <   \left\| \theta_ {2} ^ {\star} - \theta_ {1} ^ {\star} \right\| ^ {2} + (1 - \alpha \mu) ^ {T} \left\| \theta_ {1} ^ {\star} - \theta_ {0} \right\| ^ {2}. \qquad (4)
+$$
+
+Where the last inequality uses the assumption that the objective function is  $\mu$ -strongly convex. We upper bound the suboptimality gap on the second task by a quantity independent of  $\theta^{(T)}$ :
+
+$$
+J _ {2} \left(\theta^ {(2 T)}\right) - J _ {2} \left(\theta_ {2} ^ {\star}\right) <   \frac {\left\| \theta_ {2} ^ {\star} - \theta^ {(T)} \right\| ^ {2}}{\alpha T} <   \frac {\left\| \theta_ {2} ^ {\star} - \theta_ {1} ^ {\star} \right\| ^ {2} + (1 - \alpha \mu) ^ {T} \left\| \theta_ {1} ^ {\star} - \theta_ {0} \right\| ^ {2}}{\alpha T}, \tag {5}
+$$
+
+which implies that the parameter value learned on the previous task does not influence training on the new task beyond a dependence on the initial distance. This is true for an arbitrary number of tasks:
+
+$$
+J _ {\tau} \left(\theta^ {(\tau T)}\right) - J _ {\tau} \left(\theta_ {\tau} ^ {\star}\right) <   \frac {\sum_ {k = 1} ^ {\tau} (1 - \alpha \mu) ^ {T (k - \tau)} \| \theta_ {k} ^ {\star} - \theta_ {k - 1} ^ {\star} \| ^ {2}}{\alpha T} <   \frac {2 D (1 - \alpha \mu) ^ {T}}{\alpha T (1 - (1 - \alpha \mu) ^ {T})}, \tag {6}
+$$
+
+where we denote  $\theta_0^\star = \theta_0$ . The last inequality follows from our assumption that the distance between the task solutions,  $\| \theta_k^\star - \theta_{k-1}^\star \|^2 < 2D$ , is bounded and using a geometric sum in  $(1 - \alpha \mu)^T$ .
+
+Proof of Lemma 1.  $\Rightarrow$  We first prove the lemma in the forward direction:
+
+Assuming that  $\theta_{l,i}^{(t)} = \theta_{l',i}^{(t)}$  for  $l' \neq l$ , we will show that  $\theta_{l,i}^{(t-1)} = \theta_{l',i}^{(t-1)}$ .
+
+Writing the gradient update for  $\theta_{l,i}^{(t)}$  with a fixed step-size  $\alpha$ , we have that
+
+$$
+\begin{array}{l} \theta_ {l, i} ^ {(t)} = \theta_ {l, i} ^ {(t - 1)} - \alpha \nabla_ {\theta_ {l, i}} J (\theta) (7) \\ = \theta_ {l, i} ^ {(t - 1)} - \alpha \nabla_ {f _ {\theta}} \ell \left(f _ {\theta} (x), y\right) \nabla_ {\theta_ {l, i}} f _ {\theta} (x) (8) \\ = \theta^ {(t - 1)} - \alpha \nabla_ {f _ {\theta}} \ell \left(f _ {\theta} (x), y\right) \nabla_ {\theta_ {l, i}} \prod_ {j = 1} ^ {L} \theta_ {j, i} ^ {(t - 1)} x (9) \\ = \theta_ {l, i} ^ {(t - 1)} - \alpha \nabla_ {f _ {\theta}} \ell \left(f _ {\theta} (x), y\right) \prod_ {j \neq l} \theta_ {j, i} ^ {(t - 1)} x. (10) \\ \end{array}
+$$
+
+Similarly, the gradient update for  $\theta_{l',i}$  is
+
+$$
+\theta_ {l ^ {\prime}, i} ^ {(t)} = \theta_ {l ^ {\prime}, i} ^ {(t - 1)} - \alpha \nabla_ {f _ {\theta}} \ell \left(f _ {\theta} (x), y\right) \prod_ {j \neq l ^ {\prime}} \theta_ {j, i} ^ {(t - 1)} x \tag {12}
+$$
+
+(13)
+
+Using our assumption that  $\theta_{l,i}^{(t)} = \theta_{l',i}^{(t)}$ , we set the two updates equal to each other:
+
+$$
+\theta_ {l, i} ^ {(t - 1)} - \alpha \nabla_ {f _ {\theta}} \ell (f _ {\theta} (x), y) \prod_ {j \neq l} \theta_ {j, i} ^ {(t - 1)} x = \theta_ {l ^ {\prime}, i} ^ {(t - 1)} - \alpha \nabla_ {f _ {\theta}} \ell (f _ {\theta} (x), y) \prod_ {j \neq l ^ {\prime}} \theta_ {j, i} ^ {(t - 1)} x. \tag {14}
+$$
+
+We can simplify both sides of the equations, where the LHS is
+
+$$
+\begin{array}{l} \theta_ {l, i} ^ {(t - 1)} - \alpha \nabla_ {f _ {\theta}} \ell \left(f _ {\theta} (x), y\right) \prod_ {j} \theta_ {j, i} ^ {(t - 1)} \frac {x}{\theta_ {l ^ {\prime} , i} ^ {(t - 1)}} (15) \\ = \theta_ {l, i} ^ {(t - 1)} \left(1 - \alpha \nabla_ {f _ {\theta}} \ell \left(f _ {\theta} (x), y\right) \prod_ {j} \theta_ {j, i} ^ {(t - 1)} \frac {x}{\theta_ {l ^ {\prime} , i} ^ {(t - 1)} \theta_ {l , i} ^ {(t - 1)}}\right). (16) \\ \end{array}
+$$
+
+Similarly, the RHS of the equation is
+
+$$
+\theta_ {l ^ {\prime}, i} ^ {(t - 1)} \left(1 - \alpha \nabla_ {f _ {\theta}} \ell \left(f _ {\theta} (x), y\right) \prod_ {j} \theta_ {j, i} ^ {(t - 1)} \frac {x}{\theta_ {l ^ {\prime} , i} ^ {(t - 1)} \theta_ {l , i} ^ {(t - 1)}}\right). \tag {17}
+$$
+
+Notice that both expressions in the parenthesis on the LHS and RHS are equal. Thus,  $\theta_{l',i}^{(t-1)} = \theta_{l,i}^{(t-1)} \Leftarrow$  The reverse direction follows directly by following the above argument in reverse.
+
+□
+
+Proof of Lemma 2.  $\Rightarrow$  We first prove the lemma in the forward direction:
+
+Assuming that  $\bar{\theta}_i^{(t + 1)} = \bar{\theta}_i^{(t)} = 0$ , we will show that  $\theta_{l,i}^{(t)} = \theta_{l',i}^{(t)} = 0$ .
+
+We proceed by contradiction, and assume that only a single component is zero, that is  $\theta_{l',i}^{(t)} = 0$  and  $\theta_{l,i}^{(t)} \neq 0$  for  $l \neq l'$ . We will show that the gradient update will ensure that  $\theta_i^{(t + 1)} \neq 0$
+
+First, consider the update to  $\theta_{l',i}^{(t)}$ ,
+
+$$
+\begin{array}{l} \theta_ {l ^ {\prime}, i} ^ {(t + 1)} = \theta_ {l ^ {\prime}, i} ^ {(t)} - \alpha \nabla_ {f _ {\theta}} \ell \left(f _ {\theta} (x), y\right) \prod_ {j \neq l ^ {\prime}} \theta_ {j, i} ^ {(t - 1)} x (18) \\ = - \alpha \nabla_ {f _ {\theta}} \ell \left(f _ {\theta} (x), y\right) \prod_ {j \neq l ^ {\prime}} \theta_ {j, i} ^ {(t - 1)} x (19) \\ \end{array}
+$$
+
+Because we assumed that  $\theta_{l,i}^{(t)}\neq 0$  for  $l\neq l^{\prime}$ , we have that  $\prod_{j\neq l}\theta_{j,i}^{(t - 1)}\neq 0$ . Thus  $\theta_{l',i}^{(t + 1)}\neq 0$  Next consider the update to  $\theta_{l,i}^{(t)}$
+
+$$
+\begin{array}{l} \theta_ {l, i} ^ {(t + 1)} = \theta_ {l ^ {\prime}, i} ^ {(t)} - \alpha \nabla_ {f _ {\theta}} \ell \left(f _ {\theta} (x), y\right) \prod_ {j \neq l} \theta_ {j, i} ^ {(t - 1)} x (20) \\ = \theta_ {l ^ {\prime}, i} ^ {(t)} (21) \\ \end{array}
+$$
+
+Where the last line follows from the fact that  $\prod_{j\neq l}\theta_{j,i}^{(t - 1)} = 0$  because  $\theta_{l^{\prime},i}^{(t)} = 0$
+
+Thus, we have shown that  $\theta_{l,i}^{(t + 1)}\neq 0$  for all  $l$ , and hence,  $\bar{\theta}^{(t + 1)}\neq 0$  which is a contradiction.
+
+$\Leftarrow$  The reverse direction follows from the assumption directly. If two components are both equal to zero,  $\theta_{l,i}^{(t)} = \theta_{l',i}^{(t)} = 0$ , then every sub-product is zero,  $\prod_{j \neq l} \theta_{j,i}^{(t-1)}$  and so is the entire product,  $\prod_{j=1}^{L} \theta_{j,i}^{(t-1)}$ .
+
+Proof of Theorem 2. We now show that a diagonal linear network maintains a positive minimum singular value under continual learning. This is a simplified setting for analysis, where we assume that the weight matrices are diagonal and thus the input, hidden, and output dimension are all equal. Let  $f_{\theta}(x)$  be a diagonal linear network, defined by a set of diagonal weight matrices,  $\theta_l = \mathrm{Diag}(\theta_{l,1},\dots ,\theta_{l,d})$ . The output of the diagonal linear network is the product of the diagonal matrices,  $f_{\theta}(x) = \theta_L\theta_{L - 1}\dots \theta_1x$ . Then the product matrix is also a diagonal matrix, whose diagonals are the products of the parameters of each layer,  $\bar{\theta} = \mathrm{Diag}(\prod_{l = 1}^{L}\theta_{l,1},\ldots ,\prod_{l = 1}^{L}\theta_{l,d}):= \mathrm{Diag}(\bar{\theta}_1,\ldots ,\bar{\theta}_d)$ . The minimum singular value of a diagonal matrix is the minimum of its absolute values,  $\sigma_{min}(\bar{\theta}) = \min_i|\bar{\theta}_i|$ . Thus, we must show that the minimum absolute value of the product matrix is never zero.
+
+This follows immediately from Lemma 1 and Lemma 2. Taken together, these two lemmas state that with a random initialization and under gradient dynamics, a diagonal linear network will not have more than one parameter equal to zero. This means that the minimum singular value of the product matrix will never be zero. Thus, we have shown that a diagonal linear network trained with gradient descent, if initialized appropriately, will be able to converge on any given task in a sequence.
+
+Proof of Proposition 1. We prove this by considering the remainder of a Taylor series on the given interval. Due to periodicity of  $\sin (z)$  and  $\cos (z)$ , we can consider  $z\in [-\pi ,\pi ]$  without loss of generality. We can further consider two cases, either  $z\in [-\pi , - 3\pi /4]\cup [-\pi /4,\pi /4]\cup [3\pi /4,\pi ]$  or  $h\in [-3\pi /4, - \pi /4]\cup [\pi /4,3\pi /4]$ . In the first case,  $z$  is near a critical point of  $\cos (z)$  and in the second case  $z$  is near a critical point of  $\sin (z)$ .
+
+We focus on a particular subcase, where  $z \in [-\pi/4, \pi/4]$ , which is close to a critical point of  $\cos(z)$  but far from a critical point of  $\sin(h)$  (the other cases follow a similar argument).
+
+Because we know that  $z \in [-\pi/4, \pi/4]$ , by Taylor's theorem it follows that  $\sin(z) = z + R_{1,0}(z)$ , where  $R_{1,0}(z) = \frac{\sin^{(2)}(c)}{2} z^2$  is the 1st degree Taylor remainder centered at  $a = 0$  for some  $c \in [-\pi/4, \pi/4]$ . In the case of a sinusoid, this can be upperbounded,  $|R_{1,0}(z)| = |\frac{-\sin(c)}{2} z^2| < \frac{1}{8\sqrt{2}} (\pi/4)^2$ , using the fact that  $|z| < \pi/4$  and  $\sin(c) < 1/\sqrt{2}$ .
+
+Thus, when  $\cos(z)$  is close to a critical point,  $\sin(z)$  is approximately linear. A similar argument holds for the other case, when  $\sin(z)$  is close to a critical point,  $\cos(z)$  is approximately linear. In this other case, the error incurred is the same.
+
+![](images/feb5310dc87d9d55c2887919937fca306be7266f7927f10cbe39d5320208697d.jpg)
+
+Proof of Corollary 1. We prove this claim using induction.
+
+Base case: We want to show that a single layer that outputs Fourier features embeds a deep linear network. Using Proposition 1, there exists one unit for each pre-activation that is approximately linear. Because each pre-activation is used in an approximately-linear unit, the single layer approximately embeds a deep linear network using all of its parameters.
+
+Induction step: Assume a deep Fourier network with depth  $L - 1$  embeds a deep linear network, we prove that adding an additional deep Fourier layer retains the embedded deep linear network. There are two cases to consider, corresponding to the units of the additional deep Fourier layer which are approximately-linear and the other units that are not approximately-linear
+
+Case 1 (approximately-linear units): For the additional deep Fourier layer, the set of approximately-linear units already embeds a deep linear network. Because linearity is closed under composition, the composition of the additional deep Fourier layer and the deep Fourier network with depth  $L - 1$  simply adds an additional linear layer to the embedded deep linear network, increasing its depth to  $L$ .
+
+Case 2 (other units): For the units that are not well-approximated by a linear function, we can treat them as if they were separate inputs to the deep Fourier network with depth  $L - 1$ . The network's parameters associated with those inputs are, by the inductive hypothesis, already embedded in the deep linear network.
+
+Note that case 1 embeds the parameters of the additional deep Fourier layer into the deep Fourier network. Case 2 states that the parameters of the network associated with the nonlinear units of the additional deep Fourier layer are already embedded in the deep Fourier network by construction.
+
+Thus, a neural network composed of deep Fourier layers embeds a deep linear network.
+
+# C EMPIRICAL DETAILS
+
+All of our experiments use 10 seeds and we report the standard error of the mean in the figures. The optimiser used for all experiments was Adam, and after a sweep on each of the datasets over  $[0.005, 0.001, 0.0005]$ , we found that  $\alpha = 0.0005$  was most performant.
+
+We used the Adam optimizer (Kingma and Ba, 2015) for all experiments, settling on the default learning rate of 0.001 after evaluating [0.005, 0.001, 0.0005]. Results are presented with standard error of the mean, indicated by shaded regions, based on 10 random seeds.
+
+Dataset specifications and non-stationarity conditions:
+
+- For MNIST, Fashion MNIST and EMNIST: we use a random sample of 25600 of the observations and a batch size of 256 (unless otherwise indicated, such as the linearly separable experiment).  
+- For CIFAR10 and CIFAR1100: Full 50000 images for training, 1000 test images for validation, rest for testing. The batch size used was 250. Labelnoise non-stationarity: 60 epochs, 10 tasks. Class incremental learning: 6000 iterations per task, 80 tasks. Note that the datasets on different tasks in the class incremental setting can have different sizes, and so epochs are not comparable.  
+- tiny-[ImageNet]: All 100000 images for training, 10000 for validation, 10000 for testing as per predetermined split. The batch size used was 250. Labelnoise experiment non-stationarity: 80 epochs per task, 10 tasks total. Class incremental learning: 10000 iterations per task, 80 tasks. Note that the datasets on different tasks in the class incremental setting can have different sizes, and so epochs are not comparable.
+
+Neural Network Architectures For tiny-ImageNet, CIFAR10, CIFAR100, and SVHN2: We utilized standard ResNet-18 with batch normalization and a standard tiny Vision Transformer. The smaller datasets use an MLP with different widths and depths, as specified in the scaling section.
+
+Experiment Metrics All figures reporting accuracy evaluate the accuracy on the distribution given by the current task. Figures 2, 3, 4 and 7 (top) report the training accuracy on the current task. Figures 5 and 6 report the test accuracy on the current task. Figure 5 shows the accuracy at the end of each epoch, whereas Figure 6 shows the accuracy at the end of each task (due to too many tasks). The accuracy reported in Figure 7 is the final accuracy at the end of the last task.
+
+# D ADDITIONAL EXPERIMENTS
+
+# D.1 ADDITIONAL DEEP LINEAR NETWORK RESULTS
+
+![](images/c92138c0ef875de07fc26b0f33fee47398adc42406bd89fa919ba129b7a9bd70.jpg)  
+Figure 8: Minimum singular value of the product matrix for a deep general linear network on a linearly-separable task.
+
+![](images/d8e449beba36d4238a3cfaf776eb3cd3f26b0203e29f199cb6cfce7e79cfc5e4.jpg)  
+Figure 9: Average unit sign entropy on a non linearly-separable task. Deep Fourier features and other periodic activation functions have high average unit sign entropy compared to piecewise linear activations like ReLU and leaky-ReLU.
+
+# D.2 ABLATING ARCHITECTURE WITH WIDE RESIDUAL NETWORKS
+
+![](images/52ac35a0d0aa37fec73c0ad4533a9bcf310fd571aa4d2e83e45c2b216d6e1a38.jpg)  
+Figure 10: Investigating Wide Residual Networks with different width scales on tiny-ImageNet with label noise.
+
+![](images/00192b37639130b9f04245629a3fd40c3267cfc2f0a647243fcfd6f18b3d4192.jpg)
+
+![](images/9f57ee5982ead6a2a2330a65fb4b7c675c94b1f23cf2e479810a1c8fcdf754f7.jpg)
+
+# D.3 INVESTIGATING SINGLE TASK PERFORMANCE
+
+![](images/09f0d26fb6b42fd044da71975eeeb41ec7096850a2b33f62adad5f2006829790.jpg)
+
+![](images/fc6fba912e58e582e2cbc45411451ee0f0be7013da4dae2a227588227fe6159c.jpg)
+
+![](images/1cece4ba7218dd42fe1f3ae697599127bdd4428034cf9dc01fd797a6db9901e3.jpg)  
+Figure 11: Investigating single task performance of ResNet-18 using different activation functions on tiny/ImageNet.
+
+![](images/282be5c25171d0e63fc26a91e31f3b67c1108c508625c4d053231fbae737d129.jpg)
+
+# D.4 CONTINUAL IMAGENET RESULTS
+
+![](images/e63e652820ebfb191c14c4a81437667896d822b46a07d0841ca7add53ef66990.jpg)  
+Figure 12: Investigating sustained performance on many tasks using ResNet-18 on Continual ImageNet. Note that loss of trainability does not occur on this problem, whereas loss of generalization does occur.
+
+![](images/750feb58dc05d2cc3d09c3f224bd6a2ee6bb70792a04afb91a4fb65617c3da5d.jpg)
+
+![](images/9ead330d5d8b1bf07e87c557ac5ae4bb6483e8bd4c7a6a5b22fdfbf3132e93b5.jpg)  
+D.5 COMPARING CONTINUAL BACKPROP TO RECYCLING DORMANT NEURONS
+
+![](images/ee5108acf3208b1843119c3244ff9a863db27ae3e825fc5a1cf6da707a11b1d7.jpg)  
+Figure 13: Recycling dormant neurons and continual backprop are both weight reinitialization methods that perform similarly.
+
+![](images/aa51fa739629ae490f562bb7c1e4cb70df386df2554f1f6b50a4f91186a8f86d.jpg)  
+D.6 ADDITIONAL SENSITIVITY RESULTS
+
+![](images/92876e91e2254a4e54910b5424ed02e79a210d56dd918bf2b393e033764d177f.jpg)
+
+![](images/ca839127360c4465ab1b41e57559fb6a27bc211e00c912df317098d8c468b6b8.jpg)
+
+![](images/a3406602ced8b48799433b9c67c2401872707adc26d290c2325205fa38c73081.jpg)  
+Figure 14: Sensitivity analysis on tiny-ImageNet, CIFAR10, and CIFAR100. Networks with deep Fourier features are highly trainable, but have a tendency to overfit without regularization, leading to high training accuracy but low test accuracy. Due to deep Fourier features being highly trainable, they are able to train with much higher regularization strengths leading to ultimately better generalization.
+
+![](images/d73f608341f96c4aa9764dc8c5661f3a1f6ba06a4c33a6ba5785acc8d69dfb85.jpg)
+
+![](images/807d7cd4d14dab5d76c40374acd7cf03da6305569c8e07349858ec5a62aa7c1a.jpg)
+
+# D.7 FORGETTING RESULTS
+
+![](images/f62f93972477a5a18be3197e8504de9275761252848c25714448b609a5760235.jpg)
+
+![](images/44fc2eecb45fd28bb9a75252541c3c029d1870b4fb377da7f4f292a4375d8fb6.jpg)
+
+![](images/51f9534a3b4c158e0e1a3d8405da3245c53645ca90d75c6b3d92961e5ac4663d.jpg)
+
+![](images/49827b94e279a8e4b74462de850e18ce06cded1b2cd74342154d4f81ab9ec8f0.jpg)
+
+![](images/77e1faeee2b2958df11fd85f37071bb3a1c47f33a4a234d10d86d397f7c1895f.jpg)
+
+![](images/f2b233134b88df7ee206a85841cd8c48d890c12e59d6a5968648808d98fa7b7c.jpg)
+
+![](images/b3b28a47223c450db9a4a70f5441bc7e30c9f83786ca0d4e4dd933241872ca0f.jpg)
+
+![](images/629caf474bc5587ecbf9d03d13f6a2cf8ee0f75b793026c7ebeec452d6dde72d.jpg)
+
+![](images/267f73c96a1af93d2f4b15615a0d2a496f92735d9896de99ce81c162b7d6325b.jpg)
+
+![](images/bb791d280d851d644e7b2319815bb05b6315366ab1d35d559ead6424ce8f61ba.jpg)  
+Figure 15: Forgetting on online label-permuted tiny-[ImageNet, CIFAR10, and CIFAR100]. All networks are capable of continual online learning within a task, indicating that they maintain plasticity and succeed in avoiding catastrophic forgetting data early within a single task. However, deep Fourier features are particularly capable of maintaining performance on previous tasks.
+
+![](images/69f6df8bc89ebac2fd57a3e34f2cf00bda290c643f639991cd71fb4369c69ba1.jpg)
+
+![](images/c556740e4a573a8c1a55947f4604c8b1ad98202ca6411744de53fd418903c798.jpg)
+
+# D.8 ADDITIONAL TRAINABILITY RESULTS USING DEEP FOURIER FEATURES
+
+These additional experiments validate the benefits of deep Fourier features as a means of improving trainability. The experiments use the following datasets for continual supervised learning: MNIST (LeCun et al., 1998), Fashion MNIST (Xiao et al., 2017), and EMNIST (Cohen et al., 2017). We focus primarily on the problem of trainability, and thus consider random label non-stationarity, in which the labels are randomly assigned to each observation and must be memorized on each task. This type of non-stationarity is particular difficulty in sustaining trainability in continual learning (Lyle et al., 2023; Kumar et al., 2023b). We compare our network with deep Fourier feature against a corresponding feed-forward neural network with ReLU activations with the same depth. Because deep Fourier features use a concatenation of two different activation functions, it has half the width of the ReLU network and less parameters, which provides an advantage to the ReLU baseline.
+
+# D.8.1 DEEP FOURIER FEATURES ARE HIGHLY TRAINABLE
+
+The main result of this appendix is presented in Figure 16. Across different datasets, deep Fourier feature networks are highly trainable, either achieving high accuracy and maintaining it on easier tasks, such as MNIST, or improving their trainability on new tasks, such as on Fashion MNIST. In contrast, the ReLU network suffered from loss of trainability in each of the problems that we studied. This is not surprising, as loss of trainability is a well-documented issue for ReLU networks without some additional method designed to mitigate it (Dohare et al., 2021; Lyle et al., 2022; Kumar et al., 2023b; Elsayed and Mahmood, 2024).
+
+![](images/ea6d2a67da716fd26bc31036a9c9f4a80698e344203c9e212aac460f0bdf7fdc.jpg)
+
+![](images/19fbf16cefac8063c04c523e6c8464e06a838e90bf209dc55d9ee1ce8b6fd25b.jpg)
+
+![](images/85d69aeccd4119c82564725999413e190cd44bdb16c45a8cf9d50d97d35de5bb.jpg)
+
+![](images/93e8143f8baae7c10b465f5caf26f6b32c894c3a0995e78ecafb1e91b1070d28.jpg)  
+Figure 16: Trainability across different datasets and epochs per tasks. ReLU networks lose their trainability, whereas networks with deep Fourier features improve and sustain their trainability
+
+![](images/c57941c43a791203a760452de1fc0cdfc8828143c8d1e11be19d9ab4eb59d1d0.jpg)
+
+![](images/dc1cd93ec7f1d3bb7a685abe0dd18da68b56817eeda90c8f36253d7b16066534.jpg)
+
+# D.8.2 METHODS FOR IMPROVING TRAINABILITY
+
+Given that a ReLU network is unable to maintain its trainability in isolation, we investigate whether recently proposed methods for mitigating loss of trainability are able to make up for the difference in performance between a network with deep Fourier features and a network with ReLU activations. We investigate two categories of mitigators for loss of plasticity: (i) regularization and (ii) normalization layers.
+
+![](images/3fb27e4cc0382626f9ada65872f9105cb24b3a7108cb8397c98e9aa2b42a4f3c.jpg)  
+Figure 17: Hyperparameter Sensitivity Analysis. Deep Fourier features seem to not benefit from regularization for trainability. While ReLU networks are more trainable with regularization, their performance is still worse than the deep Fourier feature network. Note that the experiments in Section 5.2 indicate that deep Fourier features do benefit from regularization for generalization.
+
+![](images/c986d084c81a7d8eaa6a9d49844962e8981aca44694f1e397ede835493a555be.jpg)
+
+Regularization Loss of plasticity occurs in ReLU networks when they are not regularized. Thus, we compare the performance of the ReLU network and the deep Fourier feature network with varying regularization strengths. In particular, we use the recently proposed L2 regularization towards the initialization (Kumar et al., 2023b), because it addresses the issue of sensitivity towards zero common to L2 regularization towards zero. In Figure 17, we find that regularization does improve the trainability of ReLU networks, validating previous empirical findings. However, we found that deep Fourier feature networks do not benefit substantially from regularization. That is, deep Fourier feature network with a smaller regularization strength always outperformed the ReLU network.
+
+Layer Normalization Training deep neural networks typically involve normalization layers, either Batch Normalization (Ioffe and Szegedy, 2015) or Layer Normalization (Ba et al., 2016). Recently, it was demonstrated that layer normalization is an effective mitigator for loss of trainability (Lyle et al., 2024). We investigate whether trainability can be improved with the addition of normalization layers, for both the ReLU and deep Fourier feature network. In Figure 18, we found that layer normalization increases performance but that loss of trainability can still occur with a ReLU network. In addition to Layer Normalization, we also tried a linear version of LayerNorm which uses a stop-gradient on the standard deviation to maintain linearity, which improved training speed in some instances.
+
+![](images/7441ccd8cd44963d833677b622962609c7986a859622bd5f234d8da1c7a3884d.jpg)  
+Figure 18: Comparison of trainability with Layer Normalization. ReLU networks are more trainable with Layer Normalization, but deep Fourier feature networks learn faster and achieve better accuracy, particularly with linearized Layer Norm.
+
+![](images/5059465e0e716fb968be313cf7696977b3e3b74ac103003c1974d8bc39711cd0.jpg)
+
+![](images/53a1172287c5faa15828e9d1a47d2a11b5637f1114da876b56629dcee10365d9.jpg)
+
+# D.8.3 SCALING PROPERTIES OF DEEP FOURIER FEATURE NETWORKS
+
+![](images/e3ce8f916b6fb9d087f0958131ad37852ed0d1ff3cbe7fd9d16e0eb05ff98e40.jpg)
+
+![](images/88b51a64e3d3a6bbe5947ca85a119fe1fd6af60e7a7c3f1941563af1e5468a28.jpg)
+
+![](images/17f7e7c9b621a081998f4826f88d776f4c1d71f5e02dfc93b767266bfd560374.jpg)
+
+![](images/0c0fdbc382eb5b792b5712df655f014d1bce62517a8184187d2223ac7123ff55.jpg)  
+Figure 19: Scaling Neural Network Width and Depth. (Top) Due to the concatenation used by the activation function in deep Fourier feature networks, they scale particularly well with width. (Bottom) Deeper Fourier features also lead to improved average end of task performance.
+
+![](images/cd58618d36a0341f503afcf176b9702b350c2dfeac20ba7e9d02385c9a5626a1.jpg)
+
+![](images/0bc97be378eb110c5444f67b8aab625ba47e70c7028000a8ad2b1896bc672312.jpg)
+
+Width Scaling Another source of linearity recently proposed is an increasing width of the neural network, causing their parameter dynamics evolves as linear models in the limit (Lee et al., 2019). We investigate whether an increase in width can close the gap between the trainability of the ReLU network and the deep Fourier feature network. In Figure 19 (Top), we found that deep Fourier feature networks scale particularly well with width, whereas width seems to have little effect on the trainability of ReLU networks. Thus, our results suggest that increasing the width of a neural network does not necessarily impact its trainability, at least not to the width values we considered.
+
+Depth Scaling Neural networks in supervised learning tend to scale with depth, allowing them to learn more complex predictions. We investigate whether the depth scaling of deep Fourier feature networks also leads to similar improvements in continual learning. In Figure 19 (Bottom), we found that deep Fourier feature networks do improve with additional depth, but the degree of improvement was not as pronounced as scaling the width.

2025/Plastic Learning with Deep Fourier Features/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2b93347b5622cb0b86369fc82adbcfa2cc0a43cbde453b23a315b0267592f93c
+size 1008039

2025/PnP-Flow_ Plug-and-Play Image Restoration with Flow Matching/4058e62a-583f-44ae-9829-581a0bcb8635_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1d51856c91e056033ba3f440fb385e501a4d93687ff189922f717e59fe5cac9e
+size 19403157

2025/PnP-Flow_ Plug-and-Play Image Restoration with Flow Matching/full.md

@@ -0,0 +1,614 @@
+# PNP-FLOW: PLUG-AND-PLAY IMAGE RESTORATION WITH FLOW MATCHING
+
+Ségolène Martin $^{1,*}$  Anne Gagneux $^{2,*}$  Paul Hagemann $^{1,*}$  Gabriele Steidl $^{1}$
+
+1 Technische Universität Berlin
+
+2 ENS de Lyon, CNRS, Université Claude Bernard Lyon 1, Inria, LIP, UMR 5668
+
+* Equal contributions
+
+# ABSTRACT
+
+In this paper, we introduce Plug-and-Play (PnP) Flow Matching, an algorithm for solving imaging inverse problems. PnP methods leverage the strength of pretrained denoisers, often deep neural networks, by integrating them in optimization schemes. While they achieve state-of-the-art performance on various inverse problems in imaging, PnP approaches face inherent limitations on more generative tasks like inpainting. On the other hand, generative models such as Flow Matching pushed the boundary in image sampling yet lack a clear method for efficient use in image restoration. We propose to combine the PnP framework with Flow Matching (FM) by defining a time-dependent denoiser using a pre-trained FM model. Our algorithm alternates between gradient descent steps on the data-fidelity term, reprojections onto the learned FM path, and denoising. Notably, our method is computationally efficient and memory-friendly, as it avoids backpropagation through ODEs and trace computations. We evaluate its performance on denoising, super-resolution, deblurring, and inpainting tasks, demonstrating superior results compared to existing PnP algorithms and Flow Matching based state-of-the-art methods. Code available at https://github.com/annegnx/PnP-Flow.
+
+# 1 INTRODUCTION
+
+Image restoration aims to recover an unknown image  $x \in \mathbb{R}^d$  from a degraded observation  $y \in \mathbb{R}^m$
+
+$$
+y = H x + \xi ,
+$$
+
+where  $H: \mathbb{R}^d \to \mathbb{R}^m$  is a (linear) degradation operator and  $\xi$  describes the underlying additive noise model. Since the problem is usually ill-posed and high dimensional, its treatment is challenging. We assume that the image  $x$  is sampled from a random variable  $X \in \mathbb{R}^d$  with a density  $p_X$ , and the observation  $y$  from a random variable  $Y \in \mathbb{R}^m$  with a density  $p_Y$ . Then the maximum a posteriori estimator searches for the value with the highest probability of the posterior
+
+$$
+\underset {x \in \mathbb {R} ^ {d}} {\arg \max } \left\{\log p _ {X | Y = y} (x) \right\} = \underset {x \in \mathbb {R} ^ {d}} {\arg \max } \left\{\log p _ {Y | X = x} (y) + \log p _ {X} (x) \right\}, \tag {1}
+$$
+
+where the first term of the right hand side is the fidelity to the data and the second term represents the prior distribution of the image. Since  $p_{X}$  is generally unknown, and in the absence of training data, it is standard to instead consider a regularized optimization problem of the form:
+
+$$
+\underset {x \in \mathbb {R} ^ {d}} {\arg \min } \left\{F (x) + R (x) \right\}, \tag {2}
+$$
+
+where  $F(x) \coloneqq -\log p_{Y|X = x}(y)$  and  $R: \mathbb{R}^d \to \mathbb{R}$  usually enforces some assumptions on the solution and ensures the existence of a (unique) minimizer. For instance, for Gaussian noise  $\mathcal{N}(0, \sigma^2 I_d)$ , the data-fidelity corresponds to  $F(x) = \frac{1}{2\sigma^2} \| Hx - y \|^2$ . The regularized minimization problem in (2) can be efficiently handled using proximal splitting methods (Boyd et al., 2011; Combettes & Pesquet, 2011).
+
+Plug-and-Play (PnP) (Venkatakrishnan et al., 2013) methods build upon the insight that the proximal step on the regularization term is effectively a denoising operation. They can be performed with nonlearned denoisers such as BM3D (Chan et al., 2016), however recently these were outperformed by
+
+neural network-based approaches (Meinhardt et al., 2017; Zhang et al., 2017; 2021). While PnP methods have demonstrated considerable effectiveness, recent advances in generative models offer a more sophisticated framework for learning priors directly from data, surpassing the limitations of hand-crafted or neural denoisers.
+
+In particular, generative models are used as regularizers in Bora et al. (2017); Asim et al. (2020); Altekruger et al. (2023); Wei et al. (2022). Most of these references learn an invertible transport map  $T$  between a Gaussian latent distribution and the data distribution, using the change of variables formula to obtain the estimated prior  $p_{X}$  as an explicit function of the known latent density and the map  $T$ . In particular the normalizing flow network with the Real-NVP (Dinh et al., 2017) architecture is by default invertible and has an easy-to-evaluate log determinant of the Jacobian of  $T$ . However, this architecture yields rather poor results on large images. Therefore, recent work (Ben-Hamu et al., 2024; Zhang et al., 2024b; Pokle et al., 2024; Chung et al., 2023; Song et al., 2023) focuses more on learning diffusion models (Song et al., 2021; Ho et al., 2020; Sohl-Dickstein et al., 2015) or Flow Matching models (Lipman et al., 2023; Liu et al., 2023), which do not have these architectural constraints and scale better to large images. The main idea is to establish a path between the latent and the target distribution, which can be optimized in a simulation-free way.
+
+In this paper, we focus on Flow Matching models, which learn a velocity field  $v:[0,1]\times \mathbb{R}^d\longrightarrow \mathbb{R}^d$  going from the latent to the data distribution. Once the velocity field  $v$  is learned, sampling from the target distribution can be done by solving an ODE (Chen et al., 2018)
+
+$$
+\partial_ {t} f (t, x) = v _ {t} (f (t, x)), \quad f (0, x) = x,
+$$
+
+where  $x$  is sampled from the latent distribution.
+
+However, using Flow Matching to regularize inverse problems such as image restoration tasks is nontrivial. Indeed, when directly solving the MAP problem (1) using the change of variables formula, one faces numerical challenges due to the backpropagation through the ODE. We circumvent this problem by integrating the implicit Flow Matching prior into a custom denoiser that we plug into a Forward-Backward algorithm.
+
+Contributions. In this paper, we combine PnP methods with Flow Matching and propose a PnP Flow Matching algorithm. Our contributions are as follows:
+
+- Inspired by optimal transport Flow Matching, we design a time-dependent denoiser based on a pre-trained velocity field  $v$  learned through Flow Matching.  
+- This denoiser is integrated into an adapted Forward-Backward Splitting PnP framework, which cycles through a gradient step on the data-fidelity term, an interpolation step to reproject iterates onto flow trajectories, and a denoising step.  
+- Our algorithm is simple to implement, requires no backpropagation through the network, and is highly efficient in both memory usage and execution time compared to existing Flow Matching-based methods for solving inverse problems.  
+- We show that, on two datasets and across denoising, deblurring, inpainting, and superresolution tasks, our method consistently outperforms state-of-the-art Flow Matching-based and PnP methods in terms of both PSNR and SSIM metrics. The code with all benchmark methods is available at: https://github.com/annegnx/PnP-Flow and is now included the DeepInv library<sup>1</sup>.
+
+# 2 BACKGROUND
+
+We next provide background on both Plug-and-Play algorithms and Flow Matching models.
+
+# 2.1 PLUG AND PLAY
+
+PnP algorithms were introduced as extensions of proximal splitting methods like Forward-Backward Splitting (FBS) or the alternating direction method of multipliers (ADMM). These are first-order
+
+![](images/f9c6ccdc6d16d8831f78092098778436b6bb2b15cb3a710be9ca82981d5dfb09.jpg)  
+Figure 1: Our method on a 2D denoising task  $(\sigma = 1.5)$  with Gaussian distributions. An OT Flow Matching model is trained to sample from  $P_{1} = \mathcal{N}(m, s^{2}\mathrm{Id})$ , with  $m = 7$  and  $s = 0.5$ . At each time step, it performs a standard gradient step on the datafit, followed by a projection onto flow trajectories at time  $t$ , and finally applies the time-dependent denoiser  $D_{t}$ .
+
+optimization algorithms for solving problem (2), alternating between a proximal step on the regularization and a gradient and/or proximal step on the data-fidelity function. The proximal operator  $\mathrm{prox}_{\gamma R}:\mathbb{R}^d\to \mathbb{R}^d$ $\gamma >0$  is defined by  $\mathrm{prox}_{\gamma R}(y)\coloneqq \arg \min_{x\in \mathbb{R}^d}\left\{\frac{1}{2}\| y - x\| ^2 +\gamma R(x)\right\}$ . In this paper, we assume that the data-fidelity  $F$  is differentiable with Lipschitz continuous gradient and we focus on the FBS algorithm, corresponding to Algorithm 1. The convergence of FBS to a minimizer of  $F + R$  is guaranteed as long as the stepsize is chosen as  $\gamma \in (0,1 / \mathrm{Lip}(\nabla F))$ , where  $\mathrm{Lip}(\nabla F)$  is the Lipschitz constant of  $\nabla F$  (Combettes & Wajs, 2005).
+
+# Algorithm 1: FBS
+
+Initialization:  $x^{(0)}\in \mathbb{R}^d$ $\gamma >0$  
+for  $r = 0,1,\ldots$  do
+
+$$
+\left\lfloor \begin{array}{l} z ^ {(r + 1)} = x ^ {(r)} - \gamma \nabla F (x ^ {(r)}) \\ x ^ {(r + 1)} = \operatorname {p r o x} _ {\gamma R} (z ^ {(r + 1)}) \end{array} \right.
+$$
+
+# Algorithm 2: PnP-FBS
+
+Initialization:  $x^{(0)}\in \mathbb{R}^d$ $\gamma >0$  
+for  $r = 0,1,\dots$  do
+
+$$
+\begin{array}{l} z ^ {(r + 1)} = x ^ {(r)} - \gamma \nabla F (x ^ {(r)}) \\ x ^ {(r + 1)} = D \left(z ^ {(r + 1)}\right) \quad \triangleright \mathrm {P n P} \\ \end{array}
+$$
+
+Observing that computing  $\mathrm{prox}_{\gamma R}$  corresponds to solving a Gaussian denoising problem with regularization  $\gamma R$ , Venkatakrishnan et al. (2013) proposed to replace it with an off-the-shelf denoiser  $D: \mathbb{R}^d \longrightarrow \mathbb{R}^d$ , which can be independently designed or learned (see Algorithm 2).
+
+Many PnP-FBS methods, such as Meinhardt et al. (2017); Zhang et al. (2017; 2021); Sun et al. (2019); Terris et al. (2020); Hertrich et al. (2021); Hurault et al. (2022a); Tan et al. (2024) use neural network denoisers. Most of these PnP-FBS algorithms do not converge (Zhang et al., 2017; Sommerhoff et al., 2019) and convergence guarantees usually assume non-expansiveness of the denoiser (Pesquet et al., 2021; Ryu et al., 2019; Hurault et al., 2022b). Yet, constraining the Lipschitz constant of a neural network, e.g., through weight clipping, spectral normalization or averaged operator constructions harms in general its expressiveness (Ryu et al., 2019).
+
+# 2.2 FLOW MATCHING
+
+Let  $\mathcal{P}_2(\mathbb{R}^d)$  denote the subspace of probability measures on  $\mathbb{R}^d$  having a finite second moment. Let  $P_0$  denote the latent probability measure and  $P_{1}$  the target probability measure (i.e., the data distribution). We denote  $\ddagger$  the push-forward operation. We denote  $\Gamma (P_0,P_1)$  the set of couplings  $\pi \in \mathcal{P}(\mathbb{R}^d\times \mathbb{R}^d)$  having marginals  $P_{i}\in \mathcal{P}_{2}(\mathbb{R}^{d})$ ,  $i = 0,1$ , and  $\Gamma_0(P_0,P_1)$  the set of optimal couplings minimizing the Wasserstein-2 distance (Villani, 2008). We now fix a coupling  $\pi \in \Gamma (P_0,P_1)$ .
+
+The core idea behind Flow Matching is to define a specific target probability path  $t \mapsto P_t$ ,  $t \in [0,1]$ , between  $P_0$  and  $P_1$ . Let  $P_t \coloneqq (e_t)_{\sharp} \pi$ , where the map  $e_t(x_0,x_1) \coloneqq (1 - t)x_0 + tx_1$  interpolates between (a latent sample)  $x_0$  and (a data sample)  $x_1$ . The path  $P_t$  can be shown to be an absolutely continuous curve, so there exists a Borel vector field  $v:[0,1] \times \mathbb{R}^d \to \mathbb{R}^d$  such that the curve satisfies the continuity equation
+
+$$
+\partial_ {t} P _ {t} + \nabla \cdot \left(P _ {t} v _ {t}\right) = 0 \tag {CE}
+$$
+
+in the sense of distributions (Ambrosio et al., 2008). In addition, there exists a solution  $f:[0,1]\times \mathbb{R}^d\to \mathbb{R}^d$  of the ODE
+
+$$
+\partial_ {t} f (t, x) = v _ {t} (f (t, x)), \quad f (0, x) = x \tag {3}
+$$
+
+such that  $P_{t} = f(t,\cdot)_{\sharp}P_{0}$ . If  $f$  is known, then a sample  $x_{1}$  from the target distribution  $P_{1}$  can be drawn by first sampling  $x_{0}\sim P_{0}$  and then defining  $x_{1} = f(1,x_{0})$ .
+
+The goal of Flow Matching is to learn the velocity field of the flow ODE (3). This learning process consists in minimizing the loss function
+
+$$
+\mathcal {L} _ {\mathrm {F M}} (\theta) := \mathbb {E} _ {t \sim \mathcal {U} [ 0, 1 ], x \sim P _ {t}} \left[ \| v _ {t} ^ {\theta} (x) - v _ {t} (x) \| ^ {2} \right],
+$$
+
+where  $v^{\theta}$  is parametrized by a neural network with weights  $\theta$ . Unfortunately, in practice, the true velocity field  $v_{t}(x)$  is not available. However, Lipman et al. (2023) showed that minimizing  $\mathcal{L}_{\mathrm{FM}}(\theta)$  is equivalent to minimizing the Conditional Flow Matching (CFM) loss:  $\mathcal{L}_{\mathrm{CFM}}(\theta) = \mathcal{L}_{\mathrm{FM}}(\theta) + \mathrm{const}$ , where
+
+$$
+\mathcal {L} _ {\mathrm {C F M}} (\theta) := \mathbb {E} _ {t \sim \mathcal {U} [ 0, 1 ], (x _ {0}, x _ {1}) \sim \pi} \left[ \| v _ {t} ^ {\theta} \left(e _ {t} \left(x _ {0}, x _ {1}\right)\right) - \left(x _ {1} - x _ {0}\right) \| ^ {2} \right]. \tag {4}
+$$
+
+Minimizing this loss only requires sampling from the coupling  $\pi$ . For example, if  $\pi = P_0 \times P_1$  is the independent coupling, we only need samples from the latent and target distributions. If instead,  $\pi \in \Gamma_{\mathrm{o}}(P_0, P_1)$  is an optimal coupling, then we can use a standard optimal transport solver to (approximately) sample from  $\pi$ , as proposed in Pooladian et al. (2023); Tong et al. (2024).
+
+Straight-line flows Let  $v \colon [0,1] \times \mathbb{R}^d \longrightarrow \mathbb{R}$  and let  $f$  be a solution to the flow ODE associated with  $v$ . Given  $(X_0, X_1) \sim \pi$ , we call  $(f, v)$  a straight-line Flow Matching pair connecting  $X_0$  and  $X_1$  if  $X_t = f(t, X_0)$  almost surely, where  $X_t$  is defined as  $X_t \coloneqq e_t((X_0, X_1)) = tX_1 + (1 - t)X_0$  (Liu et al., 2023; Pooladian et al., 2023). Note that this directly implies  $v_t(f(t, X_0)) = \partial_t f(t, X_0) = \partial_t X_t = X_1 - X_0$  almost surely.
+
+We will focus on such straight-line flows later in the paper. Straight or nearly straight paths are preferred since they represent the shortest distance between two points and can be simulated with fewer steps of an ODE solver. This has been explored in many recent works that want to speed up sampling, such as Kornilov et al. (2024); Lee et al. (2023); Yang et al. (2024). A particular case of straight-line pair is given by OT couplings (Pooladian et al., 2023; Tong et al., 2024). Indeed, for  $\pi \in \Gamma_{\mathrm{o}}(P_0,P_1)$  with Monge map  $T:\mathbb{R}^d\to \mathbb{R}^d$  such that  $X_{1} = T(X_{0})$ , there exists a velocity field  $v$  for which the pair  $(P_t,v_t)$  is a solution of the continuity equation (CE), and such that for all  $x\in \mathbb{R}^d$
+
+$$
+v _ {t} (f (t, x)) = T (x) - x, \tag {5}
+$$
+
+where  $f$  is the solution of the ODE (3), see Ambrosio et al. (2008); Pooladian et al. (2023); Chemsed-dine et al. (2024) for the precise conditions. By integrating (5), it is clear that  $(f, v)$  is a straight-line Flow Matching for the coupling  $(X_0, T(X_0))$ , since  $X_t = (1 - t)X_0 + tX_1 = (1 - t)X_0 + tT(X_0) = X_0 + \int_0^t v_s(f(s, X_0)) ds = f(t, X_0)$ .
+
+# 3 PNP MEETS FLOW MATCHING
+
+# 3.1 DENOISING OPERATOR FROM FLOW MATCHING
+
+Let  $X_0 \sim P_0$  and  $X_1 \sim P_1$  with joint distribution  $(X_0, X_1) \sim \pi$ . Assume we have access to a pre-trained velocity field  $v^\theta$ . Then we define, for  $t \in [0, 1]$ , the following time-dependent denoiser
+
+$$
+D _ {t} := \operatorname {I d} + (1 - t) v _ {t} ^ {\theta}, \tag {6}
+$$
+
+and we propose to use it within a PnP framework.
+
+To motivate the choice of the denoiser in (6), recall that for a fixed time  $t \in [0,1]$ , the minimizer  $v_{t}^{*}$  of the CFM loss (4) over all possible vector fields reads
+
+$$
+v _ {t} ^ {*} (x) = \mathbb {E} [ X _ {1} - X _ {0} | X _ {t} = x ],
+$$
+
+where  $X_{t} \coloneqq e_{t}(X_{0}, X_{1}) = (1 - t)X_{0} + tX_{1}$  (Benton et al., 2024; Liu et al., 2023). Assume we are in the ideal case where  $v_{t}^{\theta} = v_{t}^{*}$ . Then it follows that, for any  $x \in \mathbb{R}^{d}$  and  $t \in [0,1]$ ,
+
+$$
+\begin{array}{l} D _ {t} (x) = x + (1 - t) v _ {t} ^ {*} (x) \\ = \mathbb {E} _ {\left(X _ {0}, X _ {1}\right) \sim \pi} \left[ X _ {t} + (1 - t) \left(X _ {1} - X _ {0}\right) \mid X _ {t} = x \right] \\ = \mathbb {E} _ {\left(X _ {0}, X _ {1}\right) \sim \pi} \left[ X _ {1} \mid X _ {t} = x \right]. \\ \end{array}
+$$
+
+Hence, the operator  $D_{t}$  can be understood at the best approximation of  $X_{1}$  given the knowledge of  $X_{t}$ , which is also used in Pokle et al. (2024) and Zhang et al. (2024b).
+
+Just as standard PnP denoisers minimize the MSE loss between noisy and clean samples, the operator  $D_{t}$  is the minimizer of  $L^2$ -problem  $\min_g\mathbb{E}_{(X_0,X_1)\sim \pi}[||X_1 - g(X_t)||^2]$ , projecting any noisy point taken along the path onto the target distribution. In particular, the following proposition holds, demonstrating that the best denoising performance is achieved with straight-line flows.
+
+Proposition 1. Assume  $v \coloneqq v^{\theta}$  is continuous and assume that, given  $v$ , the Flow ODE (3) has a unique solution  $f$ . Then the denoising loss  $\mathbb{E}_{(X_0,X_1)\sim \pi}[\| D_t(X_t) - X_1\|^2]$  is equal to 0 for all  $t \in [0,1]$ , if and only if the couple  $(f,v)$  is a straight-line Flow Matching pair between  $X_0$  and  $X_1$ .
+
+The proof can be found in Appendix A.1.
+
+As stated in Section 2.2, a special case of straight-line flows is given by OT Flow Matching for which we recover  $D_{t}(X_{t}) = X_{1}$ . Indeed, in this case,  $D_{t}$  reduces to  $T(X_0) = X_1$ , with  $T$  the Monge map between  $P_0$  and  $P_{1}$ . We next discuss whether diffusion models induce straight-line flows.
+
+Remark 2 (Flow Matching versus diffusion models). Contrary to OT Flow Matching, diffusion models, which are also flow ODE methods, do not generally induce straight paths. Indeed, during diffusion training, the target probability path takes the form  $\tilde{X}_t = \alpha_t X_0 + \beta_t X_1$  with  $\alpha_t \in (0,1)$  and  $\beta_t = \sqrt{1 - \alpha_t^2}$  (Song et al., 2021; Liu et al., 2023), which clearly does not match the desired straight path  $X_t = (1 - t) X_0 + t X_1$ . The non-straightness of the flow generated by diffusion models is illustrated in Liu et al. (2023); Liu (2022, Figure 5). On the other hand, a non-straight path obtained with Flow Matching (for instance, using an independent coupling  $\pi$ ) can be rectified to a straighter one following the procedure described in Liu et al. (2023).
+
+# 3.2 PNP FLOW MATCHING ALGORITHM
+
+In the previous section, we built a denoiser  $D_{t}$  (defined in (6)). We now want to plug it in a Forward-Backward Splitting algorithm in order to solve inverse problems. Yet, our algorithm differs from the classical PnP-FBS (Algorithm 2) in two key aspects. First, the iterations of the our algorithm depend on time because of the definition of  $D_{t}$ . Second, we introduce an intermediate reprojection step between the gradient step on the data-fidelity term and the denoising step. More precisely, at each time  $t \in [0,1]$ , given the current iterate  $x$ , our algorithm does the following updates:
+
+1. Gradient step: a gradient step on the data-fidelity term, mapping  $x$  to  $z = x - \gamma \nabla F(x)$  for a given learning rate  $\gamma > 0$ .  
+2. Interpolation step: In a standard Forward-Backward scheme, the denoiser operator is applied right after the gradient step. Yet, as discussed earlier, our operator  $D_{t}$  was specifically designed to correctly denoise inputs drawn from the straight path  $X_{t} = (1 - t)X_{0} + tX_{1}$ . If the output  $z$  from the gradient step at time  $t$  does not lie in the support of  $X_{t}$ , there is a high chance that the denoising will not be effective, hence the need to "reproject" it along the flow paths before applying  $D_{t}$ . To achieve this, we perform a linear interpolation on  $z$ , as illustrated in Figure 2: at time  $t$ , we define  $\tilde{z} = (1 - t)\varepsilon + tz$ , where  $\varepsilon$  is a noise sample drawn from  $P_{0}$ . A similar idea can be found in (Cheng et al., 2025). Note that while  $\varepsilon$  is sampled from  $P_{0}$ , it is not necessarily coupled to  $z \sim P_{1}$  via  $\pi$ . If it were,  $D_{t}$  would map  $\tilde{z}$  directly back to  $z$ , annihilating the denoising effect.  
+3. PnP Denoising step: the operator  $D_{t}$  is applied to the output  $\tilde{z}$  of the interpolation step, regularizing the current image by pushing it towards the distribution of  $X_{1}$ .
+
+![](images/530955d4501fcafdcb2bdc3ff598ebec1d3e01c1e8b1a847e83ecf40b2d4f013.jpg)
+
+Interpolation step
+
+$$
+\tilde {z} = (1 - t) \varepsilon + t z
+$$
+
+![](images/2b876fa45abbf28059d3468511a617cd9fc80481c0f3b29b016b9ea463f405cf.jpg)
+
+PnP denoising step
+
+![](images/03f488acfa316fad504bea5dbab64d9622aa66977139e3754890440baf8e8a0c.jpg)  
+Figure 2: Illustration of the interpolation step.
+
+The resulting discrete-time algorithm is given in Algorithm 3. Figure 1 illustrates the three steps of the algorithm on a denoising problem with a Gaussian prior.
+
+# Algorithm 3: PnP Flow Matching
+
+Input: Pre-trained network  $v^{\theta}$  by Flow Matching, time sequence  $(t_n)_n$  either finite with  $t_n = n / N$ ,  $N \in \mathbb{N}$  or infinite with  $\lim_{n \to +\infty} t_n = 1$ , adaptive step sizes  $(\gamma_n)_n$ .
+
+Initialize:  $x_0\in \mathbb{R}^d$
+
+for  $n = 0,1,\ldots ,$  do  $\begin{array}{l}\triangleright \mathrm{Gradient~step~on~the~data - fidelity~term}\\ \triangleright \mathrm{Linear~interpolation}\\ \triangleright \mathrm{PnP~step~with~denoiser~(6)} \end{array}$
+
+return  $x_{n + 1}$
+
+Remark 3 (Averaging in the denoising step). Instead of drawing one noise realization, we can also average over multiple samples  $\varepsilon \sim P_0$  in the last step of the algorithm:  $x_{n + 1}\coloneqq \mathbb{E}_{\varepsilon \sim P_0}[D_{t_n}(\tilde{z}_n(\varepsilon))]$  with  $\tilde{z}_n(\varepsilon)\coloneqq (1 - t_n)\varepsilon +t_nz_n$ . The algorithm's output is then deterministic. In practice, averaging over a few realizations slightly improves the numerical results.
+
+Time dependent learning rate Using a constant learning rate independent of time can give too much importance to the data fit. For example, if  $\gamma_{t} = 1$  for all  $t\in [0,1]$  on a simple denoising task, the algorithm will return the noisy sample  $y$  since  $D_{1} = \mathrm{Id}$ . To prevent this,  $\gamma_{t}$  should decrease with  $t$  to balance the contributions of the datafit and the denoiser. We set  $\gamma_{t} = (1 - t)^{\alpha}$  with  $\alpha \in (0,1]$  for the remainder of the paper. This choice yields good numerical results in our experiments, but alternative values for  $\gamma_{t}$  could also be explored.
+
+Convergence Assuming that the sequence produced by the algorithm is bounded in the infinite time regime, we have the following convergence result. The proof can be found in Appendix A.2.
+
+Proposition 4. Assume that  $F: \mathbb{R}^d \to \mathbb{R}$  is continuously differentiable and that the learned vector field  $v:[0,1] \times \mathbb{R}^d \to \mathbb{R}^d$  is continuous. Assume  $P_0$  has bounded support. Let the time sequence  $(t_n)_{n\in \mathbb{N}}$  satisfy  $\sum_{n = 0}^{\infty}(1 - t_n) < +\infty$  and let  $\gamma_{n} := 1 - t_{n}$ ,  $n \in \mathbb{N}$ . If the sequence  $(x_{n})_{n\in \mathbb{N}}$  obtained by Algorithm 3 is bounded, then it converges.
+
+# 4 RELATED WORK
+
+Our approach combining PnP with Flow Matching relates to several existing methods.
+
+Pre-trained Flow Matching methods Using pre-trained Flow Matching models for regularizing image inverse problems has been the focus of several recent works.
+
+OT-ODE (Pokle et al., 2024) assumes a Gaussian latent distribution and uses Tweedie's formula to derive a new velocity field  $\tilde{v}_t(x,y) = \mathbb{E}[X_1 - X_0\mid X_t = x,Y = y]$  from the original velocity field  $v_{t}(x) = \mathbb{E}[X_{1} - X_{0}\mid X_{t} = x]$ , all without requiring retraining. In practice, they sample from the posterior distribution  $X_{1}\mid Y$  by solving the ODE with the new velocity field using an Euler scheme.
+
+In Zhang et al. (2024b), the authors introduce Flow-Priors, a method to tackle the MAP problem by approximating it as a sequence of time-dependent MAP subproblems. Using Tweedie's formula, they show that for  $t < 1$ , the gradient of the distribution  $P_{t}$  of  $X_{t}$  can be computed in closed form, allowing for the use of gradient descent to optimize these subproblems. However, the closed-form expression for the gradient relies on the assumption of an independent coupling  $\pi$  and a Gaussian latent distribution. Besides, their method requires computing  $\mathrm{Tr}\nabla v^{\theta}$ , which is expensive.
+
+In Ben-Hamu et al. (2024), an implicit regularization approach called D-Flow is considered: instead of minimizing the Gaussian data-fidelity function  $x \mapsto \| Hx - y\| ^2$ , they minimize the latent loss  $z \mapsto \| H(f(1,z)) - y\| ^2$ , where  $f$  is a solution to the flow ODE given the pre-trained network. The two problems are theoretically equivalent since  $f(1,\cdot)$  is invertible. However, since the latent loss is not convex, first- or second-order optimization methods may not find the global minimizer. Interestingly, this is beneficial because the true minimizer of the original problem is simply the
+
+pseudo-inverse, which may not be desirable. The authors optimize the latent loss by backpropagating through the ODE solution with a few Euler steps, though this remains computationally expensive.
+
+(PnP) Diffusion methods While we present the first PnP method based on Flow Matching, related works combine diffusion models with the PnP framework. We refer to Appendix A.12 for further discussion.
+
+In Zhu et al. (2023), a Half Quadratic Splitting algorithm is used, alternating between a proximal step on the data-fidelity term and a proximal step on the regularization. Following the PnP strategy, the proximal step for the regularization term is replaced with a denoising step using a pre-trained diffusion model. The denoiser they use is reminiscent of the one we use, where the velocity is replaced by the gradient of the score function. Their method also includes an interpolation step with random noise, mapping the estimated image at each iteration back to the diffusion path. In Garber & Tirer (2024), the proximal step on the data-fidelity term in Zhu et al. (2023) is traded for a gradient step. Finally, conditional image restoration is explored in (Graikos et al., 2022), which uses a more relaxed definition of "plug and play". They integrate a pre-trained diffusion model under different conditions, leading to a variational objective similar to methods like Mardani et al. (2024).
+
+# 5 NUMERICAL EXPERIMENTS
+
+# 5.1 BASELINES
+
+We benchmark our method against three state-of-the-art Flow Matching-based restoration methods: OT-ODE (Pokle et al., 2024), D-Flow (Ben-Hamu et al., 2024) and Flow Priors (Zhang et al., 2024b). As no official implementations were publicly available for these methods, we developed our own based on the descriptions provided in their respective publications. We have made every effort to ensure faithful implementations, included in the code attached to this paper. We also benchmark our method against PnP-Diff (Zhu et al., 2023), a PnP algorithm based on diffusion models. Additionally, we compare our approach with the state-of-the-art PnP-FBS (Hurault et al., 2022a).
+
+# 5.2 EXPERIMENTAL SETUP
+
+Datasets We evaluate all methods on two datasets: CelebA (Yang et al., 2015), with images resized to  $128 \times 128$ , and AFHQ-Cat, a subset of the Animal FacesHQ dataset (Choi et al., 2020) focused on the cat class, with images resized to  $256 \times 256$ . All images are normalized to the range  $[-1, 1]$ . For CelebA, we use the standard training, validation, and test splits. For AFHQ-Cat, as no validation split is provided, we randomly select 32 images from the test set to create a validation set.
+
+Models For each dataset, we trained a Flow Matching model from scratch using the Mini Batch OT Flow Matching approach (Tong et al., 2024), as this choice of coupling usually leads to straight paths. We used a standard Gaussian as the latent distribution and a U-Net (Ronneberger et al., 2015) taken from Huang et al. (2021); Ho et al. (2020) as the model. The training parameters were a learning rate of  $10^{-4}$ , 200 epochs with a batch size of 128 for CelebA, and 400 epochs with a batch size of 64 for AFHQ-Cat. We train the denoising network for the PnP method PnP-GS (Hurault et al., 2022a) employing the same U-Net architecture. For PnP-Diff, because training a diffusion model with the same U-Net architecture as Flow-Matching yielded poor results due to insufficient number parameters, we used the pre-trained model from Choi et al. (2021), implemented in the DeepInv library (Tachella et al., 2023). Note that the pre-trained diffusion model was trained on the FFHQ dataset (Karras et al., 2019), making the comparison indirect, but we had no alternative.
+
+Settings for the experiments We evaluate the methods using 100 test images across five restoration problems: denoising with Gaussian noise  $(\sigma = 0.2)$ ; deblurring using a  $61 \times 61$  Gaussian kernel  $(\sigma_{b} = 1.0$  for CelebA,  $\sigma_{b} = 3.0$  for AFHQ-Cat); super-resolution  $(2 \times$  downsampling for CelebA,  $4 \times$  for AFHQ-Cat); box-inpainting with a centered  $s \times s$  mask  $(s = 40$  for CelebA,  $s = 80$  for AFHQ-Cat); and random pixel inpainting (70% masked pixels). For deblurring, super-resolution, and box-inpainting, we add Gaussian noise with  $\sigma = 0.05$ , and for random inpainting  $\sigma = 0.01$ .
+
+Hyper-parameters We optimize the hyper-parameters for each method using a grid search on the validation set, selecting the configuration that yields the highest Peak Signal-to-Noise Ratio (PSNR). The optimal values identified for each dataset and problem scenario are detailed in Appendix A.8. Our proposed method has two hyper-parameters: the exponent  $\alpha$  in the learning rate schedule  $\gamma_{n} = (1 - t_{n})^{\alpha}$ , and the number of uniformly spaced time steps was set to  $N = 100$  for most experiments. We averaged the results of the denoising step over 5 realizations of the interpolation step.
+
+# 5.3 MAIN RESULTS
+
+We report benchmark results for all methods across several restoration tasks, measuring average PSNR and Structural Similarity (SSIM) on 100 test images. To ensure reproducibility, all experiments were seeded. Results are presented in Table 1 for CelebA and Table 2 for AFHQ-Cat. "N/A" indicates cases where the method is inapplicable; for instance, PnP-GS (Hurault et al., 2022a) is not designed for generative tasks like box inpainting, and PnP-Diff relies on a diffusion model trained on another dataset, making its evaluation on box inpainting inappropriate.
+
+The tables show that our method consistently ranks first or second in both reconstruction metrics across all tasks and datasets. More importantly, it demonstrates stability across tasks, unlike other methods. For example, D-Flow and Flow-Priors perform well in box inpainting but struggle with denoising, while PnP-GS, PnP-Diff, and OT-ODE excel in denoising and deblurring but perform worse on pseudo-generative tasks.
+
+In terms of visual quality (Fig. 3, Fig. 4, and Appendix A.7), our method produces realistic, artifact-free images, though sometimes slightly over-smoothed. While D-Flow generates realistic images, it occasionally suffers from hallucinations (e.g., eye color shifts in CelebA denoising tasks). Flow-Priors introduces noise and artifacts, while OT-ODE captures textures well but struggles with image generation. Finally, Appendix A.6 shows the progression of the reconstruction with respect to time.
+
+Table 1: Comparisons of state-of-the-art methods on different inverse problems on the dataset CelebA. Results are averaged across 100 test images.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Denoising σ = 0.2</td><td colspan="2">Deblurring σ = 0.05, σb = 1.0</td><td colspan="2">Super-res. σ = 0.05, ×2</td><td colspan="2">Rand. inpaint. σ = 0.01, 70%</td><td colspan="2">Box inpaint. σ = 0.05, 40 × 40</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Degraded</td><td>20.00</td><td>0.348</td><td>27.67</td><td>0.740</td><td>10.17</td><td>0.182</td><td>11.82</td><td>0.197</td><td>22.12</td><td>0.742</td></tr><tr><td>PnP-Diff</td><td>31.00</td><td>0.883</td><td>32.49</td><td>0.911</td><td>31.20</td><td>0.893</td><td>31.43</td><td>0.917</td><td>N/A</td><td>N/A</td></tr><tr><td>PnP-GS</td><td>32.45</td><td>0.908</td><td>33.65</td><td>0.924</td><td>30.69</td><td>0.889</td><td>28.45</td><td>0.848</td><td>N/A</td><td>N/A</td></tr><tr><td>OT-ODE</td><td>30.50</td><td>0.867</td><td>32.63</td><td>0.915</td><td>31.05</td><td>0.902</td><td>28.36</td><td>0.865</td><td>28.84</td><td>0.914</td></tr><tr><td>D-Flow</td><td>26.42</td><td>0.651</td><td>31.07</td><td>0.877</td><td>30.75</td><td>0.866</td><td>33.07</td><td>0.938</td><td>29.70</td><td>0.893</td></tr><tr><td>Flow-Priors</td><td>29.26</td><td>0.766</td><td>31.40</td><td>0.856</td><td>28.35</td><td>0.717</td><td>32.33</td><td>0.945</td><td>29.40</td><td>0.858</td></tr><tr><td>PnP-Flow (ours)</td><td>32.45</td><td>0.911</td><td>34.51</td><td>0.940</td><td>31.49</td><td>0.907</td><td>33.54</td><td>0.953</td><td>30.59</td><td>0.943</td></tr></table>
+
+Table 2: Comparisons of state-of-the-art methods on different inverse problems on the dataset AFHQ-Cat. Results are averaged across 100 test images.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Denoising σ = 0.2</td><td colspan="2">Deblurring σ = 0.05, σb = 3.0</td><td colspan="2">Super-res. σ = 0.05, ×4</td><td colspan="2">Rand. inpaint. σ = 0.01, 70%</td><td colspan="2">Box inpaint. σ = 0.05, 80 × 80</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Degraded</td><td>20.00</td><td>0.319</td><td>23.77</td><td>0.514</td><td>11.59</td><td>0.216</td><td>13.35</td><td>0.234</td><td>21.50</td><td>0.744</td></tr><tr><td>PnP-Diff</td><td>30.27</td><td>0.835</td><td>27.97</td><td>0.764</td><td>23.22</td><td>0.601</td><td>31.08</td><td>0.882</td><td>N/A</td><td>N/A</td></tr><tr><td>PnP-GS</td><td>32.34</td><td>0.895</td><td>27.33</td><td>0.749</td><td>21.86</td><td>0.619</td><td>29.61</td><td>0.855</td><td>N/A</td><td>N/A</td></tr><tr><td>OT-ODE</td><td>29.90</td><td>0.831</td><td>26.43</td><td>0.709</td><td>25.17</td><td>0.711</td><td>28.84</td><td>0.838</td><td>23.88</td><td>0.874</td></tr><tr><td>D-Flow</td><td>26.22</td><td>0.620</td><td>27.49</td><td>0.740</td><td>24.10</td><td>0.595</td><td>31.37</td><td>0.888</td><td>26.69</td><td>0.833</td></tr><tr><td>Flow-Priors</td><td>29.32</td><td>0.768</td><td>25.78</td><td>0.692</td><td>23.34</td><td>0.573</td><td>31.76</td><td>0.909</td><td>25.85</td><td>0.822</td></tr><tr><td>PnP-Flow (ours)</td><td>31.65</td><td>0.876</td><td>27.62</td><td>0.763</td><td>26.75</td><td>0.774</td><td>32.98</td><td>0.930</td><td>26.87</td><td>0.904</td></tr></table>
+
+![](images/9f6ad5534239d7ba0524c1e5ade2c8f713d044ecc0d8bf8dc53f66be5b5d6774.jpg)  
+Figure 3: Comparison of image restoration methods on CelebA: denoising (1st row), Gaussian deblurring (2nd row), super-resolution (3rd row), random pixel inpainting (4th row), box-inpainting (5th row). N/A means "method not applicable". Zoom in to see that PnP-Flow performs consistently well across all tasks compared to the baselines.
+
+![](images/505f9d67da626d0eb08fd2f9126dfd3b701b67b866c4e668695d59453476f14c.jpg)  
+Figure 4: Comparison of image restoration methods on AFHQ-Cat: denoising (1st row), Gaussian deblurring (2nd row), super-resolution (3rd row), random pixel inpainting (4th row), box-inpainting (5th row). N/A means "method not applicable".
+
+# 5.4 PRACTICAL ASPECTS
+
+Computation time and memory All experiments in this section are conducted on a single NVIDIA RTX 6000 Ada Generation with 48GB RAM. We measure the averaged time per image to complete a deblurring task on the CelebA dataset. We also compute the peak GPU memory load per image. The results are averaged over 100 images (25 batches of 4 images each). We use the same settings as those used for reporting performance metrics on CelebA. Results are in Table 3.
+
+Table 3: Time and memory metrics per image on the deblurring task on CelebA  $(128\times 128)$  
+
+<table><tr><td>Method</td><td>Computation time</td><td>GPU peak memory load</td></tr><tr><td>OT-ODE</td><td>1.50s</td><td>0.65GB</td></tr><tr><td>Flow-Priors</td><td>16.01s</td><td>2.96GB</td></tr><tr><td>D-Flow</td><td>32.19s</td><td>5.91GB</td></tr><tr><td>PnP-Flow (ours)</td><td>3.40s</td><td>0.10GB</td></tr></table>
+
+Sensitivity to initialization Notably, our algorithm does not rely on a good initialization. By starting the algorithm at time  $t_0 = 0$ , the linear interpolation step initially outputs a pure noise, which is then given to the denoiser. As a result, the algorithm's performance is independent of the initialization. In Appendix A.4, we stress that this is not the case for other methods.
+
+**Choice of the latent distribution** Our algorithm can be used with any latent distribution. This is in the spirit of Flow Matching models (Lipman et al., 2023), which do not rely on a Gaussian latent as opposed to diffusion models. In particular, recently there has been a trend of modelling categorical data with Flow Matching (Boll et al., 2024; Stark et al., 2024; Davis et al., 2024). Our method does not rely on a Gaussian latent, contrary to the other Flow Matching restoration methods. As an example to illustrate the performance of our approach in a non-Gaussian case, in Appendix A.3 we conduct an experiment with a Dirichlet latent distribution, inspired by (Stark et al., 2024).
+
+Adaptability to any straight-line flows Our denoiser is rooted in the straight-line flow framework, which motivates our use of OT Flow Matching. However, other choices of FM models are possible. Notably, Rectified Flows (Liu et al., 2023) are of particular interest, as the method described allows for the straightening of any flow model. In Appendix A.5, we show how our method performs similarly to what we observed in Section 5.3 using pre-trained Rectified Flows.
+
+# 6 CONCLUSION
+
+We introduced PnP-Flow Matching, and compared it to recent Flow Matching and PnP methods. A great strength of our method is its versatility: it requires few hyperparameters, uses minimal memory and computational resources, delivers very good performance across various inverse problems, and supports different latent distributions as well as flexible initialization. Regarding limitations, our reconstructions seem to be more on the smooth side, which relates to the fact that our denoiser is mean square estimator. This however is a common tradeoff in regularization of inverse problems (Blau & Michaeli, 2018). Next, it would be valuable to apply our method to other types of measurement noise, such as Poisson noise (Hurault et al., 2023). Preliminary results with Laplace noise are provided in Appendix A.11. Furthermore, our method could be explored in the context of blind inverse problems, as suggested by the promising results in Appendix A.14. Another potential improvement would be to leverage different latent distributions (Boll et al., 2024; Gat et al., 2024; Stark et al., 2024) for modeling categorical distributions. In particular, Dirichlet distributions are widely used in biological and molecular data analysis, making this a relevant domain for further investigation. Finally, an ambitious question for future research would be to explore how the PnP-flow algorithm could be adapted into a posterior sampling algorithm.
+
+Acknowledgements Ségolène Martin's work is carried out in the framework of the DFG funded Cluster of Excellence EXC 2046 MATH+ (project ID: AA5-8). Anne Gagneux thanks the RT IASIS for supporting her work through the PROSSIMO grant. Anne Gagneux also acknowledges support from the Berlin Mathematical School of MATH+, which funded her research visit to TU Berlin. Paul Hagemann acknowledges funding by the German Research Foundation (DFG) within the project SPP 2298 "Theoretical Foundations of Deep Learning". The authors thank the Blaise Pascal Center for the computational means. It uses the SIDUS (Quemener & Corvellec, 2013) solution.
+
+The authors would like to thank Yasi Zhang for helping with the Flow priors code, Bastian Boll for helpful discussions and Marien Renaud for spotting some mistakes.
+
+Ethics Statement Our method makes use of pre-trained generative models, such that the general concerns of such models apply. In particular, these generative models carry inherent biases and can potentially be misused.
+
+However, our work is foundational and we believe there are many relevant applications, such as medical imaging or scientific applications which outweigh the negative usages. Furthermore our work involves only small Flow Matching models, which carry very low risk of being used for malicious purposes.
+
+Reproducibility Statement We implemented all the baselines and release the code in the supplementary material. In Appendix A.8 we state the hyper-parameter search procedure and the found values. Further, all our theoretical results state the precise assumptions and contain full proofs.
+
+# REFERENCES
+
+Fabian Altekrüger, Alexander Denker, Paul Hagemann, Johannes Hertrich, Peter Maass, and Gabriele Steidl. PatchNR: learning from very few images by patch normalizing flow regularization. Inverse Problems, 39(6):064006, 2023.  
+Luigi Ambrosio, Nicola Gigli, and Giuseppe Savare. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser, 2nd edition, 2008.  
+Muhammad Asim, Max Daniels, Oscar Leong, Ali Ahmed, and Paul Hand. Invertible generative models for inverse problems: mitigating representation error and dataset bias. In International Conference on Machine Learning (ICML), 2020.  
+Heli Ben-Hamu, Omri Puny, Itai Gat, Brian Karrer, Uriel Singer, and Yaron Lipman. D-Flow: Differentiating through flows for controlled generation. In International Conference on Machine Learning (ICML), 2024.  
+Joe Benton, George Deligiannidis, and Arnaud Doucet. Error bounds for flow matching methods. Transactions on Machine Learning Research, 2024.  
+Yochai Blau and Tomer Michaeli. The perception-distortion tradeoff. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, 2018.  
+Bastian Boll, Daniel Gonzalez-Alvarado, Stefania Petra, and Christoph Schnorr. Generative assignment flows for representing and learning joint distributions of discrete data. arXiv preprint arXiv:2406.04527, 2024.  
+Ashish Bora, Ajil Jalal, Eric Price, and Alexandros G Dimakis. Compressed sensing using generative models. In International Conference on Machine Learning (ICML), pp. 537-546, 2017.  
+Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends® in Machine learning, 3(1):1-122, 2011.  
+Stanley H Chan, Xiran Wang, and Omar A Elgendy. Plug-and-play admm for image restoration: Fixed-point convergence and applications. IEEE Transactions on Computational Imaging, 3(1): 84-98, 2016.
+
+Jannis Chemseddine, Paul Hagemann, Gabriele Steidl, and Christian Wald. Conditional Wasserstein distances with applications in Bayesian OT flow matching. arXiv preprint arXiv:2024, 2024.  
+Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David K. Duvenaud. Neural ordinary differential equations. Advances in Neural Information Processing Systems (NeurIPS), 2018.  
+Chaoran Cheng, Boran Han, Danielle C. Maddix, Abdul Fatir Ansari, Andrew Stuart, Michael W. Mahoney, and Bernie Wang. Gradient-free generation for hard-constrained systems. In International Conference on Learning Representations (ICLR), 2025. URL https://openreview.net/forum?id=teE4pl9ftK.  
+Jooyoung Choi, Sungwon Kim, Yonghyun Jeong, Youngjune Gwon, and Sungroh Yoon. Ilvr: Conditioning method for denoising diffusion probabilistic models. In CVF International Conference on Computer Vision (ICCV), volume 1, pp. 2, 2021.  
+Yunjey Choi, Youngjung Uh, Jaejun Yoo, and Jung-Woo Ha. StarGAN v2: Diverse image synthesis for multiple domains. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8188-8197, 2020.  
+Hyungjin Chung, Jeongsol Kim, Michael Thompson McCann, Marc Louis Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. In International Conference on Learning Representations (ICLR), 2023.  
+Patrick L. Combettes and Jean-Christophe Pesquet. Proximal splitting methods in signal processing. In Fixed-point Algorithms for Inverse Problems in Science and Engineering, pp. 185-212. Springer, 2011.  
+Patrick L Combettes and Valérie R Wajs. Signal recovery by proximal forward-backward splitting. Multiscale Modeling & Simulation, 4(4):1168-1200, 2005.  
+Oscar Davis, Samuel Kessler, Mircea Petrache, Ismail Ilkan Ceylan, Michael Bronstein, and Avishek Joey Bose. Fisher flow matching for generative modeling over discrete data. arXiv preprint arXiv:2405.14664, 2024.  
+Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real NVP. In International Conference on Learning Representations (ICLR), 2017.  
+Tomer Garber and Tom Tirer. Image restoration by denoising diffusion models with iteratively preconditioned guidance. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 25245-25254, 2024.  
+Itai Gat, Tal Remez, Neta Shaul, Felix Kreuk, Ricky T. Q. Chen, Gabriel Synnaeve, Yossi Adi, and Yaron Lipman. Discrete flow matching. arXiv preprint arXiv:2407.15595, 2024.  
+Alexandros Graikos, Nikolay Malkin, Nebojsa Jojic, and Dimitris Samaras. Diffusion models as plug-and-play priors. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances in Neural Information Processing Systems (NeurIPS), 2022.  
+Jingyu Guo, Hongliang Qi, Yuan Xu, Zijia Chen, Shulong Li, and Linghong Zhou. Iterative image reconstruction for limited-angle CT using optimized initial image. Computational and Mathematical Methods in Medicine, 2016(1):5836410, 2016.  
+Yutong He, Naoki Murata, Chieh-Hsin Lai, Yuhta Takida, Toshimitsu Uesaka, Dongjun Kim, Wei Hsiang Liao, Yuki Mitsufuji, J Zico Kolter, Ruslan Salakhutdinov, and Stefano Ermon. Manifold preserving guided diffusion. In The Twelfth International Conference on Learning Representations, 2024. URL https://openreview.net/forum?id=o3BxOloxm1.  
+Johannes Hertrich, Sebastian Neumayer, and Gabriele Steidl. Convolutional proximal neural networks and plug-and-play algorithms. Linear Algebra and its Applications, 631:203--234, 2021.  
+Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems (NeurIPS), volume 33, pp. 6840-6851, 2020.
+
+Chin-Wei Huang, Jae Hyun Lim, and Aaron C Courville. A variational perspective on diffusion-based generative models and score matching. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan (eds.), Advances in Neural Information Processing Systems (NeurIPS), volume 34, pp. 22863–22876, 2021.  
+Samuel Hurault, Arthur Leclaire, and Nicolas Papadakis. Gradient step denoiser for convergent plug-and-play. In International Conference on Learning Representations (ICLR), 2022a.  
+Samuel Hurault, Arthur Leclaire, and Nicolas Papadakis. Proximal denoiser for convergent plug-and-play optimization with nonconvex regularization. In International Conference on Machine Learning (ICML), 2022b.  
+Samuel Hurault, Ulugbek Kamilov, Arthur Leclaire, and Nicolas Papadakis. Convergent bregman plug-and-play image restoration for poisson inverse problems. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (eds.), Advances in Neural Information Processing Systems (NeurIPS), volume 36, pp. 27251-27280, 2023.  
+Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. In International Conference on Learning Representations (ICLR), 2018.  
+Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4401-4410, 2019.  
+Bahjat Kawar, Michael Elad, Stefano Ermon, and Jiaming Song. Denoising diffusion restoration models. In Advances in Neural Information Processing Systems, 2022.  
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2017.  
+Nikita Kornilov, Alexander Gasnikov, and Alexander Korotin. Optimal flow matching: Learning straight trajectories in just one step. arXiv preprint arXiv:2403.13117, 2024.  
+Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278-2324, 1998. doi: 10.1109/5.726791.  
+Sangyun Lee, Beomsu Kim, and Jong Chul Ye. Minimizing trajectory curvature of ODE-based generative models. In International Conference on Machine Learning (ICML), 2023.  
+Anat Levin, Yair Weiss, Fredo Durand, and William T Freeman. Understanding and evaluating blind deconvolution algorithms. In 2009 IEEE conference on computer vision and pattern recognition, pp. 1964-1971. IEEE, 2009.  
+Xiang Li, Soo Min Kwon, Ismail R. Alkhouri, Saiprasad Ravishankar, and Qing Qu. Decoupled data consistency with diffusion purification for image restoration. arXiv preprint arXiv:2403.06054, 2024.  
+Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In International Conference on Learning Representations (ICLR), 2023.  
+Qiang Liu. Rectified flow: A marginal preserving approach to optimal transport. arXiv preprint arXiv:2209.14577, 2022.  
+Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. In International Conference on Learning Representations (ICLR), 2023.  
+Raunak Manekar, Zhong Zhuang, Kshitij Tayal, Vipin Kumar, and Ju Sun. Deep learning initialized phase retrieval. In Neural Information Processing Systems (NeurIPS) Workshop, 2020.
+
+Morteza Mardani, Jiaming Song, Jan Kautz, and Arash Vahdat. A variational perspective on solving inverse problems with diffusion models. In International Conference on Learning Representations (ICLR), 2024.  
+Tim Meinhardt, Michael Moeller, Caner Hazirbas, and Daniel Cremers. Learning proximal operators: Using denoising networks for regularizing inverse imaging problems. In IEEE International Conference on Computer Vision (ICCV), pp. 1799-1808. IEEE Computer Society, 2017.  
+Jean-Christophe Pesquet, Audrey Repetti, Matthieu Terris, and Yves Wiaux. Learning maximally monotone operators for image recovery. SIAM Journal on Imaging Sciences, 14(3):1206-1237, 2021.  
+Ashwini Pokle, Matthew J. Muckley, Ricky T. Q. Chen, and Brian Karrer. Training-free linear image inverses via flows. Transactions on Machine Learning Research, 2024.  
+Aram-Alexandre Pooladian, Heli Ben-Hamu, Carles Domingo-Enrich, Brandon Amos, Yaron Lipman, and Ricky T. Q. Chen. Multisample flow matching: Straightening flows with minibatch couplings. In International Conference on Machine Learning (ICML), 2023.  
+Emmanuel Quemener and Marianne Corvellec. Sidus—the solution for extreme dedduplication of an operating system. *Linux J.*, 2013(235), November 2013. ISSN 1075-3583.  
+Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In Medical Image Computing and Computer-Assisted Intervention (MIC-CAI), pp. 234-241. Springer, 2015.  
+Ernest Ryu, Jialin Liu, Sicheng Wang, Xiaohan Chen, Zhangyang Wang, and Wotao Yin. Plug-and-play methods provably converge with properly trained denoisers. In International Conference on Machine Learning (ICML), 2019.  
+Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In Francis Bach and David Blei (eds.), Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pp. 2256-2265, 2015.  
+Hendrik Sommerhoff, Andreas Kolb, and Michael Moeller. Energy dissipation with plug-and-play priors. In Advances in Neural Information Processing Systems (NeurIPS) Workshop, 2019.  
+Jiaming Song, Arash Vahdat, Morteza Mardani, and Jan Kautz. Pseudoinverse-guided diffusion models for inverse problems. In International Conference on Learning Representations (ICLR), 2023.  
+Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations (ICLR), 2021.  
+Hannes Stark, Bowen Jing, Chenyu Wang, Gabriele Corso, Bonnie Berger, Regina Barzilay, and Tommi Jaakkola. Dirichlet flow matching with applications to DNA sequence design. In International Conference on Machine Learning (ICLR), 2024.  
+Yu Sun, Brendt Wohlberg, and Ulugbek S Kamilov. An online plug-and-play algorithm for regularized image reconstruction. IEEE Transactions on Computational Imaging, 5(3):395-408, 2019.  
+Julian Tachella, Dongdong Chen, Samuel Hurault, Matthieu Terris, and Andrew Wang. DeepInverse: A deep learning framework for inverse problems in imaging, 2023.  
+Hong Ye Tan, Subhadip Mukherjee, Junqi Tang, and Carola-Bibiane Schonlieb. Provably convergent plug-and-play quasi-Newton methods. SIAM Journal on Imaging Sciences, 17(2):785-819, 2024.  
+Matthieu Terris, Audrey Repetti, Jean-Christophe Pesquet, and Yves Wiaux. Building firmly nonexpansive convolutional neural networks. In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2020.
+
+Alexander Tong, Kilian Fatras, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector-Brooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport. Transactions on Machine Learning Research, 2024.  
+Singanallur V. Venkatakrishnan, Charles A. Bouman, and Brendt Wohlberg. Plug-and-play priors for model based reconstruction. In IEEE Global Conference on Signal and Information Processing (GlobalSIP), pp. 945-948, 2013.  
+Cédric Villani. Optimal Transport - Old and New, volume 338. Springer, 2008.  
+Yinhuai Wang, Jiwen Yu, and Jian Zhang. Zero-shot image restoration using denoising diffusion null-space model. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=mRieQgMtNTQ.  
+Xinyi Wei, Hans van Gorp, Lizeth Gonzalez-Carabarin, Daniel Freedman, Yonina C. Eldar, and Ruud J. G. van Sloun. Deep unfolding with normalizing flow priors for inverse problems. IEEE Transactions on Signal Processing, 70:2962-2971, 2022.  
+Zihui Wu, Yu Sun, Yifan Chen, Bingliang Zhang, Yisong Yue, and Katherine Bouman. Principled probabilistic imaging using diffusion models as plug-and-play priors. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. URL https://openreview.net/forum?id=Xq9HQf7VNV.  
+Xingyu Xu and Yuejie Chi. Provably robust score-based diffusion posterior sampling for plug-and-play image reconstruction. arXiv preprint arXiv:2403.17042, 2024.  
+Ling Yang, Zixiang Zhang, Zhilong Zhang, Xingchao Liu, Minkai Xu, Wentao Zhang, Chenlin Meng, Stefano Ermon, and Bin Cui. Consistency flow matching: Defining straight flows with velocity consistency. arXiv preprint arXiv:2407.02398, 2024.  
+Shuo Yang, Ping Luo, Chen-Change Loy, and Xiaou Tang. From facial parts responses to face detection: A deep learning approach. In IEEE International Conference on Computer Vision (ICCV), pp. 3676-3684, 2015.  
+Bingliang Zhang, Wenda Chu, Julius Berner, Chenlin Meng, Anima Anandkumar, and Yang Song. Improving diffusion inverse problem solving with decoupled noise annealing. arXiv preprint arXiv:2407.01521, 2024a.  
+Kai Zhang, Wangmeng Zuo, Yunjin Chen, Deyu Meng, and Lei Zhang. Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising. IEEE Transactions on Image Processing, 26(7):3142-3155, 2017.  
+Kai Zhang, Yawei Li, Wangmeng Zuo, Lei Zhang, Luc Van Gool, and Radu Timofte. Plug-and-play image restoration with deep denoiser prior. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(10):6360-6376, 2021.  
+Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 586-595, 2018.  
+Yasi Zhang, Peiyu Yu, Yaxuan Zhu, Yingshan Chang, Feng Gao, Ying Nian Wu, and Oscar Leong. Flow priors for linear inverse problems via iterative corrupted trajectory matching. arXiv preprint arXiv:2405.18816, 2024b.  
+Yuanzhi Zhu, Kai Zhang, Jingyun Liang, Jiezhang Cao, Bihan Wen, Radu Timofte, and Luc Van Gool. Denoising diffusion models for plug-and-play image restoration. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshop, pp. 1219-1229, 2023.
+
+# A APPENDIX
+
+# A.1 PROOF OF PROPOSITION 1
+
+Proof. We have by the definition of the denoiser (6) (it was also observed in Lee et al. (2023))
+
+$$
+\mathbb {E} _ {(X _ {0}, X _ {1}) \sim \pi} [ \| D _ {t} (X _ {t}) - X _ {1} \| ^ {2} ] = (1 - t) ^ {2} \mathbb {E} _ {(X _ {0}, X _ {1}) \sim \pi} [ \| v _ {t} (X _ {t}) - (X _ {1} - X _ {0}) \| ^ {2} ].
+$$
+
+If we assume that the denoising loss is zero, this yields for all  $t \in [0,1)$  that  $v_{t}(X_{t}) = X_{1} - X_{0}$  almost surely. By continuity this also follows for  $t = 1$ . By the same arguments as in (Liu et al., 2023, Theorem 3.6 iii)+ iv) we have that both  $t \mapsto f(t,X_0)$  and  $t \mapsto X_{t}$  are solution to the flow ODE initialized at  $X_0$ , since
+
+$$
+X _ {t} = X _ {0} + t v _ {t} (X _ {t}) = X _ {0} + \int_ {0} ^ {t} v _ {s} (X _ {s}) d s \quad \mathrm {a . s .}
+$$
+
+where we used that  $v$  is constant with time. This implies that  $\partial_t X_t = v_t(X_t)$  almost surely. Together with the uniqueness of the ODE solution, it follows that  $f(t, X_0) = X_t$  almost surely.
+
+On the other hand, if  $(f,v)$  is a straight-line Flow Matching pair connecting  $X_0$  and  $X_{1}$ , then we obtain that
+
+$$
+v _ {t} \left(X _ {t}\right) = v _ {t} \left(f \left(t, X _ {0}\right)\right) = \partial_ {t} f (t, X _ {0}) = \partial_ {t} X _ {t} = X _ {1} - X _ {0} \quad \text {a . s .}
+$$
+
+Therefore the denoising loss is zero.
+
+![](images/910360dad2cf0ed9493ec45f1814b1150090c4f46fcc100bf21e299cafdb00aa.jpg)
+
+# A.2 PROOF OF PROPOSITION 4
+
+We provide the proof for Proposition 4 here below. Note that the assumption of noise boundedness can be easily satisfied by clipping the noise. Additionally, it is naturally ensured numerically since floating-point numbers have finite precision.
+
+Proof. By the algorithm and definition of  $D_{t_n}$ , we have
+
+$$
+x _ {n + 1} = D _ {t _ {n}} (u _ {n}) = u _ {n} + (1 - t _ {n}) v _ {t _ {n}} ^ {\theta} (u _ {n}),
+$$
+
+where together with the definition of  $\gamma_{n}$
+
+$$
+u _ {n} := \left(1 - t _ {n}\right) \varepsilon_ {n} + t _ {n} \left(x _ {n} - \gamma_ {n} \nabla F (x _ {n})\right) = t _ {n} x _ {n} + \left(1 - t _ {n}\right) \left(\varepsilon_ {n} - t _ {n} \nabla F (x _ {n})\right).
+$$
+
+Hence we obtain
+
+$$
+\left\| x _ {n + 1} - x _ {n} \right\| = \left(1 - t _ {n}\right) \left\| \varepsilon_ {n} - x _ {n} - t _ {n} \nabla F (x _ {n}) + v _ {t _ {n}} ^ {\theta} (u _ {n}) \right\|.
+$$
+
+By assumption on  $F$  and  $v$  and since both  $(x_{n})_{n}$  and the noise  $\varepsilon_{n}$  are bounded, the expression in the norm is bounded as well, say by  $M > 0$ . Then, by assumption on  $t_n$ , we conclude
+
+$$
+\sum_ {n = 0} ^ {\infty} \left\| x _ {n + 1} - x _ {n} \right\| = M \sum_ {n = 0} ^ {\infty} (1 - t _ {n}) <   \infty ,
+$$
+
+so that  $(x_{n})_{n}$  is a Cauchy sequence and converges.
+
+![](images/043b2ef54591005c9aaa36dc725d55bffcdcf44a2579416ede216a3c4b2d9b6a.jpg)
+
+# A.3 NUMERICAL RESULTS USING A LATENT DIRICHLET DISTRIBUTION
+
+One of the main practical advantages of Flow Matching over diffusion is that one can choose a latent distribution with is not Gaussian. Motivated by DNA design and other discrete data, there has been growing interest in using a Dirichlet latent distribution in Flow Matching (Stark et al., 2024; Boll et al., 2024; Davis et al., 2024). Here, we want to show that applying PnP-Flow with a Flow Matching model trained on a Dirichlet latent distribution still yields good reconstructions. For this, we use the MNIST dataset (LeCun et al., 1998), rescaling each image to lie on the simplex, and train a Flow Matching model. We then apply this model to inpainting and super-resolution tasks and present the results. Importantly, the goal of this experiment is not to achieve state-of-the-art performance, but to illustrate that our algorithm generalizes to different latent distributions.
+
+For this, we train a Dirichlet Flow Matching model for 200 epochs with a standard OT Flow Matching loss and a Dirichlet distribution with parameters  $(1,\dots,1)\in \mathbb{R}^{784}$ . This latent distribution is also used in Stark et al. (2024) and amounts to the uniform distribution on the 784-dimensional simplex. Note that the interpolation step now involves Dirichlet noise instead of Gaussian, as this noise is drawn according to  $P_0$ . We reconstruct our images with  $\gamma = 1$  and 300 steps. Remarkably, almost no modifications for the algorithm are needed. In particular, the generated images  $x$  almost lie perfectly on the simplex without any normalization. We show the images in Figure 5. We compare to (Ben-Hamu et al., 2024), where we adapt the regularization in the algorithm so that the optimized variable  $z$  lies on the simplex, i.e., we use the loss
+
+$$
+L (z) = \| y - H (f (1, z)) \| ^ {2} + \lambda \| (\sum_ {i, j} z _ {i, j}) - 1) \| ^ {2},
+$$
+
+for a single latent image  $z$ , where  $f(1,z)$  is realized doing 5 Euler steps using the mid point rule and  $\lambda$  is a regularization constant.
+
+As one can see, the PnP Flow outperforms D-Flow on all tasks. Note that the other Flow Matching baselines (Pokle et al., 2024; Zhang et al., 2024b) are not usable since their algorithm heavily depends on Gaussian paths.
+
+![](images/bbd46e40fb00c7fc508f05674e3cb15fc0a5df4dfa4cd33283d45ffc795cfa62.jpg)  
+Figure 5: Dirichlet Flow Matching experiment on Simplex-MNIST, for denoising (1st row), superresolution (2nd row), box-inpainting (3rd row). We measure the reconstruction error as the mean L2 distance (called MSE) between ground truth and reconstruction averaged over the 16 images.
+
+# A.4 SENSITIVITY TO INITIALIZATION
+
+Interestingly, our method is inherently independent of the algorithm's initialization. For the image restoration problems considered in this work, initialization is not so much of a concern because a reasonable starting point for the solution  $x$  is given by  $H^{\top}y$ , where  $y$  is the observation and  $H$  is the degradation operator. However, for more complex problems where  $H^{\top}y$  is far from resembling a natural image, such as in CT reconstruction (Guo et al., 2016) or phase retrieval (Manekar et al., 2020), this property becomes much more relevant.
+
+In contrast, competing methods are more sensitive to initialization and cannot be started from any value. As recommended in the paper, our implementation of OT-ODE is initialized with  $t_0y + (1 - t_0)\epsilon$  where  $\epsilon \sim \mathcal{N}(0, I_d)$  and  $t_0$  is the initialization time. The latent variable in D-Flow is initialized
+
+as  $\alpha T^{-1}(H^{\top}y) + (1 - \alpha)\epsilon$ , where  $T^{-1}$  is the reverse flow,  $\epsilon \sim \mathcal{N}(0, I_d)$  is a random Gaussian noise, and  $\alpha \in (0, 1)$  is a blending coefficient. Flow-Priors, on the other hand, is initialized with random noise and, like our method, does not depend on a "good" initialization.
+
+In Figure 6, we illustrate the impact of changing the standard initialization for all methods to a black image on a Gaussian deblurring task, comparing the robustness of each approach to poor initialization.
+
+![](images/5c06eee46c259e53618f5b10755ba400a1bbc6e379a97a6e213d2abb68d8ee7f.jpg)  
+Figure 6: Comparison of restoration methods on the CelebA dataset, for a Gaussian deblurring task and for two different initializations: default initialization recommended for each method (1st row), initialization set to the zero image (2rd row).
+
+# A.5 EXPERIMENTS WITH RECTIFIED FLOWS
+
+The image restoration experiments we carry out in the core of the paper use a pre-trained OT Flow Matching model. Yet, the theory behind our method holds for any straight-line Flow Matching model. In this section, we show how our method PnP-Flow compares to Flow Priors and OT-ODE using, for all three methods, a pre-trained Rectified Flow model Liu et al. (2023) on CelebA-HQ dataset (Karras et al., 2018) with image dimension  $256 \times 256$ . We use the checkpoint provided by the Rectified Flow repository<sup>2</sup>. We were not able to run D-Flow using the Rectified Flow model. Quantitative results are in Table 4 and qualitative results on paintbrush inpainting are displayed in Figure 7.
+
+Table 4: Comparisons of state-of-the-art methods on different inverse problems on the dataset CelebA-HQ. Results are averaged across 100 test images.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Denoising σ = 0.2</td><td colspan="2">Deblurring σ = 0.05, σb = 3.0</td><td colspan="2">Super-res. σ = 0.05, ×4</td><td colspan="2">Rand. inpaint. σ = 0.01, 70%</td><td colspan="2">Box inpaint. σ = 0.05, 80 × 80</td></tr><tr><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td><td>PSNR</td><td>SSIM</td></tr><tr><td>Degraded</td><td>20.00</td><td>0.277</td><td>23.94</td><td>0.514</td><td>9.98</td><td>0.040</td><td>12.28</td><td>0.197</td><td>22.14</td><td>0.726</td></tr><tr><td>OT-ODE</td><td>29.38</td><td>0.731</td><td>25.47</td><td>0.589</td><td>24.24</td><td>0.578</td><td>26.59</td><td>0.639</td><td>26.68</td><td>0.683</td></tr><tr><td>Flow-Priors</td><td>25.73</td><td>0.524</td><td>28.09</td><td>0.804</td><td>23.64</td><td>0.494</td><td>33.26</td><td>0.935</td><td>29.89</td><td>0.817</td></tr><tr><td>PnP-Flow (ours)</td><td>32.65</td><td>0.891</td><td>28.57</td><td>0.815</td><td>25.92</td><td>0.762</td><td>33.60</td><td>0.935</td><td>31.11</td><td>0.928</td></tr></table>
+
+# A.6 PROGRESSION OF THE PNP-FLOW RECONSTRUCTION WITH TIME
+
+Figure 8 presents the progression of the reconstruction output by the PnP-Flow with respect to time.
+
+![](images/8d561169453eaa9ca11f56148a31cb6b15d84ba44f6ff2041dd45475cab39ca6.jpg)  
+Figure 7: Comparison of restoration methods on the CelebA-HQ dataset for paintbrush inpainting using Rectified Flow model.
+
+![](images/f971fb592c8891cfe443f9df04e4c90bd5833eff16f307a183c60774b5781b57.jpg)  
+Figure 8: Results for random inpainting using PnP-Flow across different iterations (time steps) with corresponding PSNR values. As expected, in the early iterations, the output resembles a natural face but diverges from the noisy observation. PSNR improves progressively with each iteration.
+
+# A.7 ADDITIONAL VISUAL RESULTS
+
+Here, we provide additional visual results, where we take the same ground truth image for the different inverse problems, see Fig. 9 and Fig. 10 for CelebA dataset and Fig. 11 and Fig. 12 for AFHQ-cat dataset.
+
+![](images/ccf33bad15af455e3c2508ab73bb33a1307ee565584109d530ebfe3206c39700.jpg)  
+Figure 9: Comparison of image restoration methods on CelebA: denoising (1st row), Gaussian deblurring (2nd row), super-resolution (3rd row), free-form inpainting (4th row).
+
+![](images/3e82abb1c551e9cd4930688e7de28eafa743a5167f6d2dd9288aca392f0fbd65.jpg)  
+Figure 10: Comparison of image restoration methods on CelebA: denoising (1st row), Gaussian deblurring (2nd row), super-resolution (3rd row), free-form inpainting (4th row).
+
+![](images/3c739932f9f34f723c28c81739586f81b56b55c20f037a33126b184cd5633db2.jpg)  
+Figure 11: Comparison of restoration methods on AFHQ-Cat: denoising (1st row), Gaussian deblurring (2nd row), super-resolution (3rd row), free-form inpainting (4th row).
+
+![](images/17b1da9988a6d9001bf2f5c0c6b220bf6786703f3a45e73317e9868abf4a36ea.jpg)  
+Figure 12: Comparison of image restoration methods on AFHQ-Cat: denoising (1st row), Gaussian deblurring (2nd row), super-resolution (3rd row), free-form inpainting (4th row).
+
+# A.8 OPTIMAL HYPER-PARAMETERS VALUES
+
+We implemented the Gradient Step denoiser from Hurault et al. (2022a;b) following the repository<sup>3</sup>. For random inpainting, the PnP algorithm is the Half Quadratic Splitting with the parameters given in the paper (Hurault et al., 2022a). For denoising, we only apply the trained denoiser once, feeding it with the true noise level  $\sigma$ . For all other methods, we use Proximal Gradient Descent (PGD), as prescribed in Hurault et al. (2022b). Note that we do not constrain the Lipschitz constant of the denoiser. For the PGD case, we tune three hyper-parameters: the learning rate  $\gamma \in \{0.99, 2.0\}$  in the gradient step (which is related to the regularization parameter  $\lambda$  in the paper), the inertia parameter  $\alpha \in \{0.3, 0.5, 0.8, 1.0\}$  which corresponds to the relaxed denoiser  $D_{\sigma}^{\alpha} = \alpha D_{\sigma} + (1 - \alpha)\mathrm{Id}$ , and the factor  $\sigma_f \in \{1., 1.2, 1.5, 1.8, 2., 3., 4., 5.\}$  so that the denoiser gets as noise map  $\sigma_f \times \sigma$  (where  $\sigma$  is the true noise level). We also considered the number of iterations as a hyperparameter that we tuned on the validation set (with maximum number of iterations fixed to 100).
+
+For PnP-Diff, we tuned two parameters: the regularization parameter  $\lambda \in \{1.0, 5.0, 10.0, 100.0, 1000.0\}$  and the blending parameter  $\zeta \in \{0.1, 0.3, 0.5, 1.0\}$ . The number of iterations was fixed to 100.
+
+For D-Flow, we adjusted the blending parameter for initialization  $\alpha \in \{0.1, 0.3, 0.5\}$  and the regularization parameter  $\lambda \in \{0.1, 0.01, 0.001\}$ . We observed that the PSNR of the reconstruction did not consistently increase across iterations (the method does not always converge); thus, we finetuned the value of the last iteration while keeping it below 20 for computational efficiency. The number of iterations for the inner LBFGS optimization was set to 20, and the number of Euler step when solving the ODE to 5, as recommended in the paper Ben-Hamu et al. (2024).
+
+For OT-ODE, we tuned the initial time  $t_0 \in \{0.1, 0.2, 0.3, 0.4\}$  and the type of learning step  $\gamma$  (either  $\sqrt{t}$  or constant). The number of iterations was set to 100.
+
+For Flow-Priors, we considered -as described in the paper- the two hyper-parameters  $\eta \in \{10^{-3}, 10^{-2}, 10^{-1}\}$  and  $\lambda \in \{10^{2}, 10^{3}, 10^{4}, 10^{4}\}$  which respectively correspond to the step size for the gradient descent and the guidance weight (weight put on the data likelihood). The number of iterations was set to 100 and the number of inner iterations to  $K = 1$ .
+
+Finally, for our method PnP-Flow, we adjusted the exponent in the learning rate  $\alpha \in \{0.01, 0.1, 0.3, 0.5, 0.8, 1.0\}$  and the number of time steps  $N \in \{100, 200, 500\}$ . When increasing  $N$  beyond 100 resulted in less than a  $0.2\mathrm{dB}$  improvement in PSNR, we set  $N = 100$  for computational efficiency. We average the output of the denoising step in Algorithm 3 over 5 realizations of the interpolation step. To speed up the algorithm, this number can be reduced to 1 with only a minor impact on performance.
+
+# A.9 VISUALIZATION OF EACH STEP OF THE ALGORITHM
+
+In Figure 13, we show the output of the PnP-Flow algorithm at different stages for an image from the CelebA dataset: after the gradient step, the interpolation step, and the denoising step.
+
+# A.10 ADDITIONAL EVALUATION METRIC
+
+Here, we add a comparison of the models in the LPIPS metric Zhang et al. (2018), which measures the feature distance between the ground truth and generated images via some pretrained classification network. The results are contained in Table 7 and Table 8. None of the methods stands out as a clear winner with respect to the LPIPS metric. However, our PnP-Flow shows a good performance, often being the second or third best method. In Blau & Michaeli (2018), it is argued that perceptual and distortion seem to be at odds with one another: indeed, one can show that the optimal distortion performance (optimizing the PNSR) does not yield optimal perceptual performance (more in line with LPIPS).
+
+Table 5: Hyper-parameters used for all methods on the CelebA dataset. The values were selected based on the highest PSNR on the validation split.  
+
+<table><tr><td></td><td>Denoising</td><td>Deblurring</td><td>Super-res.</td><td>Rand. inpaint.</td><td>Box inpaint.</td></tr><tr><td>PnP-Diff</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ζ (blending)</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>N/A</td></tr><tr><td>λ (regularization)</td><td>1.0</td><td>1000.0</td><td>100.0</td><td>1.0</td><td>N/A</td></tr><tr><td>PnP-GS</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>γ (learning rate)</td><td>-</td><td>2.0</td><td>2.0</td><td>-</td><td>N/A</td></tr><tr><td>α (inertia param.)</td><td>1.0</td><td>0.3</td><td>1.0</td><td>-</td><td>N/A</td></tr><tr><td>σf (factor for noise input)</td><td>1.0</td><td>1.8</td><td>3.0</td><td>-</td><td>N/A</td></tr><tr><td>niter (number of iter.)</td><td>1</td><td>35</td><td>20</td><td>23</td><td>N/A</td></tr><tr><td>OT-ODE</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>t0 (initial time)</td><td>0.3</td><td>0.4</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>γ</td><td>√t</td><td>√t</td><td>constant</td><td>constant</td><td>√t</td></tr><tr><td>Flow-Priors</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>λ (regularization)</td><td>100</td><td>1,000</td><td>10,000</td><td>10,000</td><td>10,000</td></tr><tr><td>η (learning rate)</td><td>0.01</td><td>0.01</td><td>0.1</td><td>0.01</td><td>0.01</td></tr><tr><td>D-Flow</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>λ (regularization)</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.01</td><td>0.001</td></tr><tr><td>α (blending)</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>niter (number of iter.)</td><td>3</td><td>7</td><td>10</td><td>20</td><td>9</td></tr><tr><td>PnP-Flow</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>α (learning rate factor)</td><td>0.8</td><td>0.01</td><td>0.3</td><td>0.01</td><td>0.5</td></tr><tr><td>N (Number of time steps)</td><td>100</td><td>100</td><td>100</td><td>100</td><td>100</td></tr></table>
+
+Table 6: Hyper-parameters used for all methods on the AFHQ-Cat dataset. The values were selected based on the highest PSNR on the validation split.  
+
+<table><tr><td></td><td>Denoising</td><td>Deblurring</td><td>Super-res.</td><td>Rand. inpaint.</td><td>Box inpaint.</td></tr><tr><td>PnP-Diff</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ζ (blending)</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>N/A</td></tr><tr><td>λ (regularization)</td><td>1.0</td><td>1000.0</td><td>100.0</td><td>1.0</td><td>N/A</td></tr><tr><td>PnP-GS</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>γ (learning rate)</td><td>-</td><td>2.0</td><td>2.0</td><td>-</td><td>N/A</td></tr><tr><td>α (inertia param.)</td><td>1.0</td><td>0.3</td><td>1.0</td><td>-</td><td>N/A</td></tr><tr><td>σf (factor for noise input)</td><td>1.0</td><td>1.8</td><td>5.0</td><td>-</td><td>N/A</td></tr><tr><td>niter (number of iter.)</td><td>1</td><td>60</td><td>50</td><td>23.</td><td>N/A</td></tr><tr><td>OT-ODE</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>t0 (initial time)</td><td>0.3</td><td>0.3</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>γ</td><td>√t</td><td>√t</td><td>constant</td><td>constant</td><td>√t</td></tr><tr><td>Flow-Priors</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>λ (regularization)</td><td>100</td><td>1,000</td><td>10,000</td><td>10,000</td><td>10,000</td></tr><tr><td>η (learning rate)</td><td>0.01</td><td>0.01</td><td>0.1</td><td>0.01</td><td>0.01</td></tr><tr><td>D-Flow</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>λ (regularization)</td><td>0.001</td><td>0.01</td><td>0.001</td><td>0.001</td><td>0.01</td></tr><tr><td>α (blending)</td><td>0.1</td><td>0.5</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>niter (number of iter.)</td><td>3</td><td>20</td><td>20</td><td>20</td><td>9</td></tr><tr><td>PnP-Flow</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>α (learning rate factor)</td><td>0.8</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.5</td></tr><tr><td>N (Number of time steps)</td><td>100</td><td>500</td><td>500</td><td>200</td><td>100</td></tr></table>
+
+![](images/da3eb09e648c02318824165fed7a001dfb1df839b0507e621deded91fc1e98a2.jpg)  
+Figure 13: Progression of the output of the algorithm at intermediate steps (log-likelihood step, reprojection onto the path, and denoising) for a random inpainting task.
+
+Table 7: Comparisons of state-of-the-art methods on different inverse problems on the dataset CelebA using the LPIPS perceptual metric Zhang et al. (2018). Results are averaged across 100 test images.  
+
+<table><tr><td>Method</td><td>Denoising σ = 0.2</td><td>Deblurring σ = 0.05, σb = 1.0</td><td>Super-res. σ = 0.05, ×4</td><td>Rand. inpaint. σ = 0.01, 70%</td><td>Box inpaint. σ = 0.05, 40 × 40</td></tr><tr><td>Degraded</td><td>0.373</td><td>0.125</td><td>0.827</td><td>1.034</td><td>0.213</td></tr><tr><td>PnP-Diff</td><td>0.060</td><td>0.060</td><td>0.034</td><td>0.025</td><td>N/A</td></tr><tr><td>PnP-GS</td><td>0.035</td><td>0.040</td><td>0.053</td><td>0.085</td><td>N/A</td></tr><tr><td>OT-ODE</td><td>0.110</td><td>0.096</td><td>0.091</td><td>0.137</td><td>0.102</td></tr><tr><td>D-Flow</td><td>0.078</td><td>0.053</td><td>0.027</td><td>0.021</td><td>0.035</td></tr><tr><td>Flow-Priors</td><td>0.137</td><td>0.056</td><td>0.101</td><td>0.018</td><td>0.048</td></tr><tr><td>PnP-Flow (ours)</td><td>0.056</td><td>0.046</td><td>0.055</td><td>0.021</td><td>0.043</td></tr></table>
+
+# A.11 HANDLING NON-GAUSSIAN NOISE MODELS
+
+Our approach is capable of handling various types of degradation noise, beyond Gaussian. In general, any noise model corresponding to a differentiable likelihood can be handled by our method effortlessly. Note that this is not the case of all Flow-Matching methods. In particular, the methods D-Flow and OT-ODE are designed under the assumption of standard Gaussian noise. Although it may be possible to adapt these methods to other kinds of noises, doing so would require additional modifications, which we leave for future work.
+
+In the following experiment, we consider a super-resolution task with a degradation model incorporating Laplace noise:
+
+$$
+y = \operatorname {L a p l a c e} (H x),
+$$
+
+Table 8: Comparisons of state-of-the-art methods on different inverse problems on the dataset AFHQ-Cat using the LPIPS perceptual metric Zhang et al. (2018). Results are averaged across 100 test images.  
+
+<table><tr><td>Method</td><td>Denoising σ = 0.2</td><td>Deblurring σ = 0.05, σb = 3.0</td><td>Super-res. σ = 0.05, ×4</td><td>Rand. inpaint. σ = 0.01, 70%</td><td>Box inpaint. σ = 0.05, 80 × 80</td></tr><tr><td>Degraded</td><td>0.524</td><td>0.460</td><td>0.888</td><td>1.069</td><td>0.206</td></tr><tr><td>PnP-Diff</td><td>0.194</td><td>0.354</td><td>0.438</td><td>0.064</td><td>N/A</td></tr><tr><td>PnP-GS</td><td>0.075</td><td>0.360</td><td>0.384</td><td>0.116</td><td>N/A</td></tr><tr><td>OT-ODE</td><td>0.190</td><td>0.226</td><td>0.193</td><td>0.226</td><td>0.187</td></tr><tr><td>D-Flow</td><td>0.177</td><td>0.178</td><td>0.207</td><td>0.049</td><td>0.091</td></tr><tr><td>Flow-Priors</td><td>0.159</td><td>0.191</td><td>0.279</td><td>0.047</td><td>0.122</td></tr><tr><td>PnP-Flow (ours)</td><td>0.170</td><td>0.341</td><td>0.258</td><td>0.043</td><td>0.118</td></tr></table>
+
+where  $H: \mathbb{R}^d \to \mathbb{R}^m$  is 2-downsampling linear operator. Laplace noise, which has heavier tails than Gaussian noise, is well-suited for modeling data with outliers. In this context, the data-fidelity function is defined as  $f(x) = \| Hx - y\|_1$  and is differentiable almost everywhere.
+
+For each method, we tune the hyper-parameters via grid search, selecting the configuration that yields the highest PSNR, similar to our previous experiments. We report the results in terms of PSNR, SSIM, and LIPPS in Table 9. We also show the visual results in Figure 14.
+
+Table 9: Comparisons of state-of-the-art methods on Laplace super-resolution experiments on the dataset CelebA. Results are averaged across 100 test images.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Laplace super-resolution σ = 0.1</td><td colspan="3">Laplace super-resolution σ = 0.3</td></tr><tr><td>PSNR</td><td>SSIM</td><td>LPIPS</td><td>PSNR</td><td>SSIM</td><td>LPIPS</td></tr><tr><td>Degraded</td><td>8.82</td><td>0.086</td><td>0.845</td><td>8.76</td><td>0.040</td><td>0.865</td></tr><tr><td>PnP-Diff</td><td>29.67</td><td>0.866</td><td>0.059</td><td>24.82</td><td>0.730</td><td>0.268</td></tr><tr><td>PnP-GS</td><td>28.50</td><td>0.828</td><td>0.079</td><td>25.31</td><td>0.725</td><td>0.185</td></tr><tr><td>Flow-Priors</td><td>27.53</td><td>0.721</td><td>0.102</td><td>21.72</td><td>0.454</td><td>0.376</td></tr><tr><td>PnP-Flow (ours)</td><td>30.30</td><td>0.895</td><td>0.063</td><td>26.50</td><td>0.809</td><td>0.108</td></tr></table>
+
+![](images/f0fdebaf38c004d3366bb6579a2c61db9aaf227e22ebe730c8ef0b687d674cc2.jpg)  
+Figure 14: Example of reconstructions obtained on a super-resolution tasks with Laplace noise  $(\sigma = 0.3)$ , on the CelebA dataset.
+
+# A.12 DISCUSSION OF RELATED DIFFUSION APPROACHES
+
+In He et al. (2024), based on the seminal work Chung et al. (2023), a similar algorithm is proposed in the diffusion case (although for latent models): Our algorithm closely resembles their pixel space algorithm with a different noise update rule. However, note that they do not relate it to PnP methods, and that we outlined several theoretical and practical advantages of flow matching over diffusion models. Further similar ideas are explored in Wu et al. (2024); Zhang et al. (2024a); Xu & Chi (2024); Li et al. (2024), which focus on posterior sampling using diffusion models. These works
+
+employ a similar decoupling of the likelihood (data-consistency) sampling step and a denoising diffusion sampler (referred to as the decoder in Li et al. (2024)) that acts as an implicit image prior. Additionally, Wang et al. (2023) introduces a particularly efficient null-space approach. Instead of making steps on the log likelihood like Chung et al. (2023), they optimize in the null space of the linear operator directly. Such ideas could in principle also be applied to our algorithm, which we leave for future work.
+
+# A.13 ADDITIONAL EXPERIMENTS WITH DIFFUSION-BASED METHODS
+
+Table 10: Comparison of PnP-Flow with diffusion-based methods on various inverse problems using the CelebA dataset. Results are averaged over 100 test images.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Denoising σ = 0.2</td><td colspan="3">Deblurring σ = 0.05, σb = 1.0</td><td colspan="3">Super-resolution σ = 0.05, ×2</td><td colspan="3">Random Inpainting σ = 0.01, 70% mask</td></tr><tr><td>PSNR</td><td>SSIM</td><td>LPIPS</td><td>PSNR</td><td>SSIM</td><td>LPIPS</td><td>PSNR</td><td>SSIM</td><td>LPIPS</td><td>PSNR</td><td>SSIM</td><td>LPIPS</td></tr><tr><td>Degraded</td><td>20.00</td><td>0.348</td><td>0.373</td><td>27.67</td><td>0.740</td><td>0.125</td><td>7.53</td><td>0.012</td><td>0.827</td><td>11.82</td><td>0.197</td><td>1.034</td></tr><tr><td>PnP-Diff</td><td>31.00</td><td>0.883</td><td>0.060</td><td>32.49</td><td>0.911</td><td>0.060</td><td>31.20</td><td>0.893</td><td>0.034</td><td>31.43</td><td>0.917</td><td>0.025</td></tr><tr><td>DDRM</td><td>31.46</td><td>0.895</td><td>0.071</td><td>31.44</td><td>0.901</td><td>0.080</td><td>31.56</td><td>0.909</td><td>0.046</td><td>30.31</td><td>0.902</td><td>0.040</td></tr><tr><td>DPS</td><td>31.52</td><td>0.887</td><td>0.068</td><td>31.42</td><td>0.917</td><td>0.038</td><td>29.18</td><td>0.841</td><td>0.058</td><td>26.83</td><td>0.858</td><td>0.122</td></tr><tr><td>PnP-Flow (Ours)</td><td>32.45</td><td>0.911</td><td>0.056</td><td>34.51</td><td>0.940</td><td>0.046</td><td>31.49</td><td>0.907</td><td>0.055</td><td>33.54</td><td>0.953</td><td>0.021</td></tr></table>
+
+Table 11: Comparison of PnP-Flow with diffusion-based methods on various inverse problems using the AFHQ-Cat dataset. Results are averaged over 100 test images.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Denoising σ = 0.2</td><td colspan="3">Deblurring σ = 0.05, σb = 3.0</td><td colspan="3">Super-resolution σ = 0.05, ×4</td><td colspan="3">Random Inpainting σ = 0.01, 70% mask</td></tr><tr><td>PSNR</td><td>SSIM</td><td>LPIPS</td><td>PSNR</td><td>SSIM</td><td>LPIPS</td><td>PSNR</td><td>SSIM</td><td>LPIPS</td><td>PSNR</td><td>SSIM</td><td>LPIPS</td></tr><tr><td>Degraded</td><td>20.00</td><td>0.319</td><td>0.524</td><td>23.77</td><td>0.514</td><td>0.460</td><td>10.74</td><td>0.042</td><td>0.888</td><td>13.35</td><td>0.234</td><td>1.069</td></tr><tr><td>PnP-Diff</td><td>30.27</td><td>0.835</td><td>0.194</td><td>27.97</td><td>0.764</td><td>0.354</td><td>23.22</td><td>0.601</td><td>0.438</td><td>31.08</td><td>0.882</td><td>0.064</td></tr><tr><td>DDRM</td><td>30.60</td><td>0.838</td><td>0.240</td><td>25.89</td><td>0.680</td><td>0.430</td><td>27.50</td><td>0.746</td><td>0.329</td><td>30.24</td><td>0.857</td><td>0.109</td></tr><tr><td>DPS</td><td>27.64</td><td>0.743</td><td>0.327</td><td>24.31</td><td>0.673</td><td>0.390</td><td>27.83</td><td>0.755</td><td>0.322</td><td>27.23</td><td>0.739</td><td>0.334</td></tr><tr><td>PnP-Flow (Ours)</td><td>31.65</td><td>0.876</td><td>0.170</td><td>27.62</td><td>0.763</td><td>0.341</td><td>26.75</td><td>0.774</td><td>0.258</td><td>32.98</td><td>0.930</td><td>0.043</td></tr></table>
+
+In Table 10 and Table 11, we provide extensive benchmarks using diffusion-based methods (PnP-Diff Zhu et al. (2023), DDRM Kawar et al. (2022) and DPS Chung et al. (2023)), respectively on the CelebA and AFHQ-Cat datasets. Qualitative results are given in Figure 15. All diffusion-based methods use the same pre-trained model from Choi et al. (2021), implemented in the DeepInv library $^4$  (Tachella et al., 2023). Note that the pre-trained diffusion model was trained on the FFHQ dataset (Karras et al., 2019), making the comparison indirect. For all methods, a gridsearch over hyperparameters has been conducted (same methodology as followed for the other experiments, see Appendix A.8). More precisely, for DDRM, we adjusted the value of  $\eta$  and  $\eta_B$  with a grid search over  $\{0.7, 0.8, 0.9, 1.0\}$  following the ablation study in Kawar et al. (2022). For DPS, we adjusted the value of  $\eta \in \{0.5, 0.7, 0.8, 0.85, 0.9, 0.95\}$  which controls the stochasticity.
+
+# A.14 EXTENSION TO BLIND INVERSE PROBLEMS
+
+In this Section, we describe how our algorithm can be used for blind deconvolution Levin et al. (2009), i.e., in a deblurring problem, where the forward operator is unknown. Here, we assume that we know the kernel size and that the kernel is the same across channels. The true forward operator is given by a Gaussian blur and we attempt to simultaneously reconstruct blur and "true" image. This we do as follows: We loop over a discrete time sequence and after each time-step in Algorithm 3, we run a gradient descent update of the "learned" kernel on the log likelihood loss  $\| y - H_{\text{learned}}(x) \|^2$  with the Adam optimizer (Kingma & Ba, 2017). The result can be seen for one exemplary image in Figure 16, where we see that our reconstruction is clearly improving upon the blurry measurement. In Figure 17, we repeat this experiment for a random inpainting task, learning
+
+![](images/b715264067be0e129304eb969ad694604bc0955ee94ccd6eefc0f1223d3c2a43.jpg)  
+Figure 15: Up: example of reconstructions obtained on a Gaussian denoising task  $(\sigma = 0.2)$  on the CelebA dataset. Bottom: example of reconstructions obtained on a random inpainting task (70% of masked pixels) on the AFHQ-Cat dataset.
+
+both the reconstruction and the parameterized mask. This seems to be an interesting avenue for future research.
+
+![](images/89614d7cbe8c5a1f9a9ae4f8b5303d56b71268c211815d16949e36f24631b9b5.jpg)  
+Clean ground truth.
+
+![](images/dfda79f0be5b967622c12456a9157192c0bb3ff69bfb8aedac3eb9b8bb8108cd.jpg)  
+Blurry measurement.
+
+![](images/9eccbdcf9445cd8c6f832c16ff62b23b85beab63cd575aaa9fef2e1a39f3f732.jpg)  
+Reconstruction with our method.
+
+![](images/d4f5a87f0529bfc5654e346a4cdba1371093e2254dd70c3021bd9a836affdc29.jpg)  
+Figure 16: Application of our method to a blind deconvolution problem.  
+Clean ground truth.  
+Figure 17: Application of our method to a blind masking problem.
+
+![](images/e7b0c358538b3139393e28b0bb527854a26bef1b546d983c907bab2675243d70.jpg)  
+Blurry measurement.
+
+![](images/e833bb50e762b3b21db18a0d251af06b995159c5ff4628a89b229f785172c6d3.jpg)  
+Reconstruction with our method.

2025/PnP-Flow_ Plug-and-Play Image Restoration with Flow Matching/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:aa8470da4811d32b362e526f93d0626b700d1f537d7e2f73bfc6b50013f8d2f1
+size 2294403

2025/Point Cluster_ A Compact Message Unit for Communication-Efficient Collaborative Perception/ed2c2ba3-1518-434b-a59b-d326eb00ba86_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:45d9399126f9d10ca469083691190ff63cbfdad74c2d6240a8a034a75c370a81
+size 12609072

2025/Point Cluster_ A Compact Message Unit for Communication-Efficient Collaborative Perception/full.md

@@ -0,0 +1,388 @@
+# POINT CLUSTER: A COMPACTMESSAGE UNIT FOR COMMUNICATION-EFFICIENTCOLLABORATIVE PERCEPTION
+
+Zihan Ding $^{1}$ , Jiahui Fu $^{1}$ , Si Liu $^{1*}$ , Hongyu Li $^{1}$ , Siheng Chen $^{2}$ , Hongsheng Li $^{3,4}$ , Shifeng Zhang $^{5}$ , Xu Zhou $^{5}$
+
+$^{1}$  Institute of Artificial Intelligence, Beihang University,  
+$^{2}$  School of Artificial Intelligence, Shanghai Jiao Tong University,  
+<sup>3</sup> MMLab, CUHK, <sup>4</sup> Centre for Perceptual and Interactive Intelligence, <sup>5</sup> Sangfor Technologies dingzihan737@gmail.com
+
+# ABSTRACT
+
+The objective of the collaborative perception task is to enhance the individual agent's perception capability through message communication among neighboring agents. A central challenge lies in optimizing the inherent trade-off between perception ability and communication cost. To tackle this bottleneck issue, we argue that a good message unit should encapsulate both semantic and structural information in a sparse format, a feature not present in prior approaches. In this paper, we innovatively propose a compact message unit, namely point cluster, whose core idea is to represent potential objects efficiently with explicitly decoupled low-level structure information and high-level semantic information. Building upon this new message unit, we propose a comprehensive framework CPPC for communication-efficient collaborative perception. The core principle of CPPC is twofold: first, through strategical point sampling, structure information can be well preserved with a few key points, which can significantly reduce communication cost; second, the sequence format of point clusters enables efficient message aggregation by set matching and merging, thereby eliminating unnecessary computation generated when aligning squared BEV maps, especially for long-range collaboration. To handle time latency and pose errors encountered in real-world scenarios, we also carefully design parameter-free solutions that can adapt to different noisy levels without finetuning. Experiments on serval widely recognized collaborative perception benchmarks showcase the superior performance of our method compared to the previous state-of-the-art approaches.
+
+# 1 INTRODUCTION
+
+Collaborative perception aims to enhance the perception capabilities of individual agents by utilizing complementary information exchanged between surrounding agents. This approach offers a novel strategy to address several inherent challenges of single-agent perception He et al. (2017); Cheng et al. (2022), including occlusion and long-range limitations. There is a pressing need for related methods and systems across a wide spectrum of practical applications, such as vehicle-to-everything autonomous driving Yu et al. (2022), automated multi-robot warehouse systems Li et al. (2020), and multi-UAVs for search and rescue Scherer et al. (2015). Recent efforts have made valuable contributions in terms of high-quality real and simulated datasets Yu et al. (2022); Xu et al. (2022b); Li et al. (2022); Xu et al. (2022a; 2023b), as well as effective solutions Xu et al. (2023a); Li et al. (2021a); Wang et al. (2020); Xu et al. (2022a); Hu et al. (2022); Chen et al. (2019a); Xu et al. (2022b) for collaborative perception.
+
+The paramount challenge in this field involves optimizing perceptual ability in the face of limited communication bandwidth in real-world scenarios, i.e., communication-efficiency. Based on the type of observation transmission medium (i.e., message unit), previous studies can be categorized into three types: early collaboration with raw point cloud data Chen et al. (2019b); Arnold et al. (2020), late collaboration using bounding boxes Shi et al. (2022); Zeng et al. (2020); Rauch et al. (2012); Glaser & Kira (2023), and intermediate collaboration through bird's eye view (BEV) maps Xu et al. (2022b;a; 2023a); Hu et al. (2022). Early collaboration can simplify subsequent
+
+analyses and enrich information for downstream models. Despite fostering high performance, this approach incurs huge bandwidth consumption due to the transmission of complete raw observations. Late collaboration ensures efficient economic bandwidth consumption; however, it suffers from performance bottlenecks and reduced robustness due to scarce object information. Intermediate collaboration compresses representative information into BEV feature maps, resulting in reduced communication bandwidth compared to early collaboration, while also enhancing perception capabilities in comparison to late collaboration. Despite significant development, the manual projection of point clouds to BEV feature maps suffers from quantization error Shi et al. (2019) and creates a bottleneck in collaboration, hindering these methods from effectively completing 3D structural information for precise object boundary predictions.
+
+Considering the above issues, we innovatively propose a compact message unit, called point cluster, which describes objects with point coordinates representing the object structure, a cluster center representing the object position, and cluster feature representing the high-level semantics of the object. The point cluster has several merits compared to existing message units: 1) Unlike raw point clouds, point clusters intrinsically capture only the foreground objects within a scene in a sparse manner, leading to efficient communication; 2) In contrast to bounding boxes, point clusters offer sufficient information about potential objects, enabling more robust aggregation and decoding for enhanced regression accuracy; 3) Compared to BEV maps, point clusters explicitly preserve the structural information of objects in original coordinate space, enabling fine-grained point alignment and complementary structure information fusion between different agents.
+
+To fully unleash the potential of this powerful message unit, we propose CPPC, a comprehensive Collaborative Perception framework based on Point Cluster that revolutionizes existing communication mechanism, while featuring low bandwidth usage, high-performance perception, and real-noise robustness. The main body of our CPPC includes two key modules: 1) a point cluster packing module, which flexibly controls the number of points contained in each point cluster while maintaining their geometric structure to cope with different bandwidth constraints; 2) a point cluster aggregation module, which integrates point clusters from other agents with set matching and merging to achieve a comprehensive understanding of the surrounding scene for the ego agent. In addition to optimizing bandwidth and performance trade-off, our CPPC also has careful designs for two common robustness challenges, i.e., pose error Lu et al. (2023) and time delay Lei et al. (2022); Wei et al. (2024), benefited from the low-level information in point cluster. To explicitly solve the pose error, we propose to reformulate cluster centers as vertices of a graph and optimize to promote pose consistency between agents and point clusters. In order to compensate for the time latency, we link point clusters of the same object along the time dimension and measure its speed for position prediction in the current timestamp.
+
+As an intermediate collaboration method, using point cluster as message unit gives our CPPC three distinct advantages compared with previous methods based on BEV map: 1) Decoupling of structure and semantic information allows our CPPC to efficiently represent each potential object with a few key points and significantly fewer feature channels, all while avoiding the loss of high-level information during message packing. 2) The computational complexity of our CPPC during message aggregation is efficiently related to the number of possible objects in the scene, rather than quadratically related to the perception range, which is more suitable for long-range collaboration. 3) Our graph optimization and speed estimation methods possess a strong generalization capability as they do not rely on any training parameters during the optimization process. This enables our method to adapt to varying levels of noise with ease. To validate the effectiveness of our proposed CPPC, we conduct experiments on three popular collaborative perception benchmarks, V2X-Set Xu et al. (2022a), OPV2V Xu et al. (2022b), and DAIR-V2X Yu et al. (2022). Our method achieves the new state-of-the-art performances with significant performance gains of  $5.7\%$ ,  $7.3\%$  and  $12.8\%$  on the strict mAP@0.7 metric, respectively. Not only that, we also contributes this community a series of new metrics to demonstrate the benefits of different message units brings across various collaboration phases in a fine-grained manner, where our CPPC still achieves the best performance.
+
+# 2 PROBLEM FORMULATION
+
+Suppose there are  $N_{\mathrm{agent}}$  agents present in the scene, whose observations and the ground truth annotation are denoted as  $\{\mathcal{X}^i\}_{i = 1}^N$  and  $\{\mathcal{Y}^i\}_{i = 1}^N$ , respectively. Collaborative perception aims to maximize the perception performance of all agents while taking into account the constraint on available
+
+![](images/a4448eb26a42865a5ed049e2b2d9b97673b3d43f40dd1ad6e1fbad5c11ec3d9f.jpg)  
+Figure 1: Illustration of using BEV map and point cluster as message units for intermediate collaboration. bandwidth  $\beta$ , which can be formulated as:
+
+$$
+\underset {\theta , \mathcal {M}} {\arg \max } \sum_ {i = 1} ^ {N} g \left(\Phi_ {\theta} \left(\mathcal {X} ^ {i; t}, \left\{\mathcal {M} ^ {j; t - \tau^ {j \rightarrow i; t}} \right\} _ {j = 1, j \neq i} ^ {N}\right), \mathcal {Y} ^ {i; t}\right), \text {s . t .} \sum_ {i = 1} ^ {N} \sum_ {j = 1, j \neq i} ^ {N} | \mathcal {M} ^ {j; t - \tau^ {j \rightarrow i; t}} | \leq \beta . \tag {1}
+$$
+
+Here,  $\theta$  denotes trainable parameters of the network  $\Phi$  and  $g(\cdot ,\cdot)$  is the evaluation metric.  $\mathcal{M}^{j;t - \tau^{j\rightarrow i;t}}$  is the message transmitted from the  $j$ -th agent at the timestamp  $t - \tau^{j\rightarrow i;t}$  and received by the  $i$ -th agent at the timestamp  $t$ , where  $\tau^{j\rightarrow i;t}$  is the transmission latency. It includes the  $j$ -th agent's extracted collaborative message units  $M^j$  and 6DoF pose  $\xi^j$ . Note that 1) there is no collaboration when  $\beta = 0$  and the objective reflects the single-agent perception performance; 2) when accounting for pose error, it is necessary to correct the pose  $\xi^j$  for feature alignment; 3) we will omit the superscripts  $t$  related to the timestamp in the following when  $\tau = 0$  for simplicity.
+
+Existing intermediate collaboration approaches employing dense BEV maps as basic message units have several limitations: 1) Object feature destruction during message packing (Figure 1 (a)). Channel compression Xu et al. (2022a) enables the sender to preserve spatially complete scene information, while the receiver may suffer from object feature degradation during reconstruction due to heterogeneity representation across channels. In contrast, spatial selection Hu et al. (2022); Wang et al. (2023b) transmits only informative regions pointed out by spatial confidence maps, which may result in potential object loss when bandwidth constraints become more stringent. 2) Inefficient message aggregation for long-range collaboration (Figure 1 (b)). Collaboration can bring a larger perception range to the ego agent, but computational complexity also grows quadratically with the expansion of dense BEV feature maps. Moreover, limited by the square map structure and convolution operation requirements, the received BEV features are inevitably filled to the same shape for aggregation with zero paddings, resulting in unnecessary calculation on non-overlapped areas. 3) Implicit structure representation communication (Figure 1 (c)). The voxelization operation sacrifices 3D geometric details in comparison to the raw point clouds. While aggregating BEV representations from different agents can enhance the response of potential object regions, the precision of predicted box boundaries may be constrained by incomplete object structure modeling.
+
+Our CPPC based on point clusters for communication can overcome these issues: 1) Object feature preservation during message packing. (Figure 1 (d)). Point clusters inherently contain only the information of foreground objects present in the scene, eliminating the need for filtering out irrelevant backgrounds by handcrafted rules. Moreover, we can control the transmission bandwidth by explicitly reducing the number of points, as opposed to implicitly compressing feature channels. 2) Range-irrelevant efficient message aggregation (Figure 1 (e)). The number of point clusters is more related to the number of objects in the scene rather than the collaboration range. Furthermore, point clusters can be conveniently associated with the same object and aggregated through set merging, without the need for padding to the same shape of joint field of view. 3) Explicit structure representation communication ((Figure 1 (f))). Point clusters fully preserve the geometric structural information of objects in the original coordinate space, enabling fine-grained point alignment and complementary structure information fusion between different agents, which can improve the precision of predictions.
+
+![](images/5725c14aaa5f9b8f226d54c469c355de75d058190a0738d2277d826a13b3ebb9.jpg)  
+Figure 2: Overview of our CPPC system, including a shared PCE module to extract point clusters from raw point clouds, a PCP module to selectively packing informative point clusters, and a PCA to complete object information. Pose error and time latency are revised before aggregation phase.
+
+# 3 METHOD
+
+To begin with, we will present an overview of our CPPC framework in §3.1. The we describe details about how we encode, pack, and aggregate point clusters in §3.3 and §3.4, respectively. Finally, we will address time latency and pose error problems to improve our method's robustness in §3.5.
+
+# 3.1 OVERVIEW
+
+Figure 2 illustrates the overall architecture of our proposed method. Initially, the raw point clouds of all agents are processed by a shared Point Cluster Encoder (PCE), which segments foreground points on the surface of objects and divides them into clusters based on the distance metric. From each of these point clusters, we extract and formulate the point coordinates, the center coordinates, and the cluster feature as corresponding intermediate representations. Then we propose a Point Cluster Packing (PCP) module to filter noisy background clusters and correct the involved points of foreground clusters via proposal generation. The reduction in bandwidth usage of the point cluster can be achieved by decreasing the number of included points. After receiving messages from other agents, we address the pose error and time latency with parameter-free approaches and then align the coordinate space of point clusters from multiple agents via pose transformation, which allows the ego agent to obtain an agent-cluster graph as the comprehensive scene representation from its own coordinate space. As for message aggregation, we propose a Point Cluster Aggregation (PCA) module, where point cluster matching is performed to find point clusters belonging to the same object and merge them into a new point cluster that contains complete object information with linear complexity. Finally, we refine cluster features of point clusters via point-based operators, based on which we output the final detection results.
+
+# 3.2 POINT CLUSTER ENCODER
+
+Since the original BEV representations for collaborative perception suffer from object feature destruction, inefficient message aggregation for long-range collaboration, and implicit structure representation communication, we propose a brand new representation called point cluster to address these issues. We adopt the encoder-decoder point cloud backbone Shi et al. (2020b) based on 3D sparse convolution and deconvolution as the sparse voxel feature extractor. To construct each point feature, we concatenate the voxel feature where the point is located and the corresponding offset from the point to the voxel center. These point features are then passed through a MLP for foreground segmentation. Since objects are naturally well-separated, 3D box annotations in autonomous driving scenes directly provide semantic masks for supervision Shi et al. (2019). We use focal loss Lin et al. (2017) for segmentation loss, denoted as  $\mathcal{L}_{\mathrm{seg}}$ . For foreground points, we use another MLP to predict their offsets to the corresponding object centers, which is supervised by L1 loss Ren et al. (2015), denoted as  $\mathcal{L}_{\mathrm{center}}$ . Next, we measure the distance among the predicted centers of foreground points, where two points belong to the same point cluster if their predicted centers' Euclidean distance is smaller than a certain threshold  $\epsilon_{\mathrm{point}}$ .
+
+After we can directly extract cluster features via point-based operators, such as PointNet Qi et al. (2017a), DGCNN Wang et al. (2019), Meta-Kernel Fan et al. (2021), and SIR Fan et al. (2022). We select SIR in this paper and stack  $L_{1}$  layers to encode all cluster features in parallel. To express more clearly, we take the processing process of the  $q$ -th cluster in the  $l$ -th SIR layer as an example. Concretely, assume there are  $N_{\mathrm{point}}^{q}$  foreground points, we denote the included point coordinates and features as  $\pmb{P}^{q} \in \mathbb{R}^{N_{\mathrm{point}}^{q} \times 3}$  and  $\pmb{F}_{\mathrm{point}}^{q;l} \in \mathbb{R}^{N_{\mathrm{point}}^{q} \times D}$ , respectively, where  $D$  is number of feature channels. We take the average coordinates of all predicted cluster centers as the cluster center,
+
+denoted as  $C^q \in \mathbb{R}^{1 \times 3}$ . The processing process of SIR can be formulated as:
+
+$$
+\widetilde {\boldsymbol {F}} _ {\text {p o i n t}} ^ {q; l} = \mathrm {M L P} \left(\left[ \boldsymbol {F} _ {\text {p o i n t}} ^ {q; l}; \boldsymbol {P} ^ {q} \ominus \boldsymbol {C} ^ {q} \right]\right), \boldsymbol {F} _ {\text {p o i n t}} ^ {k; l + 1} = \mathrm {M L P} \left(\left[ \widetilde {\boldsymbol {F}} _ {\text {p o i n t}} ^ {q; l}; \operatorname {m a x p o o l} \left(\widetilde {\boldsymbol {F}} _ {\text {p o i n t}} ^ {q; l}\right) \right]\right), \tag {2}
+$$
+
+where  $[\cdot ]$  denotes concatenation along the channel dimension,  $\ominus$  means applying subtraction on each point in  $P^q$ , and  $F_{\mathrm{point}}^{q;l + 1}\in \mathbb{R}^{N_{\mathrm{point}}^q\times D}$  is the processed point cluster feature. We concatenate  $\{\pmb {F}_{\mathrm{point}}^{q;l}\}_{l = 1}^{L_1}$  along the channel dimension, and apply linear transformation and max-pooling on it to obtain the final cluster feature  $F^q\in \mathbb{R}^{1\times D}$ .
+
+# 3.3 POINT CLUSTER PACKING
+
+The intermediate representations of point clusters extracted from PCE by the  $i$ -th agent can be formulated as:
+
+$$
+\boldsymbol {M} ^ {i} = \left\{\boldsymbol {m} ^ {i; q} \right\} _ {q = 1} ^ {N _ {\text {c l u s t e r}} ^ {i}} = \left\{\left(\boldsymbol {P} ^ {i; q}, \boldsymbol {C} ^ {i; q}, \boldsymbol {F} ^ {i; q}\right) \right\} _ {k = 1} ^ {N _ {\text {c l u s t e r}} ^ {i}}, \tag {3}
+$$
+
+where  $m^{i;q}$  is the representation of the  $q$ -th cluster,  $N_{\mathrm{cluster}}^i$  is the number of point clusters extracted by the  $i$ -th agent,  $P^{i;q}$  denotes the set of included point coordinates,  $C^{i;q}$  denotes the cluster center, and  $F^{i;q}$  denotes the cluster feature. However, during the point cluster grouping process, errors may occur, resulting in the loss of points on the object's surface and the inclusion of background distractors. In order to correct this issue, we propose to generate a proposal bounding box for each cluster. To achieve this, we feed the cluster features to two separate MLPs for proposal classification and regression. During the training phase, clusters are classified as positive if their predicted centers are located in the ground truth bounding boxes. We adopt L1 loss Ren et al. (2015) and focal loss Lin et al. (2017) as regression loss  $\mathcal{L}_{\mathrm{reg}}$  and classification loss  $\mathcal{L}_{\mathrm{cls}}$ , respectively. After proposal generation, we only retain point clusters with positive proposals. The point coordinates in these clusters are overridden with the coordinates of points within the positive proposals. Consider there are  $N_{\mathrm{cluster + }}^i$  positive clusters, we modify the formulation of  $M_i$  as following:
+
+$$
+\boldsymbol {M} ^ {i} = \left\{\boldsymbol {m} ^ {i; q} \right\} _ {q = 1} ^ {N _ {\text {c l u s t e r +}} ^ {i}} = \left\{\left(\boldsymbol {P} ^ {i; q}, \boldsymbol {C} ^ {i; q}, \boldsymbol {F} ^ {i; q}, \boldsymbol {B} ^ {i; q}\right) \right\} _ {q = 1} ^ {N _ {\text {c l u s t e r +}} ^ {i}}, \tag {4}
+$$
+
+where  $B^{i;q} = (\hat{x},\hat{y},\hat{z},\hat{h},\hat{w},\hat{l},\alpha ,\hat{c})$  includes the center coordinates, the size, the yaw angle and the confidence score of the proposal bounding box of the  $q$ -th cluster. The final message distributed can be formulated as  $\mathcal{M}^i = (M^i,\xi^i)$ , where  $\xi^i$  is the 6DoF pose used for pose transforming in the later aggregation phase.
+
+Optimizing the trade-off between perception performance and communication bandwidth is vital in collaborative perception. Unlike BEV maps, our point clusters are inherently sparse in the spatial dimensions. Therefore, the communication cost in our method is mainly on the point coordinates (i.e. thousands of 3-dimensional coordinates) rather than the object features (i.e. one 128-dimensional features). Considering that the geometric structure of the object can be represented by keypoints, we propose Semtanic and Distribution guided Farthest Point Sampling (SD-FPS) to compress the transmission data, which can effectively exclude raw points with ambiguous semantic features or redundant structural features. Specifically, not only considering the distance  $d_{\mathrm{point}}$  between points like naive farthest point sampling, but also we select keypoints based on the object's semantic confidence score  $s_{\mathrm{f}}$  and distribution density score  $s_{\mathrm{d}}$ . The semantic confidence score is derived from segmentation head in PCE, with a higher score indicating richer semantic information for distinguishing objects. The distribution density score for point  $p$  is inversely proportional to its density estimation:  $\frac{1}{|\mathcal{N}(p)|} \sum_{q \in \mathcal{N}(p)} K(p, q)$ , where  $\mathcal{N}(p)$  is a point set in the nearby area of  $p$ , and  $K(\cdot, \cdot)$  is a gaussian kernel function to measure the similarity between position of two points. Thus, a lower  $s_{\mathrm{d}}$  indicates removing it will not significantly affect the object's shape. We show the algorithm details of applying SD-FPS on the  $q$ -th cluster of the  $i$ -th agent in Appendix A.1.
+
+# 3.4 POINT CLUSTER AGGREGATION
+
+After message communication, we need to appropriately aggregate point clusters from surrounding agents to form a holistic perception. We take the aggregation process from the  $j$ -th agent to the  $i$ -th agent as an example, which can be extended to all agents easily. Firstly, we align the coordinate space of  $M^j$  to that of  $M^i$  through the transform matrix calculated from  $\xi^i$  and  $\xi^j$ . The transformed message units from the  $j$ -th agent are denoted as  $M^{j\rightarrow i}$ . Similar to the foreground point grouping process, we match  $M^i$  and  $M^{j\rightarrow i}$  based on their clusters' centers, where the  $q$ -th point cluster of the  $i$ -th agent  $m^{i;q}$  and the  $r$ -th point cluster of the  $j$ -th agent  $m^{j\rightarrow i;r}$  belong to the same object if
+
+their centers' distance  $\| C^{i;q} - C^{j\rightarrow i;r}\| _2$  is less than a predefined threshold  $\epsilon_{\mathrm{agg}}$ . After matching, we organize all point clusters as two disjoint sets  $M_{\mathrm{unique}}$  and  $M_{\mathrm{share}}$ . The set  $M_{\mathrm{unique}}$  comprises point clusters exclusively observed by a single agent, which do not need to be involved in the following aggregation process. Differently,  $M_{\mathrm{share}}$  is the set of tuples including point clusters belonging to the same object in the scene. We combine each tuple to form a novel point cluster that encompasses comprehensive low-level and high-level object information. In detail, assume that  $\pmb{m}^{i;q}$  and  $\pmb{m}^{j\rightarrow i;r}$  belong to the  $s$ -th object in the scene, the aggregated point cluster  $\ddot{m}^s = (\ddot{P}^s,\ddot{C}^s,\ddot{F}^s,\ddot{B}^s)$  can be formulated as follows:
+
+$$
+\ddot {\boldsymbol {P}} ^ {s} = \boldsymbol {P} ^ {i; q} \cup \boldsymbol {P} ^ {j \rightarrow i; r}, \quad \ddot {\boldsymbol {C}} ^ {s} = \frac {\boldsymbol {C} ^ {i ; q} + \boldsymbol {C} ^ {j \rightarrow i ; r}}{2}, \tag {5}
+$$
+
+$$
+\ddot {\boldsymbol {F}} ^ {s} = \operatorname {a v g p o o l} \left(\boldsymbol {F} ^ {i; q}, \boldsymbol {F} ^ {j \rightarrow i; r}\right), \quad \ddot {\boldsymbol {B}} ^ {s} = \left\{\begin{array}{l l}\boldsymbol {B} ^ {i; q},&\text {i f} \hat {\boldsymbol {c}} ^ {i; q} - \hat {\boldsymbol {c}} ^ {j \rightarrow i; r} \geq 0\\\boldsymbol {B} ^ {j \rightarrow i; r},&\text {o t h e r w i s e}\end{array}, \right. \tag {6}
+$$
+
+Since there are no convolution operations and unnecessary zero-padding, the computational complexity of aggregating point clusters in our PCA is only strongly related to the number of potential objects, which is more efficient for long-range collaboration than BEV-based aggregation methods.
+
+# 3.5 ROBUSTNESS
+
+In realistic communication settings, the presence of pose error and time latency is unavoidable and it leads to misalignment of point clouds that significantly affect the reliability of transferred information in collaborative perception. Benefiting from the low-level object information in point clusters, we propose parameter-free approaches that can generalize to different noise settings. For simplicity, we take the correction processes that happen between the  $i$ -th (ego) and  $j$ -th agents as an example.
+
+Pose Correction. To address this issue, we propose to align the clusters from different agents belonging to the same object. After receiving the message from the  $j$ -th agent, we first align the coordinate space of  $M^j$  to the  $i$ -th agent with relative pose  $\xi^{j\rightarrow i} = (\xi^i)^{-1}\circ \xi^j$ , where  $\circ$  means multiplying their homogeneous transformation matrices. We denote  $m^{i;q}$  and  $m^{j\rightarrow i;r}$  as point clusters belonging to the  $s$ -th unique object in the scene after spatial cluster matching with threshold  $\epsilon_{\mathrm{pose}}$ . In the following, we simplify each pose in 2D space. The pose of the  $s$ -th unique object is defined as  $\chi^s = \xi^i\circ (C^{i;q} + C^{j\rightarrow i;r}) / 2$ . Inspired by CoAlign Lu et al. (2023), we define the pose consistency error vector as  $e^{js} = C^{j;r}\circ ((\xi^j)^{-1}\circ \chi^s)$ , which is zero when there is no pose error. The overall optimization problem can be formatted as follows:
+
+$$
+\left\{\left(\chi^ {s}\right) ^ {\prime}, \left(\xi^ {j}\right) ^ {\prime} \right\} = \underset {\{\xi^ {j}, \chi^ {s} \}} {\arg \min } \sum_ {j = 1} ^ {N _ {\text {a g e n t}}} \sum_ {s = 1} ^ {N _ {\text {o b j e c t}}} \left(e ^ {j s}\right) ^ {T} e ^ {j s}. \tag {7}
+$$
+
+Different from existing BEV-based methods that need additional detection results for pose correction, we can directly utilize low-level information contained in the point cluster.
+
+Latency Compensation. SyncNet Lei et al. (2022) proposes to complete BEV maps at the current timestamp using locally stored historical BEV maps from other agents. Differently, we propose to directly predict the location of point clusters in the current timestamp via speed estimation based on the low-level coordinate information, which is more efficient and interpretable. We denote the received point clusters from the  $j$ -th agent at the  $t$ -th timestamp as  $M^{j; t - \tau^{j \rightarrow i}; t}$ , where  $\tau^{j \rightarrow i; t}$  is the time latency. The stored point clusters from the  $j$ -th agent during the last communication round are denoted as  $M^{j; t'}$ , where  $t' < t - \tau^{j \rightarrow i; t}$ . Similar to the spatial matching process in 3.4, we match point clusters along the time dimension by measuring whether their Euclidean distance is in a predefined range  $[\underline{\epsilon}_{\mathrm{latency}}, \overline{\epsilon}_{\mathrm{latency}}]$ . Assume  $m^{j; r; t'}$  and  $m^{j; q; t - \tau^{j \rightarrow i; t}}$  belong to the same object, we can infer its speed  $v^{j; q; t - \tau^{j \rightarrow i; t}}$  and offset  $\Delta d_{\mathrm{cluster}}^{j; q; t - \tau^{j \rightarrow i; t}}$  to the current timestamp  $t$  as follows:
+
+$$
+\boldsymbol {v} ^ {j; q; t - \tau^ {j \rightarrow i; t}} = \frac {\left\| C ^ {j ; q ; t - \tau^ {j \rightarrow i ; t}} - C ^ {j ; r ; t ^ {\prime}} \right\| _ {2}}{t - \tau^ {j \rightarrow i ; t} - t ^ {\prime}}, \Delta d _ {\text {c l u s t e r}} ^ {j; q; t - \tau^ {j \rightarrow i; t}} = \boldsymbol {v} ^ {j; q; t - \tau^ {j \rightarrow i; t}} \times \tau^ {j \rightarrow i; t}. \tag {8}
+$$
+
+Finally, we can obtain the estimated point cluster  $\pmb{m}^{j;q;t}$  at the  $t$ -th timestamp by coordinates translation with  $\Delta d_{\text{cluster}}^{j;q;t - \tau^{j \to i:t}}$ .
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASETS AND EVALUATION METRICS
+
+We conducted experiments on three widely used benchmarks for collaborative perception, i.e., V2XSet Xu et al. (2022a), OPV2V Xu et al. (2022b), and DAIR-V2X-C Yu et al. (2022). We follow previous works Hu et al. (2022); Xu et al. (2022a) to select one of the agents in the scene as the ego agent, whose detection results are assessed by Average Precision (AP) at Intersection-over-Union (IoU) thresholds of 0.5 and 0.7, denoted as AP@0.5 and AP@0.7. Nevertheless, they treat all elements in the scenario uniformly, disregarding the input of collaborative participants. To compare the collaborative perception ability of different methods in a more fine-grained manner, we calculate the number of points observed by the ego agent  $N_{\mathrm{ego}}$  in all target objects and categorize them as: 1) Single-agent Perception of Other agents (SP-O), which signifies objects that are scarcely perceived by the ego agent; 2) Single-agent Perception of Ego agent (SP-E), which denotes objects that are more effectively scanned by the individual agent, with additional information provided by other agents primarily serving as supplementary; 3) Collaborative Perception (CP), which indicates objects that are partially observable by the individual agent and require assistance from other agents to obtain comprehensive information about the objects. We evaluate the AP@0.7 of each group, denoted as APSP-O, APCP, and APSP-E, respectively.
+
+# 4.2 IMPLEMENTATION DETAILS
+
+We set the perception range along the  $x$ ,  $y$ , and  $z$ -axis to  $[-140.8m, 140.8m] \times [-40m, 40m] \times [-3m, 1m]$  for V2XSet and  $[-100.8m, 100.8m] \times [-40m, 40m] \times [-3m, 1m]$  for DAIR-V2X-C, respectively. The communication results measure message size in bytes using a logarithmic scale with base 2. The thresholds  $\epsilon_{\mathrm{agg}}$ ,  $\epsilon_{\mathrm{pose}}$ ,  $\epsilon_{\mathrm{latency}}$ , and  $\bar{\epsilon}_{\mathrm{latency}}$  for cluster matching are set as 0.6, 1.5, 0.5 and 2.0, respectively. The number of SIR layers is  $L_1 = 6$  in PCE and  $L_2 = 3$  during message decoding. The channel number of cluster features is  $D = 128$ . Adam Kingma & Ba (2014) is employed as the optimizer for training our model end-to-end on NVIDIA Tesla V100 GPUs, with a total of 35 epochs. The initial learning rate is set as 0.001 and we reduce it by 10 after 20 and 30 epochs, respectively. Our method is implemented with PyTorch.
+
+Table 1: Comparison with state-of-the-art methods on the test sets of V2XSet, OPV2V, and DAIR-V2X-C on perfect setting. The second highest performance accuracy is highlighted in blue in the table.  
+Table 2: Comparison with state-of-the-art methods on the test set of DAIR-V2X-C with different target categories.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">V2XSet</td><td colspan="2">OPV2V</td><td colspan="2">DAIR-V2X-C</td></tr><tr><td>AP@0.5</td><td>AP@0.7</td><td>AP@0.5</td><td>AP@0.7</td><td>AP@0.5</td><td>AP@0.7</td></tr><tr><td>DiscoNet Li et al. (2021a)</td><td>90.78</td><td>83.81</td><td>86.2</td><td>73.3</td><td>69.28</td><td>58.56</td></tr><tr><td>V2X-ViT Xu et al. (2022a)</td><td>89.03</td><td>79.02</td><td>91.4</td><td>82.3</td><td>73.98</td><td>61.50</td></tr><tr><td>Where2comm Hu et al. (2022)</td><td>85.03</td><td>76.77</td><td>-</td><td>-</td><td>71.48</td><td>60.36</td></tr><tr><td>OPV2V Xu et al. (2022b)</td><td>91.88</td><td>84.75</td><td>89.9</td><td>76.5</td><td>66.07</td><td>50.92</td></tr><tr><td>CoBEVT Xu et al. (2023a)</td><td>90.33</td><td>82.69</td><td>-</td><td>-</td><td>63.90</td><td>51.67</td></tr><tr><td>CoAlign Lu et al. (2023)</td><td>-</td><td>-</td><td>94.5</td><td>86.8</td><td>74.60</td><td>60.40</td></tr><tr><td>Ours</td><td>92.83</td><td>89.55</td><td>94.6</td><td>93.1</td><td>76.89</td><td>69.39</td></tr></table>
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Metric</td></tr><tr><td>APSP-O</td><td>APCP</td><td>APSP-E</td></tr><tr><td>Where2comm</td><td>38.54</td><td>73.17</td><td>67.18</td></tr><tr><td>V2X-ViT</td><td>36.85</td><td>74.94</td><td>67.27</td></tr><tr><td>Ours</td><td>40.13</td><td>82.63</td><td>76.72</td></tr></table>
+
+# 4.3 COMPARISON WITH STATE-OF-THE-ART METHODS
+
+![](images/ec14e5793425f1300de33abd5a2198577a8c76c6c0731c90364ab228afbae177.jpg)  
+Figure 3: Comparison with state-of-the art methods considering (a) performance-bandwidth trade-off, (b) time latency, (c) positional error, and (d) heading error.
+
+As shown in Table 1, our CPPC outperforms previous BEV-based approaches on the test sets of all evaluated benchmarks, indicating the effectiveness of adopting point cluster as the basic collaborative message unit in both simulated and realistic scenarios. Our CPPC outperforms other methods significantly in the most stringent metric AP@0.7, and is  $5.7\%$ ,  $7.3\%$ , and  $12.8\%$  higher than the previous best OPV2V, CoAlign, and V2X-ViT on V2XSet, OPV2V, and DAIR-V2X-C, respectively, demonstrating its ability to accurately locate the correct object through collaborative perception and regress a more precise bounding box to cover the object. In addition, we compare our point cluster
+
+aggregation method with late collaboration, which directly use  $\dot{B}^s$  for point clusters matching the same potential objects, i.e., in  $M_{\mathrm{share}}$ , and  $B$  for objects exclusively observed by a single agent, i.e., in  $M_{\mathrm{unique}}$ , as the final outputs, respectively. Our method improves late collaboration by  $15.0\%$  and  $68.2\%$  on V2X-Set and DAIR-V2X on AP@0.7. This finding confirms that our PCA module can effectively utilize rich object information—specifically, semantic features and structural information—in point clusters received from surrounding agents, enabling the discovery of objects that cannot be accurately detected by individual agents.
+
+We explore the performance-bandwidth trade-off of our CPPC and previous typical methods in Figure 3 (a). Thanks to the sparse nature of point clusters, the communication volume is under 16 even if we pack all cluster points, which is close to the lower bound of bandwidth usage of other methods. Under the same communication volume, our method achieves significant performance improvements of around absolute  $10.0\mathrm{AP}@0.7$ . The little performance drops with bandwidth decrease indicate that appropriate point sampling strategies can reduce representation redundancy.
+
+Time latency poses a pervasive challenge in real-world V2X communication, leading to asynchronzation between ego features and received collaborative features. We compare the model robustness against time latency ranging from 0 to  $500ms$  in Figure 3 (b). Compared to previous intermediate methods, our CPPC achieves superior performance, with similar AP degradation as latency increases. It is worth noting that both V2X-ViT and Where2comm need to be finetuned with data on different noisy levels, while our CPPC can adapt to arbitrary noise levels in a zero-shot manner.
+
+Collaborative agents depend on precise pose data from others to transform coordinates of received messages. Despite advanced localization technologies like GPS, pose error is unavoidable. Therefore, collaborative approaches need to be resilient to localization errors. We compare different methods with pose noises following Gaussian distribution with standard deviations from  $\{0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6\}$  for positional error  $(m)$  (Figure 3 (c)) and heading error  $(^{\circ})$  (Figure 3 (d)), respectively. Results show that there is no significant AP drop, which validates that our CPPC can handle large noise disturbances without further finetuning and additional model parameters.
+
+The existing evaluation metrics in collaborative perception treat all objects equally, which is not conducive to fine-grained analysis. As shown in Table 2, we split targets into different categories based on the number of points scanned by the ego agent and evaluate AP@0.7 for each
+
+Table 3: Different numbers of feature channels in PCE.  
+
+<table><tr><td rowspan="2">Metric</td><td colspan="6">Channel Number</td></tr><tr><td>8</td><td>16</td><td>32</td><td>64</td><td>96</td><td>128</td></tr><tr><td>AP@50</td><td>74.41</td><td>75.32</td><td>77.12</td><td>76.63</td><td>77.40</td><td>77.30</td></tr><tr><td>AP@70</td><td>64.79</td><td>67.08</td><td>69.32</td><td>68.92</td><td>69.55</td><td>69.43</td></tr></table>
+
+category, denoted as  $\mathrm{AP_{SP - O}}$ ,  $\mathrm{AP_{CP}}$ , and  $\mathrm{AP_{SP - E}}$ . Experimental results demonstrate the superiority of our CPPC over the state-of-the-art BEV-based approaches across all evaluation metrics. In detail, high  $\mathrm{AP_{SP - O}}$  means that our PCP module can keep more complete object information since we avoid feature destruction caused by channel compression and spatial selection. The results in the 2-nd column demonstrate the significant improvement of our CPPC over BEV-based methods in terms of the  $\mathrm{AP_{CP}}$  metric. This indicates that utilizing point clusters as the basic collaborative message unit is advantageous for both message packing and aggregation phases, resulting in enhanced collaboration compared to earlier techniques relying on dense BEV maps.
+
+# 4.4 ABLATION STUDIES
+
+Number of Feature Channels in PCE. We assess AP@50 and AP@70 using various channel numbers of cluster features in PCE on the test set of DAIR-V2X-C in Table 3. The results indicate that there is no notable decrease in performance when the number of channels is reduced to 16. We argued that since we explicitly
+
+Table 4: Different sampling ratios and methods.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="6">Ratio</td></tr><tr><td>1/128</td><td>1/64</td><td>1/32</td><td>1/16</td><td>1/8</td><td>1/4</td></tr><tr><td>RPS</td><td>64.60</td><td>65.46</td><td>67.12</td><td>67.80</td><td>68.61</td><td>68.75</td></tr><tr><td>FPS</td><td>65.39</td><td>66.32</td><td>67.43</td><td>68.52</td><td>69.14</td><td>69.36</td></tr><tr><td>S-FPS</td><td>66.21</td><td>67.08</td><td>68.04</td><td>69.01</td><td>69.05</td><td>68.96</td></tr><tr><td>D-FPS</td><td>65.65</td><td>66.52</td><td>67.51</td><td>68.57</td><td>69.21</td><td>69.46</td></tr><tr><td>SD-FPS</td><td>66.12</td><td>67.03</td><td>68.08</td><td>69.22</td><td>69.19</td><td>69.41</td></tr></table>
+
+include structure representation in point clusters, the semantic information can be compressed to a large extent, leading to small bandwidth consumption. In contrast to prior approaches that compress solely before the message packing phase, our framework incorporates a small number of channels throughout the entire encoding phase, thereby decreasing network size and computational overhead without destroying object features across channels (3-rd and 4-st rows).
+
+Sampling Point Clusters with Different Ratios and Methods. We evaluate AP@70 of different sampling ratios and methods during message packing on the test set of DAIR-V2X-C in Table 4. The ratio is the number of sample points divided by the total number of points. The first and second
+
+Table 5: Ablation experiments on  ${\epsilon }_{\text{point }},{\epsilon }_{\text{agg }},{\epsilon }_{\text{pose }}$  ,and  ${\bar{\epsilon }}_{\text{latency }}$  .  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">0.1</td><td rowspan="2">0.2</td><td colspan="4">εpoint</td><td rowspan="2">0.3</td><td rowspan="2">0.4</td><td rowspan="2" colspan="4">εagg</td><td rowspan="2">1.0</td><td colspan="3">εpose</td><td>1.5</td><td colspan="3">εlatency</td></tr><tr><td>0.3</td><td>0.4</td><td>0.5</td><td></td><td>1.5</td><td>2.0</td><td></td><td>2.5</td><td></td><td></td><td></td></tr><tr><td>AP@0.5</td><td>91.94</td><td>92.01</td><td>91.92</td><td>91.75</td><td>91.47</td><td>91.97</td><td>91.95</td><td>91.99</td><td>92.01</td><td>91.97</td><td>86.63</td><td>90.00</td><td>89.80</td><td>72.50</td><td>72.69</td><td>72.68</td><td></td><td></td><td></td><td></td></tr><tr><td>AP@0.7</td><td>89.88</td><td>89.99</td><td>89.78</td><td>89.73</td><td>89.51</td><td>89.79</td><td>89.86</td><td>89.88</td><td>89.99</td><td>89.95</td><td>86.33</td><td>87.37</td><td>86.93</td><td>64.44</td><td>64.61</td><td>64.59</td><td></td><td></td><td></td><td></td></tr><tr><td colspan="20">rows represent the baseline methods of random sampling and basic FPS. The second, third, and fourth rows represent the results of introducing the semantic score, density score, and their joint application. It can be seen that the semantic score performs better than the baseline under smaller sampling ratios, because it preserves the semantic of object categories in extremely sparse structure. The density score performs better than the baseline under larger sampling ratios, because it can remove local redundant information. By combining the two scores, our method has significant performance advantages over the baseline at all sampling ratios. Moreover, SD-FPS does not incur significant time costs, less than 3% of time consumption in entire pipeline (Details in Appendix C.4). The reason is that our method applies SD-FPS to each point cluster, which contains limited object-level points (about average 102 points in the V2X-Set dataset) rather than large-scale scene-level points (about average 104 points in the V2X-Set dataset).</td><td></td></tr></table>
+
+Ablation on Hyperparameters We evaluate AP@0.5/AP@0.7 of different lower bounds  $\underline{\epsilon}_{\text{latency}}$  for matching during latency compensation in Table 6. In line with Figure 3 (b), an increase in time latency may result in performance degradation due to notable position shifts that complicate temporal alignment. If our method encounters relatively high time la
+
+Table 6: Lower bound for matching during latency compensation.  
+
+<table><tr><td rowspan="2">Latency</td><td colspan="5">Time Latency</td></tr><tr><td>100</td><td>200</td><td>300</td><td>400</td><td>500</td></tr><tr><td>0</td><td>76.36/67.97</td><td>75.23/66.08</td><td>74.13/64.61</td><td>73.19/63.57</td><td>71.84/62.79</td></tr><tr><td>0.1</td><td>76.34/67.93</td><td>75.20/66.09</td><td>74.15/64.60</td><td>73.20/63.53</td><td>71.82/62.80</td></tr><tr><td>0.2</td><td>76.34/67.88</td><td>75.21/66.06</td><td>74.13/64.59</td><td>73.17/63.46</td><td>72.37/64.00</td></tr><tr><td>0.3</td><td>76.34/67.86</td><td>75.19/66.00</td><td>74.14/64.60</td><td>73.19/63.50</td><td>72.72/64.46</td></tr><tr><td>0.4</td><td>76.42/68.01</td><td>75.26/66.05</td><td>74.11/64.58</td><td>73.50/64.42</td><td>72.76/64.47</td></tr><tr><td>0.5</td><td>76.44/68.01</td><td>75.35/65.54</td><td>74.36/65.18</td><td>73.79/64.82</td><td>72.75/64.48</td></tr></table>
+
+tency (e.g., 300ms, 400ms, and 500ms), performance significantly decreases when  $\epsilon_{\mathrm{latency}} = 0$  or is very low. Our research revealed that certain vehicles remain stationary, rendering the assumption of uniform motion in Section 3.5 invalid. By adjusting  $\underline{\epsilon}_{\mathrm{latency}}$  as demonstrated in the 3-rd, 4-th, and 5-th columns, we can mitigate this issue by filtering out stationary targets during latency compensation. Other involved hyperparameters i.e.,  $\epsilon_{\mathrm{point}}$ ,  $\epsilon_{\mathrm{agg}}$ ,  $\epsilon_{\mathrm{pose}}$ , and  $\bar{\epsilon}_{\mathrm{latency}}$ , in our CPPC are evaluated with different values in Table 5.
+
+As shown in Figure 4, we show qualitative evaluation of our CPPC w/ and w/o collaboration on the test sets of both V2XSet and DAIR-V2X-C datasets. We can observe that the absence of collaboration leads to the overlooking of many objects to the ego agent's inability to perceive sufficient information regarding long-range or obscured targets. These issue can be effectively addressed using our message packing and aggregation mechanisms based on point clusters.
+
+We also visualize the detection results before and after latency compensation on the test set of the DAIR-V2X-C dataset in Figure 5. The latency results in a delay of point clusters from road infrastructure (marked by yellow dots) compared to the point clusters from the ego car (marked by blue dots) at the current timestamp. Consequently, the ego car erroneously detects
+
+![](images/67c69e48a8f37d7fc02ff08573eb49b0c21e9904e707119acf08331f20dc7310.jpg)  
+Figure 4: Qualitative comparison results of our CPPC with and without collaboration on the test sets of (a) V2XSet and (b) DAIR-V2X-C, respectively. The green bounding boxes represent the ground-truth, and the red ones depict our predictions.
+
+them as distinct objects, resulting in an escalation of false-positive predictions. After refinement by our latency compensation module, the delayed point clusters can be adjusted to the correct positions, enhancing the detection results. Further, we illustrate the detection results before and after pose correction on the test set of the V2XSet dataset in Figure 6. By aligning clusters among all agents, we can correct the noisy poses and obtain bounding boxes with high precision.
+
+# 5 RELATED WORK
+
+# 5.1 COLLABORATIVE PERCEPTION
+
+Collaborative perception can be systematically classified into three primary types based on distinct fusion stages: early Chen et al. (2019b); Arnold et al. (2020), intermediate, and late Shi et al. (2022);
+
+![](images/8164cc62c59058293dfa002e81d9dca496aff82f0da0b92524fe048a25d24f84.jpg)
+
+![](images/b5f936eaca12c19fac202a91ee8380ea65aa56f5800ce088aafd068dc651fe74.jpg)
+
+![](images/e28142d95d8150bb0032acae3e1c6325835b97b9588de5ee3c1366867b5cd46a.jpg)
+
+![](images/99d6b32553f9bb4e8f93feb50f40bcb1c54a832a78b3f2faf4ded3384c6cd374.jpg)
+
+![](images/27bc54f441adb6e54f283b00a105b58f8fbf6a0d47f652d99e7e52aa9fdb3180.jpg)  
+Before Refinement
+
+![](images/97bc101a093ab8dfccd3349133591981108f13292923316cc7282b68a03dbc9d.jpg)  
+After Refinement
+
+![](images/56fac5312160c30432d7e1612afe9b38ce6e5eb1d4f07e916a1d04e2700ca623.jpg)  
+Before Refinement  
+Figure 5: Qualitative comparison results of our Figure 6: Qualitative comparison results before and CPPC before and after latency compensation. after pose correction.
+
+![](images/9796d0071523485a876c0e8bffecc733921cecd447b3f2788e12a53896d8125e.jpg)  
+After Refinement
+
+Zeng et al. (2020); Rauch et al. (2012); Glaser & Kira (2023) collaboration. We focus on intermediate collaboration in this paper, which enables the exchange of intermediate features created by the involved agents and has demonstrated significant potential in recent years. Considering the bandwidth constraints in practical application scenarios, various cooperation strategies are proposed to decide who Liu et al. (2020b), when Liu et al. (2020a), and where Hu et al. (2022); Wang et al. (2023b) to communicate. After receiving features from other agents, existing methods adopt attention mechanism Xu et al. (2022b;a; 2023a); Zhang et al. (2022a); Wang et al. (2023a); Yang et al. (2023), graph neural network Wang et al. (2020); Li et al. (2021a), maxout Bai et al. (2022); Guo et al. (2021); Qiao & Zulkernine (2023), and addition Marvasti et al. (2020) to aggregate complementary scene information. This paper proposes a novel collaborative message unit named point cluster and demonstrates its superiority over the BEV map for collaborative perception.
+
+# 5.2 SPARSE DETECTORS
+
+To address quantization errors caused by voxelization, point-based detectors Shi et al. (2019); Yang et al. (2020); Qi et al. (2019); Zhang et al. (2022b); Qi et al. (2017b); Fan et al. (2022); Chen et al. (2023); Huang et al. (2023); Fu et al. (2024) have emerged as a popular research topic. PointR-CNN Shi et al. (2019) is recognized as groundbreaking research in the advancement of this line of work. Taking inspiration from Hough voting, VoteNet Qi et al. (2019) initially casts votes for object centroids and subsequently generates high-quality proposals based on the voted center. FSD Fan et al. (2022) is the pioneering fully sparse 3D object detector, which treats instances as groups and gets rid of the dependence on the neighborhood query. In this paper, we study the shortcomings of BEV-based collaboration methods and propose the point cluster as the collaborative message unit, which keeps the low-level structure information to support effective and efficient collaboration.
+
+# 6 CONCLUSION AND DISCUSSION
+
+In this paper, we concentrate on the multi-agent collaborative perception task, illustrating that current approaches are hampered by the inherent limitations of existing message units leading to sub-optimal collaboration. To handle this issue, we create a brand new message unit for collaborative perception, namely point cluster, and based on this we further present a novel collaborative framework CPPC. The core idea of CPPC involves representing scenes through object-level point clusters, which are sparse and encompass comprehensive information about objects. These clusters can be compressed efficiently without losing geometric or high-level object information and can be integrated through set matching and merging. To deal with time latency and pose errors encountered in real environments, we align point clusters from spatial and temporal dimensions and propose parameter-free solutions for them. Extensive experiments on two collaborative perception benchmarks show our method outperforms previous state-of-the-art methods.
+
+Limitation and future work. By leveraging the low-level information within point clusters, we introduce parameter-free solutions to enhance the robustness of our approach. Despite good results, we may need to manually adjust hyperparameters when facing a new environment. In the future, we plan to address this issue using novel techniques specifically designed for adapting to environmental changes. In addition, we also plan to extend our CPPC to handle tasks considering temporal modeling, like tracking and forecasting Yu et al. (2023).
+
+# ACKNOWLEDGMENTS
+
+This research is supported in part by National Key R&D Program of China (2022ZD0115502), National Natural Science Foundation of China (NO. 62461160308, U23B2010), “Pioneer” and “Leading Goose” R&D Program of Zhejiang (No. 2024C01161).
+
+# REFERENCES
+
+Eduardo Arnold, Mehrdad Dianati, Robert de Temple, and Saber Fallah. Cooperative perception for 3d object detection in driving scenarios using infrastructure sensors. IEEE Transactions on Intelligent Transportation Systems, 23(3):1852-1864, 2020.  
+Zhengwei Bai, Guoyuan Wu, Matthew J Barth, Yongkang Liu, Emrah Akin Sisbot, and Kentaro Oguchi. Pillargrid: Deep learning-based cooperative perception for 3d object detection from onboard-roadside lidar. In 2022 IEEE 25th International Conference on Intelligent Transportation Systems (ITSC), pp. 1743-1749. IEEE, 2022.  
+Qi Chen, Xu Ma, Sihai Tang, Jingda Guo, Qing Yang, and Song Fu. F-cooper: Feature based cooperative perception for autonomous vehicle edge computing system using 3d point clouds. In Proceedings of the 4th ACM/IEEE Symposium on Edge Computing, pp. 88-100, 2019a.  
+Qi Chen, Sihai Tang, Qing Yang, and Song Fu. Cooper: Cooperative perception for connected autonomous vehicles based on 3d point clouds. In 2019 IEEE 39th International Conference on Distributed Computing Systems (ICDCS), pp. 514-524. IEEE, 2019b.  
+Yukang Chen, Jianhui Liu, Xiangyu Zhang, Xiaojuan Qi, and Jiaya Jia. Voxelnet: Fully sparse voxelnet for 3d object detection and tracking. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 21674-21683, 2023.  
+Bowen Cheng, Ishan Misra, Alexander G Schwing, Alexander Kirillov, and Rohit Girdhar. Masked-attention mask transformer for universal image segmentation. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 1290–1299, 2022.  
+Alexey Dosovitskiy, German Ros, Felipe Codevilla, Antonio Lopez, and Vladlen Koltun. Carla: An open urban driving simulator. In Conference on robot learning, pp. 1-16. PMLR, 2017.  
+Lue Fan, Xuan Xiong, Feng Wang, Naiyan Wang, and Zhaoxiang Zhang. Rangedet: In defense of range view for lidar-based 3d object detection. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 2918-2927, 2021.  
+Lue Fan, Feng Wang, Naiyan Wang, and ZHAO-XIANG ZHANG. Fully sparse 3d object detection. Advances in Neural Information Processing Systems, 35:351-363, 2022.  
+Jiahui Fu, Chen Gao, Zitian Wang, Lirong Yang, Xiaofei Wang, Beipeng Mu, and Si Liu. Eliminating cross-modal conflicts in bev space for lidar-camera 3d object detection. In ICRA, 2024.  
+Nathaniel Moore Glaser and Zsolt Kira. We need to talk: Identifying and overcoming communication-critical scenarios for self-driving. arXiv preprint arXiv:2305.04352, 2023.  
+Jingda Guo, Dominic Carrillo, Sihai Tang, Qi Chen, Qing Yang, Song Fu, Xi Wang, Nannan Wang, and Paparao Palacharla. Coff: Cooperative spatial feature fusion for 3-d object detection on autonomous vehicles. IEEE Internet of Things Journal, 8(14):11078-11087, 2021.  
+Kaiming He, Georgia Gkioxari, Piotr Dólar, and Ross Girshick. Mask r-cnn. In Proceedings of the IEEE international conference on computer vision, pp. 2961-2969, 2017.  
+Yue Hu, Shaoheng Fang, Zixing Lei, Yiqi Zhong, and Siheng Chen. Where2comm: Communication-efficient collaborative perception via spatial confidence maps. Advances in neural information processing systems, 35:4874-4886, 2022.  
+Zhe Huang, Yongcai Wang, Jie Wen, Peng Wang, and Xudong Cai. An object detection algorithm combining semantic and geometric information of the 3d point cloud. Advanced Engineering Informatics, 56:101971, 2023.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.  
+Zixing Lei, Shunli Ren, Yue Hu, Wenjun Zhang, and Siheng Chen. Latency-aware collaborative perception. In European Conference on Computer Vision, pp. 316-332. Springer, 2022.  
+Yiming Li, Shunli Ren, Pengxiang Wu, Siheng Chen, Chen Feng, and Wenjun Zhang. Learning distilled collaboration graph for multi-agent perception. Advances in Neural Information Processing Systems, 34:29541-29552, 2021a.  
+Yiming Li, Dekun Ma, Ziyan An, Zixun Wang, Yiqi Zhong, Siheng Chen, and Chen Feng. V2xsim: Multi-agent collaborative perception dataset and benchmark for autonomous driving. IEEE Robotics and Automation Letters, 7(4):10914-10921, 2022.  
+Zhi Li, Ali Vatankhah Barenji, Jiazhi Jiang, Ray Y Zhong, and Gangyan Xu. A mechanism for scheduling multi robot intelligent warehouse system face with dynamic demand. Journal of Intelligent Manufacturing, 31:469-480, 2020.  
+Zhichao Li, Feng Wang, and Naiyan Wang. Lidar r-cnn: An efficient and universal 3d object detector. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 7546-7555, 2021b.  
+Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dólar. Focal loss for dense object detection. In Proceedings of the IEEE international conference on computer vision, pp. 2980-2988, 2017.  
+Yen-Cheng Liu, Junjiao Tian, Nathaniel Glaser, and Zsolt Kira. When2com: Multi-agent perception via communication graph grouping. In Proceedings of the IEEE/CVF Conference on computer vision and pattern recognition, pp. 4106-4115, 2020a.  
+Yen-Cheng Liu, Junjiao Tian, Chih-Yao Ma, Nathan Glaser, Chia-Wen Kuo, and Zsolt Kira. Who2com: Collaborative perception via learnable handshake communication. In 2020 IEEE International Conference on Robotics and Automation (ICRA), pp. 6876-6883. IEEE, 2020b.  
+Yifan Lu, Quanhao Li, Baoan Liu, Mehrdad Dianati, Chen Feng, Siheng Chen, and Yanfeng Wang. Robust collaborative 3d object detection in presence of pose errors. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pp. 4812-4818. IEEE, 2023.  
+Ehsan Emad Marvasti, Arash Raftari, Amir Emad Marvasti, Yaser P Fallah, Rui Guo, and Hongsheng Lu. Cooperative lidar object detection via feature sharing in deep networks. In 2020 IEEE 92nd Vehicular Technology Conference (VTC2020-Fall), pp. 1-7. IEEE, 2020.  
+Charles R Qi, Hao Su, Kaichun Mo, and Leonidas J Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 652-660, 2017a.  
+Charles R Qi, Or LITany, Kaiming He, and Leonidas J Guibas. Deep hough voting for 3d object detection in point clouds. In proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 9277-9286, 2019.  
+Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J Guibas. Pointnet++: Deep hierarchical feature learning on point sets in a metric space. Advances in neural information processing systems, 30, 2017b.  
+Donghao Qiao and Farhana Zulkernine. Adaptive feature fusion for cooperative perception using lidar point clouds. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pp. 1186-1195, 2023.  
+Andreas Rauch, Felix Klanner, Ralph Rasshofer, and Klaus Dietmayer. Car2x-based perception in a high-level fusion architecture for cooperative perception systems. In 2012 IEEE Intelligent Vehicles Symposium, pp. 270-275. IEEE, 2012.
+
+Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. Advances in neural information processing systems, 28, 2015.  
+Jürgen Scherer, Saeed Yahyanejad, Samira Hayat, Evsen Yanmaz, Torsten Andre, Asif Khan, Vladimir Vukadinovic, Christian Bettstetter, Hermann Hellwagner, and Bernhard Rinner. An autonomous multi-uav system for search and rescue. In Proceedings of the first workshop on micro aerial vehicle networks, systems, and applications for civilian use, pp. 33-38, 2015.  
+Shaoshuai Shi, Xiaogang Wang, and Hongsheng Li. Pointcnn: 3d object proposal generation and detection from point cloud. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 770-779, 2019.  
+Shaoshuai Shi, Chaoxu Guo, Li Jiang, Zhe Wang, Jianping Shi, Xiaogang Wang, and Hongsheng Li. Pv-rcnn: Point-voxel feature set abstraction for 3d object detection. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10529–10538, 2020a.  
+Shaoshuai Shi, Zhe Wang, Jianping Shi, Xiaogang Wang, and Hongsheng Li. From points to parts: 3d object detection from point cloud with part-aware and part-aggregation network. IEEE transactions on pattern analysis and machine intelligence, 43(8):2647-2664, 2020b.  
+Shuyao Shi, Jiahe Cui, Zhehao Jiang, Zhenyu Yan, Guoliang Xing, Jianwei Niu, and Zhenchao Ouyang. Vips: Real-time perception fusion for infrastructure-assisted autonomous driving. In Proceedings of the 28th Annual International Conference on Mobile Computing And Networking, pp. 133-146, 2022.  
+Binglu Wang, Lei Zhang, Zhaozhong Wang, Yongqiang Zhao, and Tianfei Zhou. Core: Cooperative reconstruction for multi-agent perception. arXiv preprint arXiv:2307.11514, 2023a.  
+Tianhang Wang, Guang Chen, Kai Chen, Zhengfa Liu, Bo Zhang, Alois Knoll, and Changjun Jiang. Umc: A unified bandwidth-efficient and multi-resolution based collaborative perception framework. arXiv preprint arXiv:2303.12400, 2023b.  
+Tsun-Hsuan Wang, Sivabalan Manivasagam, Ming Liang, Bin Yang, Wenyuan Zeng, and Raquel Urtasun. V2vnet: Vehicle-to-vehicle communication for joint perception and prediction. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part II 16, pp. 605–621. Springer, 2020.  
+Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E Sarma, Michael M Bronstein, and Justin M Solomon. Dynamic graph cnn for learning on point clouds. ACM Transactions on Graphics (tog), 38(5): 1-12, 2019.  
+Sizhe Wei, Yuxi Wei, Yue Hu, Yifan Lu, Yiqi Zhong, Siheng Chen, and Ya Zhang. Asynchrony-robust collaborative perception via bird's eye view flow. Advances in Neural Information Processing Systems, 36, 2024.  
+Runsheng Xu, Yi Guo, Xu Han, Xin Xia, Hao Xiang, and Jiaqi Ma. Opencda: an open cooperative driving automation framework integrated with co-simulation. In 2021 IEEE International Intelligent Transportation Systems Conference (ITSC), pp. 1155-1162. IEEE, 2021.  
+Runsheng Xu, Hao Xiang, Zhengzhong Tu, Xin Xia, Ming-Hsuan Yang, and Jiaqi Ma. V2x-vit: Vehicle-to-everything cooperative perception with vision transformer. In European conference on computer vision, pp. 107-124. Springer, 2022a.  
+Runsheng Xu, Hao Xiang, Xin Xia, Xu Han, Jinlong Li, and Jiaqi Ma. Opv2v: An open benchmark dataset and fusion pipeline for perception with vehicle-to-vehicle communication. In 2022 International Conference on Robotics and Automation (ICRA), pp. 2583-2589. IEEE, 2022b.  
+Runsheng Xu, Zhengzhong Tu, Hao Xiang, Wei Shao, Bolei Zhou, and Jiaqi Ma. Cobevt: Cooperative bird's eye view semantic segmentation with sparse transformers. In Conference on Robot Learning, pp. 989-1000. PMLR, 2023a.
+
+Runsheng Xu, Xin Xia, Jinlong Li, Hanzhao Li, Shuo Zhang, Zhengzhong Tu, Zonglin Meng, Hao Xiang, Xiaoyu Dong, Rui Song, et al. V2v4real: A real-world large-scale dataset for vehicle-to-vehicle cooperative perception. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13712–13722, 2023b.  
+Kun Yang, Dingkang Yang, Jingyu Zhang, Mingcheng Li, Yang Liu, Jing Liu, Hanqi Wang, Peng Sun, and Liang Song. Spatio-temporal domain awareness for multi-agent collaborative perception. arXiv preprint arXiv:2307.13929, 2023.  
+Zetong Yang, Yanan Sun, Shu Liu, and Jiaya Jia. 3dssd: Point-based 3d single stage object detector. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 11040-11048, 2020.  
+Haibao Yu, Yizhen Luo, Mao Shu, Yiyi Huo, Zebang Yang, Yifeng Shi, Zhenglong Guo, Hanyu Li, Xing Hu, Jirui Yuan, et al. Dair-v2x: A large-scale dataset for vehicle-infrastructure cooperative 3d object detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 21361-21370, 2022.  
+Haibao Yu, Wenxian Yang, Hongzhi Ruan, Zhenwei Yang, Yingjuan Tang, Xu Gao, Xin Hao, Yifeng Shi, Yifeng Pan, Ning Sun, et al. V2x-seq: A large-scale sequential dataset for vehicleinfrastructure cooperative perception and forecasting. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5486-5495, 2023.  
+Wenyuan Zeng, Shenlong Wang, Renjie Liao, Yun Chen, Bin Yang, and Raquel Urtasun. Dsdnet: Deep structured self-driving network. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part XXI 16, pp. 156-172. Springer, 2020.  
+Hui Zhang, Guiyang Luo, Yuzhouhan Cao, Yi Jin, and Yidong Li. Multi-modal virtual-real fusion based transformer for collaborative perception. In 2022 IEEE 13th International Symposium on Parallel Architectures, Algorithms and Programming (PAAP), pp. 1-6. IEEE, 2022a.  
+Yifan Zhang, Qingyong Hu, Guoquan Xu, Yanxin Ma, Jianwei Wan, and Yulan Guo. Not all points are equal: Learning highly efficient point-based detectors for 3d lidar point clouds. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 18953-18962, 2022b.
+
+# A METHOD DETAILS
+
+A.1 ALGORITHM: SEMTANIC AND DISTRIBUTION GUIDED FARthest POINT SAMPLING
+
+# A.2 POINT CLUSTER DECODING
+
+We denote the intermediate representation of point clusters in the scene after aggregation as  $\ddot{M} = \{\ddot{m}^s\}_{s=1}^{N_{\mathrm{object}}}$ , where  $N_{\mathrm{object}} = \| M_{\mathrm{unique}}\| + \| M_{\mathrm{share}}\|$  is the potential object number in the scene observed by all involved agents. Different from BEV-based message aggregation methods,  $\ddot{P}^s$  contains complete low-level structure information, which can be utilized to enhance the precision of the proposal bounding box  $\ddot{B}$ . In detail, we apply an additional SIR module that includes  $L_2$  layers on  $\ddot{M}$ , which predicts the box residual  $\Delta_{\mathrm{res}}$  to its corresponding ground truth box. For each point cluster  $\ddot{m}^s$ , we generate its point feature  $\ddot{F}_{\mathrm{point}}^s$  by concatenating its offsets from the cluster proposal  $\ddot{B}^s$  and the cluster feature  $\ddot{F}^s$ . This introduces the proposal boundary information to the SIR module, which can handle the size ambiguity problem to a certain extent Li et al. (2021b). We define the residual loss  $\mathcal{L}_{\mathrm{res}}$  as the L1 distance between  $\Delta_{\mathrm{res}}$  and the ground-truth residual  $\hat{\Delta}_{\mathrm{res}}$ . In addition, we define the soft classification label as  $\min(1, \max(0, 2u - 0.5))$  following previous works Shi et al. (2020a;b), where  $u$  is the 3D Intersection-of-Union (IoU) between the predicted proposal and the ground truth. We adopt cross-entropy loss as the IoU loss  $\mathcal{L}_{\mathrm{iou}}$ . Considering all losses in our framework, the total training loss  $\mathcal{L}$  can be formulated as:
+
+$$
+\mathcal {L} = \mathcal {L} _ {\text {s e g}} + \mathcal {L} _ {\text {c e n t e r}} + \mathcal {L} _ {\text {r e g}} + \mathcal {L} _ {\text {c l s}} + \mathcal {L} _ {\text {r e s}} + \mathcal {L} _ {\text {i o u}}. \tag {9}
+$$
+
+Algorithm 1 Semantic and Distribution guided Farthest Point Sampling Algorithm.  $N_{\mathrm{point}}$  is the number of input points and  $N_{\mathrm{sample}} = N_{\mathrm{point}} \times \zeta$  is the number of sampled points controlled by a predefined sampling rate  $\zeta$ .  
+Input: coordinates  $P = \{p^1,\dots ,p^{N_{\mathrm{fg}}}\} \in \mathbb{R}^{N_{\mathrm{point}}\times 3}$    
+semantic scores  $S_{\mathrm{f}} = \{s_{\mathrm{f}}^{1},\ldots ,s_{\mathrm{f}}^{N_{\mathrm{point}}}\} \in \mathbb{R}^{N_{\mathrm{point}}};$  distribution scores  $S_{\mathrm{d}} = \{s_{\mathrm{d}}^{1},\dots ,s_{\mathrm{d}}^{N_{\mathrm{point}}}\} \in \mathbb{R}^{N_{\mathrm{point}}}$    
+Output: sampled key point set  $\widetilde{\boldsymbol{P}} = \{\widetilde{\boldsymbol{p}}^1,\dots ,\widetilde{\boldsymbol{p}}^{N_{\mathrm{sample}}}\}$    
+1: initialize an empty sampling point set  $\widetilde{\boldsymbol{P}}$    
+2: initialize a distance array  $D_{\mathrm{point}}$  of length  $N_{\mathrm{point}}$  with all  $+\infty$    
+3: initialize a visit array  $\mathbf{V}$  of length  $N_{\mathrm{point}}$  with all zeros   
+4: for  $n = 1$  to  $N_{\mathrm{sample}}$  do   
+5: if  $n = 1$  then   
+6:  $o = \arg \max (\{(s_{\mathrm{f}}^{k})^{\lambda_{\mathrm{s}}} \cdot (s_{\mathrm{d}}^{k})^{\lambda_{\mathrm{d}}} | V^{k} = 0\}_{k=1}^{N_{\mathrm{point}}})$    
+7: else   
+8:  $\widetilde{D}_{\mathrm{point}} = \{(s_{\mathrm{f}}^{k})^{\lambda_{\mathrm{s}}} \cdot (s_{\mathrm{d}}^{k})^{\lambda_{\mathrm{d}}} \cdot d_{\mathrm{point}}^{k} | V^{k} = 0\}_{k=1}^{N_{\mathrm{point}}}$    
+9:  $o = \arg \max (\widetilde{D}_{\mathrm{point}})$    
+10: end if   
+11: add  $P^o$  to  $\widetilde{P},V^o = 1$    
+12: for  $u = 1$  to  $N$  do   
+13:  $d_{\mathrm{point}}^u = \min (d_{\mathrm{point}}^u, \| p^u -p^o \|)$    
+14: end for   
+15: end for   
+16: return  $\widetilde{P}$
+
+# B DATASETS
+
+DAIR-V2X-C Yu et al. (2022) is the first to provide a large-scale collection of real-world scenarios for vehicle-infrastructure collaborative autonomous driving. It contains 38,845 frames of point cloud data annotated with almost 464k 3D bounding boxes representing objects in 10 different classes. Since the original DAIR-V2X-C does not include objects beyond the camera's view, we have adopted the complemented annotations encompassing the 360-degree detection range, which are relabeled by Hu et al. Hu et al. (2022).
+
+V2XSet Xu et al. (2022a) is a large-scale V2X perception dataset founded on CARLA Dosovitskiy et al. (2017) and OpenCDA Xu et al. (2021), which explicitly takes into account real-world noises like localization error and transmission latency. V2XSet has 11,447 frames (6,694/1,920/2,833 for train/validation/test respectively) captured in 55 representative simulation scenes that cover the most common driving scenarios in real life. Each scene typically involves 2-7 agents engaged in collaborative perception.
+
+OPV2V Xu et al. (2022b) is a vehicle-to-vehicle collaborative perception dataset, cosimulated by OpenCDA Xu et al. (2021) and Carla Dosovitskiy et al. (2017), which includes 12K frames of 3D LiDAR point clouds and RGB images with 230K annotated 3D boxes.
+
+# C EXPERIMENTS
+
+# C.1 DATASET STATISTICS.
+
+# C.2 COMMUNICATION VOLUME.
+
+In real-world applications, collaborative perception methods must achieve a delicate equilibrium between communication volume and precision due to the typically limited and variable communication bandwidth. The communication volume is calculated as follows:
+
+$$
+\operatorname {C o m m} = \log_ {2} (N \times C \times 1 6 / 8), \tag {10}
+$$
+
+where  $N$  represents the number of collaborative message units,  $C$  represents the number of channels, and the data is transmitted in fp16 data type, resulting in minimal performance impact. The volume in bits is then converted to bytes using the logarithm base 2.
+
+![](images/30e2a3d5bda2263d193876a26ab7a5436162409b0e7750025fa09bcee1a48e17.jpg)  
+Figure 7: Illustration of histogram of all targets and proportions of those belonging to SP-O, SP-E, and CP categories in the test set of DAIR-V2X-C.
+
+# C.3 COMPUTATIONAL COMPLEXITY OF POINT CLUSTER AGGREGATION
+
+The computational complexity discussed primarily aims to demonstrate the efficiency of message aggregation based on sparse point clusters in comparison to dense BEV map representations. The computational complexity of message aggregation based on point clusters is solely related to the number of potential objects, which is significantly smaller than the scale of the quadratic relationship with the collaboration scope.
+
+Assume there are  $N_{\mathrm{agent}}$  agents in the scene, so each tuple in  $M_{\mathrm{share}}$  contains at most  $N_{\mathrm{agent}}$  matched point clusters. According to eq (5) and eq (6), the computational complexity of aggregating single tuple in  $M_{\mathrm{share}}$  is  $\mathcal{O}(\max_{1 \leq i \leq N_{\mathrm{agent}}} |P^i| + N_{\mathrm{agent}} + CN_{\mathrm{agent}} + N_{\mathrm{agent}})$ , where  $P^i$  is the set of points in point cluster and  $C$  is the channel number of cluster features. After point sampling during point cluster packing,  $|P^i|$  can be reduced significantly as shown in Table 4. And Table 3 shows that  $C$  can also be reduced to a small constant. Overall, the computational complexity of aggregating single tuple in  $M_{\mathrm{share}}$  can be reduced to  $\mathcal{O}(D)$ , where  $D$  is a constant. Assume there are  $N_{\mathrm{object}}$  possible objects in the scene, there are at most  $N_{\mathrm{object}}$  tuples in  $M_{\mathrm{share}}$ , while  $M_{\mathrm{unique}} = \emptyset$ . The computational complexity of aggregating all tuples is  $\mathcal{O}(DN_{\mathrm{object}})$ .
+
+# C.4 ADDITIONAL ABLATION STUDIES
+
+Table 7: Features for object pose calculation during pose correction. "0.1" label of each column means experiments with standard deviation 0.1 for both heading error  $(^{\circ})$  and positional error  $(m)$ .  
+
+<table><tr><td rowspan="2">Feature</td><td colspan="5">Pose Error</td></tr><tr><td>0.1</td><td>0.2</td><td>0.3</td><td>0.4</td><td>0.5</td></tr><tr><td>Point Center</td><td>75.13/45.44</td><td>76.89/46.59</td><td>78.50/48.20</td><td>79.60/51.03</td><td>78.90/52.07</td></tr><tr><td>Cluster Center</td><td>90.22/86.95</td><td>90.34/87.01</td><td>90.04/87.06</td><td>90.10/87.06</td><td>89.97/86.94</td></tr></table>
+
+Features for Object Pose Calculation during Pose Correction. We evaluate AP@0.5/AP@0.7 of different features for object pose calculation during pose correction in Table 7. "Point Center" denotes representing the object pose with the mean coordinates of cluster points. "Cluster Center" denotes representing the object pose with the estimated cluster center. The experiments demonstrate that utilizing the "Point Center" method for determining the object's pose correction results in notable performance deterioration. Due to Lidar typically scanning objects partially, directly representing a point cluster by averaging the coordinates of all cluster points can lead to a significant offset from the true object center.
+
+Filtering background information. We evaluate the recall of points belonging to target objects after segmentation. The results demonstrate that segmentation can recall more than  $90\%$  of target object points. This finding confirms that point clusters generated from segmentation results can effectively represent potential object parts, enabling the fusion module to complete object information utilizing information from all agents and achieve better performance.
+
+Table 8: Time cost (ms) analysis.  
+
+<table><tr><td rowspan="2">V2X-ViT</td><td rowspan="2">Total</td><td colspan="5">CPPC</td></tr><tr><td>PCE</td><td>PCP</td><td>PCA</td><td>Decoder</td><td>Others</td></tr><tr><td>153.31</td><td>79.52</td><td>53.71</td><td>3.21</td><td>9.69</td><td>12.70</td><td>0.21</td></tr></table>
+
+Table 9: Different point-based operators.  
+
+<table><tr><td></td><td>AP@0.5</td><td>AP@0.7</td></tr><tr><td>PointNet</td><td>89.67</td><td>85.82</td></tr><tr><td>DGCNN</td><td>91.39</td><td>87.68</td></tr><tr><td>Meta-Kernel</td><td>91.52</td><td>88.48</td></tr><tr><td>SIR</td><td>92.01</td><td>89.99</td></tr></table>
+
+Inference time cost analysis. We evaluate the time cost of our method's entire pipeline and compare it with one of the state-of-the-art methods in Table 8, i.e., V2X-ViT. Comparing the first and second columns, the results show that our method significantly reduces the time cost compared with previous collaborative sensing methods, indicating the efficiency of communication with our new message unit point cluster. Moreover, the collaborative modules (PCP and PCA) occupy a relatively small portion of our entire pipeline, which verifies collaboration based on point cluster can impose little additional burden on existing single-vehicle detectors.
+
+Different point-based operators. We evaluate AP@70 of different existing point-based operators on V2XSet in Table 9, where SIR achieves the best performance.
+
+# C.5 QUALITATIVE COMPARISON
+
+![](images/c6023969af7a82f31c271e4eb50176bf992f4f88ea50bc77ac2c8ee19fe37efc.jpg)  
+Figure 8: Qualitative comparison results of our CPPC with state-of-the-art methods on V2XSet (a, b) and DAIR-V2X-C (c, d). The green bounding boxes represent the ground-truth, and the red ones depict predictions.

2025/Point Cluster_ A Compact Message Unit for Communication-Efficient Collaborative Perception/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c2e80d7b439d263fbc111a265856ab4a6ac7237387e9aea87d8a00ca80747c2a
+size 780873

2025/Point-SAM_ Promptable 3D Segmentation Model for Point Clouds/b6e8bccc-f4f8-4762-844b-752b678c4fe0_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:580d67dbcf86a1418ecc0dba227c80f99c4efc756acf6858c3998d4565a94ec5
+size 40715781

2025/Point-SAM_ Promptable 3D Segmentation Model for Point Clouds/full.md

@@ -0,0 +1,368 @@
+# POINT-SAM: PROMPTABLE 3D SEGMENTATION MODEL FOR POINT CLOUDS
+
+Yuchen Zhou $^{1,3}$  * Jiayuan Gu $^{2}$  * Tung Yen Chiang $^{3}$  Fanbo Xiang $^{1}$  Hao Su $^{1,3}$
+
+$^{1}$ Hillbot Inc.  $^{2}$ ShanghaiTech University  $^{3}$ UC San Diego
+
+# ABSTRACT
+
+The development of 2D foundation models for image segmentation has been significantly advanced by the Segment Anything Model (SAM). However, achieving similar success in 3D models remains a challenge due to issues such as non-unified data formats, poor model scalability, and the scarcity of labeled data with diverse masks. To this end, we propose a 3D promptable segmentation model Point-SAM, focusing on point clouds. We employ an efficient transformer-based architecture tailored for point clouds, extending SAM to the 3D domain. We then distill the rich knowledge from 2D SAM for Point-SAM training by introducing a data engine to generate part-level and object-level pseudo-labels at scale from 2D SAM. Our model outperforms state-of-the-art 3D segmentation models on several indoor and outdoor benchmarks and demonstrates a variety of applications, such as interactive 3D annotation and zero-shot 3D instance proposal.
+
+# 1 INTRODUCTION
+
+The development of 2D foundation models for image segmentation has been significantly advanced by Segment Anything (Kirillov et al., 2023). That pioneering work includes a promptable segmentation task, a segmentation model (SAM), and a data engine for collecting a dataset (SA-1B) with over 1 billion masks. SAM shows impressive zero-shot transferability to new image distributions and tasks. Thus, it has been widely used in many applications, e.g., segmenting foreground objects for image-conditioned 3D generation (Liu et al., 2023c;a), NeRF (Cen et al., 2023b), and robotic tasks (Wang et al., 2023; Chen et al., 2023).
+
+Can we just lift SAM to create 3D foundation models for segmentation? Despite a few efforts (Yang et al., 2023; Xu et al., 2023; Zhou et al., 2023b) to extend SAM to the 3D domain, those existing approaches are limited to applying SAM on 2D images and then lifting the results to 3D. This process is constrained by image quality, and thus is likely to fail for textureless or colorless shapes like CAD models (Lambourne et al., 2021). Besides, it is also affected by view selection. Too few views may not adequately cover the entire shape, while too many views can significantly increase the computational burden. Moreover, it can suffer from multi-view inconsistency when results are merged from different views, since they may conflict and be impacted by occlusions. Furthermore, multi-view images only capture surface, making it infeasible to label internal structures, essential for annotating articulated objects (e.g. drawers in a cabinet). Therefore, it is necessary to develop native 3D foundation models to address the aforementioned limitations.
+
+However, developing native 3D foundation models, or extending SAM to the 3D domain, presents several challenges: 1) There is no unified representation for 3D shapes. 3D shapes can be represented by meshes, voxels, point clouds, implicit functions, or multi-view images, while 2D images are usually represented by a dense grid of pixels. Unlike 2D images, 3D shapes can vary significantly in scale and sparsity. For example, indoor and outdoor datasets often cover different ranges and typically require different models. 2) There are no unified network architectures in the 3D domain. Due to the heterogeneity of 3D data, different network architectures have been proposed for different representations, such as PointNet (Qi et al., 2017a) for point clouds and SparseConv (Graham et al., 2018) for voxels. 3) It is more difficult to scale up 3D networks. 3D networks are natively more computationally costly. For instance, SAM utilizes deconvolution and
+
+![](images/405cb860511206385e44c97caede6fa4fe599ad2ed276df707d315f432c8d835.jpg)  
+Figure 1: We propose a 3D extension of SAM, named Point-SAM (Sec. 3), which predicts masks given the input point cloud and prompts. To scale up training data, we develop a data engine (Sec. 4) to generate pseudo labels with the help of SAM. The final models, trained on a mixture of datasets, are capable of handling data from various sources and producing results at multiple levels of granularity. We demonstrate the versatility and efficacy of our approach through multiple applications and downstream tasks, as detailed in Sec. 5.
+
+bilinear upsampling in its decoder, whereas there are no 3D operators for point clouds as efficient as their 2D counterparts. 4) High-quality 3D labels, especially those with diverse masks, are rare. SAM is initially trained on existing datasets with ground-truth labels of low diversity, and then used to facilitate annotating more masks at different granularity (e.g., part, object, semantics) to increase label diversity. However, in the 3D domain, existing datasets contain only a small number of segmentation labels. For example, the largest dataset with part-level annotations, PartNet (Mo et al., 2019), includes only about 26,671 shapes and 573,585 part instances.
+
+In this work, our goal is to build a 3D promptable segmentation model for point clouds, as a foundational step towards 3D foundation models. Point clouds are selected as our primary representation, because other representations can be readily converted into point clouds, and real-world data is often captured in this format. Following SAM, we address 3 critical components: task, model, and data. We focus on the 3D promptable segmentation task, which involves predicting valid segmentation masks in response to any given segmentation prompt. To address this task, we propose a 3D extension of SAM, named Point-SAM. We utilize a transformer-based encoder to embed the input point cloud, alongside a point prompt encoder for point prompts, and a mask prompt encoder for mask prompts. Point-cloud and prompt embeddings are fed to a transformer-based mask decoder to predict segmentation masks. To efficiently encode point clouds and pointwise prompts, we develop a novel tokenizer based on Voronoi diagram to obtain point-cloud embeddings, as input to the transformer-based encoder. Regarding data, we train Point-SAM on a mixture of heterogeneous datasets, including PartNet and ScanNet (Dai et al., 2017), with both part- and object-level annotations. To expand label diversity and leverage large-scale unlabeled datasets such as ShapeNet (Chang et al., 2015), we have developed a data engine to generate pseudo labels with the assistance of SAM. This pipeline enables us to distill knowledge from SAM, and our experiments demonstrate that these pseudo labels significantly improve zero-shot transferability. Our contributions include:
+
+- We develop a 3D promptable segmentation model Point-SAM, adept at processing point clouds from various sources in a unified way. A novel tokenizer based on Voronoi diagram is proposed to efficiently embed point clouds and dense prompts.  
+- We propose a data engine to generate pseudo labels with substantial mask diversity by distilling knowledge from SAM. It is shown to significantly enhance our model's performance on out-of-distribution (OOD) data.  
+- Our experiments demonstrate the strong zero-shot transferability of our model to unseen point cloud distributions and new tasks, positioning it as a 3D foundation model.
+
+# 2 RELATED WORK
+
+Lifting 2D foundation models for 3D segmentation Despite the growing number of 3D datasets, high-quality 3D segmentation labels remain scarce. To address this, 2D foundation models trained on web-scale 2D data, such as CLIP (Radford et al., 2021), GLIP (Li et al., 2022), and SAM (Kirillov
+
+et al., 2023), have been leveraged. A prevalent framework involves adapting these 2D foundation models for 3D applications by merging results across multiple views (Liu et al., 2023d) (Zhou et al., 2024) (Cen et al., 2024). SAM3D (Yang et al., 2023) and SAMPro3D (Xu et al., 2023) utilize RGB-D images with known camera poses to lift SAM to segment 3D indoor scenes. PartSLIP (Liu et al., 2023b; Zhou et al., 2023b), dedicated to part-level segmentation, first renders multiple views of a dense point cloud, then employs GLIP and SAM to segment parts, and finally consolidates multi-view results into 3D predictions. These methods are limited by the capabilities of 2D foundation models and the quality of multi-view rendering. Besides, they usually require complicated and slow post-processing to integrate multi-view results, which also poses challenges in maintaining multi-view consistency. Another strategy involves distilling knowledge from 2D foundation models directly into 3D models. For example, Segment3D (Huang et al., 2023) and SAL (Osep et al., 2024) both utilize SAM to generate pseudo labels given RGB images and train native 3D models on scene-level point clouds. However, these approaches can only handle surface points, making it difficult to segment internal structures that are common in part-level segmentation of articulated 3D shapes such as cabinets with drawers.
+
+3D foundation models The development of 3D foundation models has advanced notably. Point-BERT (Yu et al., 2022b) proposes a self-supervised paradigm for pretraining 3D representations for point clouds. OpenShape (Liu et al., 2024) and Uni3D (Zhou et al., 2023a) scale up 3D representations with multi-modal contrastive learning. (Hong et al., 2023) trains 3D-based Large Language models (3D-LLM) on collected diverse 3D-language data, utilizing 2D pretrained VLMs. LEO (Huang et al., 2024), sharing similar ideas, focuses on embodied ability such as navigation and robotic manipulation. Our work concentrates on 3D segmentation. Despite several initiatives aimed at open-world 3D segmentation, such as OpenScene (Peng et al., 2023) and OpenMask3D (Takmaz et al., 2024), these primarily address scene-level segmentation and are trained on relatively small datasets.
+
+3D interactive segmentation Interactive segmentation has been explored across both 2D and 3D domains. (Kirillov et al., 2023) introduces a groundbreaking project including the promptable segmentation task, the 2D foundation model (SAM), and a data engine to collect large-scale labels. In the 3D domain, InterObject3D (Kontogianni et al., 2023) and AGILE3D (Yue et al., 2023) share similar ideas to segment point clouds while their training is confined to ScanNet (Dai et al., 2017). In contrast, our model is designed to handle both object- and part-level segmentation, leveraging a wide range of datasets including CAD models and real scans. Thus, our model shows greater versatility and adaptability. Besides, 3D interactive segmentation is also explored within implicit representations. SA3D (Cen et al., 2023b) enables users to achieve 3D segmentation of any target object through a single one-shot manual prompt in a rendered view. SAGA (Cen et al., 2023a) distills SAM features into 3D Gaussian point features through contrastive training. While these methods necessitate an additional optimization process, our model operates on a feed-forward basis and can respond within seconds, offering a more efficient solution.
+
+# 3 POINT-SAM
+
+In this section, we present Point-SAM, a promptable segmentation model for point clouds. Fig. 2 provides an overview of Point-SAM. Inspired by SAM (Kirillov et al., 2023), Point-SAM consists of 3 components: a point-cloud encoder, a prompt encoder, and a mask decoder. Unlike 2D models, Point-SAM addresses unique challenges related to point clouds: computation efficiency, scalability, and irregularity. We denote the input point cloud as  $P \in \mathbb{R}^{N \times 3}$  and its point-wise feature as  $F \in \mathbb{R}^{N \times \bar{D}}$ .
+
+Point-cloud encoder with Voronoi tokenizer The point-cloud encoder transforms the input point cloud into a point-cloud embedding. Inspired by recent advancements in 3D point-cloud transformers (Zhao et al., 2021; Wu et al., 2022; Yu et al., 2022a; Zhou et al., 2023a), we employ a similar transformer-based encoder. Concretely, it first selects a fixed number of centers  $C \in \mathbb{R}^{L \times 3}$  using farthest point sampling (FPS), and groups the k-nearest neighbors of each center as a patch. A local PointNet (Qi et al., 2017a) is used to tokenize each patch  $G_{patch} \in \mathbb{R}^{L \times K \times (3 + \bar{D})}$ . The resulting patch tokens  $F_{patch} \in \mathbb{R}^{L \times D}$ , combined with the positional embeddings of the group centers, are processed by a Vision Transformer (Dosovitskiy et al., 2021) to generate the final point-cloud embedding  $F_{pc} \in \mathbb{R}^{L \times D}$ .
+
+![](images/c4f64a5acadc9c169d36b95d8fbb73e3e07a623d9ac60eb49d1006009ccfd66a.jpg)  
+(a) Network Architecture
+
+![](images/a0d75a1faf42f7d17bcacdc3b03e323c5b20f1b50f259319ca4d9dd355ebd5e7.jpg)  
+(b) Voronoi Tokenizer
+
+![](images/d61e8fc06a393729f11156e834c78e18f88100cf226d0a788bde19e30f177a05.jpg)  
+(c) Grouping Diagram  
+Figure 2: Overview of Point-SAM. (a) illustrates the overall network architecture. The model takes a point cloud along with several point prompts as inputs. Initially, the point cloud is divided into patch tokens using a Voronoi tokenizer. After that, the patch tokens are embedded through a vanilla Vision Transformer (ViT). The token features are then fused with the mask features from the previous iteration. The two-way transformer is employed to allow interaction with the features of the prompt points. Finally, a lightweight decoder generates the mask output. (b) depicts the design of the Voronoi tokenizer, where a Voronoi diagram is used for grouping the high-resolution point cloud into patch tokens, instead of relying on traditional K-nearest neighbors (KNN) methods. (c) provides a visual diagram of the grouping process within the Voronoi tokenizer.
+
+We observe that  $L \times K$  is typically much larger than  $N$ , making the process of tokenizing point clouds to obtain  $F_{patch}$  time-consuming and memory-intensive. To address this, we propose a novel tokenizer based on the Voronoi diagram, which strikes a balance between efficiency and effectiveness. Specifically, we group points by assigning each point to its nearest center, forming a Voronoi diagram where each patch corresponds to a Voronoi cell. An MLP is then used to extract pointwise features  $\hat{F}_{patch} \in \mathbb{R}^{N \times D}$  based on the relative position of each point to its nearest center. Patch tokens  $F_{patch} \in \mathbb{R}^{L \times D}$  are max-pooled within each Voronoi cell via a scatter-max operator.
+
+Prompt encoder The prompt encoder encodes various types of prompts into prompt embeddings. In this work, we focus on two types of prompts: points and masks. Point prompts are processed similarly to SAM. Each point is associated with a binary label indicating whether it is a foreground prompt. These prompts are encoded to their positional encodings (Tancik et al., 2020)  $F_{point} \in \mathbb{R}^{Q \times D}$ , summed with learned embeddings indicating their labels.  $Q$  denotes the number of point prompts. Mask prompts are represented as pointwise logits  $X_{mask} \in \mathbb{R}^{N \times 1}$ , typically derived from the model's previous predictions. These logits are concatenated with the input point cloud's coordinates and processed through a tokenizer described in the point-cloud encoder. The resulting mask prompt embeddings  $F_{mask} \in \mathbb{R}^{L \times D}$  are element-wise summed with the point-cloud embedding.
+
+Mask decoder The mask decoder efficiently maps the point-cloud embedding, prompt embeddings, and an output token  $F_{out} \in \mathbb{R}^{1 \times D}$  into a segmentation mask  $Y_{mask} \in \mathbb{R}^{N \times 1}$ . We follow SAM to employ two Transformer decoder blocks that use prompt self-attention and cross-attention in two directions (prompt-to-point-cloud and vice versa), to update all embeddings. Different from the 2D counterpart, we upsample the updated point-cloud embedding  $F_{pc} \in \mathbb{R}^{L \times D}$  to match the input resolution by using inverse distance weighted average interpolation based on 3 nearest neighbors (Qi et al., 2017b), followed by an MLP. The upsampled point-cloud embedding is denoted as  $X_{pc} \in \mathbb{R}^{N \times D}$ . Another MLP transforms the output token to the weight of a dynamic linear classifier  $X_{out} \in \mathbb{R}^{1 \times D}$ , which calculates the mask's foreground probability at each point location as  $Y_{mask} = X_{pc} \cdot X_{out}^T$ . Consistent with SAM, our model can generate multiple output masks for a single point prompt by introducing multiple output tokens. Note that multi-mask outputs are enabled only when there is only a single point prompt with no mask prompts. In addition, we also introduce another token  $F_{iou} \in \mathbb{R}^{M \times D}$  to predict the IoU score for each mask output, where  $M$  is the number of multiple mask outputs.
+
+Table 1: Summary of training datasets.  
+
+<table><tr><td>Dataset</td><td>PartNet</td><td>PartNet-Mobility</td><td>ScanNet</td><td>ScanNet-block</td><td>Fusion360</td><td>ShapeNet</td><td>Overall</td></tr><tr><td>Number of point clouds</td><td>16442</td><td>2163</td><td>1198</td><td>24328</td><td>35000</td><td>20000</td><td>99131</td></tr><tr><td>Average number of masks</td><td>13</td><td>16</td><td>30</td><td>12</td><td>2</td><td>17</td><td>10</td></tr><tr><td>Has ground truth labels</td><td>True</td><td>True</td><td>True</td><td>True</td><td>True</td><td>False</td><td>-</td></tr></table>
+
+Training Mask predictions are supervised with a weighted combination of focal loss (Lin et al., 2017) and dice loss (Milletari et al., 2016), in line with SAM. We simulate an interactive setup, detailed in Sec. 5.1, by sampling prompts across 7 iterations per mask. The loss for mask prediction is computed between the ground truth mask and the predictions at all iterations. More details are provided in App. A. For multiple mask outputs, we follow SAM to use a "hindsight" loss, where we only back-propagate only the minimum loss over masks. Additionally, the predicted IoU score is supervised using a mean squared error loss. For training, we randomly sample 10,000 points as input. Besides, we normalize the input point to fit within a unit sphere centered at zero, to standardize the inputs. The number of patches  $L$  and the patch size  $K$  are set to 512 and 64 by default.
+
+Inference with variability A significant challenge in handling 3D point clouds is their irregular input structure; the number of points can vary, necessitating a dynamic approach to group points into a varying number of patches with adjustable sizes. While previous point-based methods (Zhou et al., 2023a) are typically limited to processing a fixed number of points, our model's flexible design allows it to handle larger point sets than those used during training, by adjusting the number of patches and the patch size. Unless otherwise specified, we set the number of patches and the patch size to 2048 and 512 when the number of input points exceeds 32768. In contrast, voxelization-based methods (Yue et al., 2023) struggle with such variations as changing voxel resolution can significantly impact performance, the results with different voxel resolutions are shown in App. B.
+
+# 4 TRAINING DATASETS
+
+Integrating existing datasets Foundation models are typically data-hungry, and the diversity of segmentation masks is crucial to support "segment anything". Thus, we use a mixture of existing datasets with ground truth segmentation labels, which are summarized in Table 1. We utilize synthetic datasets including the training split of PartNet (Mo et al., 2019), PartNet-Mobility (Xiang et al., 2020), and Fusion360 (Lambourne et al., 2021). Since PartNet does not provide textured meshes, we only keep the models that are from ShapeNet where textured meshes are available. We use all part hierarchies of PartNet. For PartNet-Mobility, we hold out 3 categories (scissors, refrigerators, and doors) not included in ShapeNet, which are used for evaluation on unseen categories. For PartNet and Fusion360, we uniformly sample 32768 points from mesh faces. For each object in PartNet-Mobility, we render 12 views, fuse point clouds from rendered RGB-D images, and sample 32768 points from the fused point cloud using Farthest Point Sampling (FPS). For scene-level datasets, we use the training split of ScanNet200 (Dai et al., 2017) and augment it by splitting each scene into blocks. The augmented version is denoted as ScanNet-Block. Concretely, we use a  $3\mathrm{m}\times 3\mathrm{m}$  block with a stride of  $1.5\mathrm{m}$ . We use FPS to sample 32768 points per scene or block.
+
+Generating pseudo labels Existing datasets lack sufficient diversity in masks. Large-scale 3D datasets like ShapeNet (Chang et al., 2015) usually do not include part-level segmentation labels. Besides, most segmentation datasets only provide exclusive labels, where each point belongs to a single instance. To this end, we develop a data engine to generate pseudo labels.
+
+Figure 3 illustrates the pseudo label generation process. Initially, Point-SAM is trained on the mixture of existing datasets. Next, we utilize both pre-trained Point-SAM and SAM to generate pseudo labels. Concretely, for each mesh, we render RGB-D images at 6 fixed camera positions and fuse a colored point cloud. SAM is applied to generate diverse 2D proposals for each view. For each 2D proposal, we intend to find a 3D proposal corresponding to it. We start from the view corresponding to the 2D proposal. A 2D prompt is randomly sampled from the 2D proposal and lifted to a 3D prompt, which prompts Point-SAM to predict a 3D mask on the fused point cloud. Then, we sample the next
+
+![](images/9dba9c9058a2e92a06af1422e63429c4a7e02d3053e9e088c331335bd7a56670.jpg)  
+Figure 3: Illustration of pseudo label generation. Initially, we select one segmentation mask from the instance proposals ("segment everything") generated by SAM on the first view. Then, we prompt Point-SAM by lifting 2D prompt points to 3D (View 1 prompt). Subsequently, the 3D segmentation mask output by Point-SAM is refined using additional views. We first prompt SAM by projecting the 3D segmentation mask onto the second view (View 2), leveraging SAM's strong prior knowledge to revise the mask. Then, we sample more 2D prompt points from the revised area by SAM, and prompt Point-SAM again by lifting these points to 3D (View 2 prompt).
+
+2D prompt from the error region between the 2D proposal and the projection of the 3D proposal at this view. New 3D prompts and previous 3D proposal masks are fed to Point-SAM to update the 3D proposal. The process is repeated until the IoU between the 2D proposal and the projection of the 3D proposal is larger than a threshold. This step ensures 3D-consistent segmentation regularized by Point-SAM while retaining the diversity of SAM's predictions. We repeat the above process with a few modifications at other views to refine the 3D proposal. At other view, we first sample the initial 2D prompt from the projection of previous 3D proposal, which is used to prompt SAM to generate multiple outputs. The output 2D mask with the highest IoU relative to the projection is selected as the "2D proposal" in the previous process. If the IoU is lower than a threshold, the 3D proposal is discarded. Previous 3D proposal mask is used to prompt Point-SAM at each iteration. This step aids in refining the 3D masks by incorporating 2D priors from SAM through space carving. We use our data engine to generate pseudo labels for 20000 shapes from ShapeNet. On average, each shape is annotated with 17 masks, offering a diversity comparable to PartNet.
+
+# 5 EXPERIMENTS
+
+We have conducted the experiments showing the strong zero-shot transferability and the superior efficiency of our method. Experiments are conducted on zero-shot point-prompted segmentation (Sec. 5.1), few-shot part segmentation (App. B.2) and zero-shot object proposal generation (App. B.1). Furthermore, we showcase an application of 3D interactive annotation in our supplementary materials.
+
+# 5.1 ZERO-SHOT POINT-PROMPTED SEGMENTATION
+
+Task and metric The task is to segment instances based on 3D point prompts. For automatic evaluation, point prompts need to be selected. We adopt the same method to simulate user clicks described in Kontogianni et al. (2023). In brief, the first point prompt is selected as the "center" of the ground truth mask, which is the point farthest from the boundary. Each subsequent point is chosen from two candidates: one from the false-positive set at the farthest minimum distance to the complementary set, and the other from the false-negative set selected similarly. Then, the candidate farther from the boundary is selected. See App. C for details. This evaluation protocol is commonly
+
+Table 2: Summary of evaluation datasets.  
+
+<table><tr><td>Property</td><td>PartNet-Mobility</td><td>ScanObjectNN</td><td>S3DIS</td><td>KITTI360</td><td>Replica</td><td>Overall</td></tr><tr><td>Number of point clouds</td><td>125</td><td>2,723</td><td>68</td><td>379</td><td>17</td><td>3312</td></tr><tr><td>Average number of masks</td><td>6</td><td>3</td><td>33</td><td>9</td><td>152</td><td>5</td></tr><tr><td>Average number of points</td><td>10,000</td><td>157,669</td><td>522,058</td><td>40,000</td><td>1,308,124</td><td>-</td></tr></table>
+
+used in prior 2D (Kirillov et al., 2023; Zhang et al., 2024) and 3D (Kontogianni et al., 2023; Yue et al., 2023) works on interactive single-object segmentation. Following those prior works, we use the metric IoU@k, which is the Intersection over Union (IoU) between ground truth masks and prediction given  $k$  point prompts. The metric is averaged across instances.
+
+Datasets We evaluate on a heterogeneous collection of datasets, covering both indoor and outdoor data, along with part- and object-level labels. For part-level evaluation, we use the synthetic dataset PartNet-Mobility (Xiang et al., 2020) and the real-world dataset ScanObjectNN (Uy et al., 2019). As mentioned in Sec. 4, we hold out 3 categories of PartNet-Mobility for evaluation. In the same way as the training dataset, we render 12 views for each shape, fuse a point cloud from multi-view depth images, and sample 10,000 points for evaluation. ScanObjectNN contains 2902 objects of 15 categories collected from SceneNN (Hua et al., 2016) and ScanNet (Dai et al., 2017). For scene-level evaluation, we use S3DIS (Armeni et al., 2016) and KITTI-360 (Liao et al., 2022). Specifically, we use the processed data from AGILE3D (Yue et al., 2023), which contains scans cropped around each instance. Table 2 summarizes the datasets used for evaluation.
+
+Baselines We compare Point-SAM with a multi-view extension of SAM, named MV-SAM, and a 3D interactive segmentation method, AGILE3D (Yue et al., 2023). Inspired by previous works (Yang et al., 2023; Xu et al., 2023; Zhou et al., 2023b) that lift SAM's multi-view results to 3D, we introduce MV-SAM for zero-shot point-prompted segmentation as a strong baseline. First, we render multiview RGB-D images from the mesh of each shape. Note that mesh rendering is needed to ensure high-quality images, which are essential for good SAM's performance. Thus, this baseline actually has access to more information than ours. Then we prompt SAM at each view with the simulated click sampled from the "center" of the error region between the SAM's prediction and 2D ground truth mask. The predictions are subsequently lifted back to the sparse point cloud (10,000 points) and merged into a single mask. If a point is visible from multiple views, its foreground probability is averaged. For both MV-SAM and our method, we select the most confident prediction if there are multiple outputs. AGILE3D is close to our approach, while it uses a sparse convolutional U-Net as its backbone and is only trained on the real-world scans of ScanNet40. Besides, it does not normalize its input, and thus it is sensitive to object scales. To process CAD models without known physical scales, we adjust the scale of the input point cloud for AGILE3D, so that its axis-aligned bounding box has a maximum size of  $5\mathrm{m}$ , determined through a grid search (App. B.3).
+
+Results Table 3 presents the quantitative results. Point-SAM shows superior zero-shot transferability and effectively handle data with different numbers of points as well as from different sources. Point-SAM significantly outperforms MV-SAM, especially when only few point prompts are provided, while MV-SAM achieves reasonably good performance with a sufficient number of prompts. Notably, for IoU@k, MV-SAM actually samples  $k$  prompts per view. It indicates that our 3D native method is more prompt-efficient. Besides, it is challenging for SAM to achieve multi-view consistency without extra fine-tuning, especially with limited prompts. Moreover, Point-SAM also surpasses AGILE3D across all datasets, particularly in out-of-distribution (OOD) scenarios such as PartNet-Mobility (held-out categories) and KITTI360. It underscores the strong zero-shot transferability of our method and the importance of scaling datasets. Figure 4 shows the qualitative comparison between Point-SAM, AGILE3D and MV-SAM, where Point-SAM demonstrates superior quality with a single prompt and significantly faster convergence compared to AGILE3D and MV-SAM.
+
+Table 3 also compares the Voronoi tokenizer with the previous KNN tokenizer. We observe that the Voronoi tokenizer achieves comparable performance to the KNN tokenizer, while showing superior efficiency. We test the time and memory efficiency on a single Nvidia RTX-4090 GPU using point clouds from KITTI360. For each point cloud, 10 prompt points are sampled. The Voronoi tokenizer increases the frames per second (FPS) by  $163.4\%$  ( $5.2 \to 13.7$  shape/sec) and reduces GPU memory usage by  $18.4\%$  ( $3890 \to 3172$  MB).
+
+Table 3: Quantitative results for zero-shot point-prompted segmentation. The notation Voronoi indicates the use of the Voronoi tokenizer and KNN indicates the use of the KNN tokenizer, while all other settings remain unchanged.  
+
+<table><tr><td>Dataset</td><td>Method</td><td>IoU@1</td><td>IoU@3</td><td>IoU@5</td><td>IoU@7</td><td>IoU@10</td></tr><tr><td rowspan="4">PartNet-Mobility</td><td>AGILE3D</td><td>26.4</td><td>40.8</td><td>50.8</td><td>57.4</td><td>61.9</td></tr><tr><td>MV-SAM</td><td>29.3</td><td>57.0</td><td>69.7</td><td>74.3</td><td>76.9</td></tr><tr><td>Ours (Voronoi)</td><td>46.0</td><td>68.1</td><td>73.2</td><td>76.0</td><td>77.8</td></tr><tr><td>Ours (KNN)</td><td>47.9</td><td>67.7</td><td>74.2</td><td>77.0</td><td>78.6</td></tr><tr><td rowspan="3">ScanObjectNN</td><td>AGILE3D</td><td>34.8</td><td>52.0</td><td>61.6</td><td>67.2</td><td>72.3</td></tr><tr><td>Ours (Voronoi)</td><td>52.4</td><td>76.3</td><td>81.4</td><td>84.0</td><td>85.7</td></tr><tr><td>Ours (KNN)</td><td>49.4</td><td>75.3</td><td>82.0</td><td>84.8</td><td>86.3</td></tr><tr><td rowspan="4">S3DIS</td><td>InterObject3D++</td><td>32.7</td><td>69.0</td><td>80.8</td><td>-</td><td>89.2</td></tr><tr><td>AGILE3D</td><td>58.7</td><td>77.4</td><td>83.6</td><td>86.4</td><td>88.5</td></tr><tr><td>Ours (Voronoi)</td><td>63.6</td><td>81.8</td><td>84.8</td><td>86.4</td><td>87.3</td></tr><tr><td>Ours (KNN)</td><td>47.6</td><td>78.4</td><td>86.2</td><td>89.2</td><td>90.4</td></tr><tr><td rowspan="4">KITTI360</td><td>InterObject3D++</td><td>3.4</td><td>11.0</td><td>19.9</td><td>-</td><td>40.6</td></tr><tr><td>AGILE3D</td><td>34.8</td><td>42.7</td><td>44.4</td><td>45.8</td><td>49.6</td></tr><tr><td>Ours (Voronoi)</td><td>52.8</td><td>71.4</td><td>81.3</td><td>83.9</td><td>85.7</td></tr><tr><td>Ours (KNN)</td><td>49.4</td><td>74.4</td><td>81.7</td><td>84.3</td><td>85.8</td></tr></table>
+
+Table 4: Ablation study on training dataset. We report the IoU@k metrics for zero-shot prompt-segmentation on PartNet-Mobility (held-out categories).  
+
+<table><tr><td>Training dataset</td><td>PartNet</td><td>PartNet +ScanNet</td><td>PartNet +ShapeNet</td><td>PartNet +ShapeNet+ScanNet</td><td>Full</td></tr><tr><td>IoU@1</td><td>38.2</td><td>39.7</td><td>44.5</td><td>45.4</td><td>47.9</td></tr><tr><td>IoU@3</td><td>56.3</td><td>58.6</td><td>65.2</td><td>66.5</td><td>67.7</td></tr><tr><td>IoU@5</td><td>60.6</td><td>68.8</td><td>71.8</td><td>72.6</td><td>74.2</td></tr><tr><td>IoU@10</td><td>63.5</td><td>71.9</td><td>76.2</td><td>77.5</td><td>78.6</td></tr></table>
+
+# 5.2 ABLATIONS
+
+Scaling up datasets Previous works have been limited by the size and scope of their training datasets. For example, AGILE3D (Yue et al., 2023) was trained solely on ScanNet (Dai et al., 2017), which includes only 1,201 scenes. As detailed in Table 1, our training dataset encompasses 100,000 point clouds, 100 times larger than ScanNet. To verify the effectiveness of scaling up training data, we conduct an ablation study on dataset size and composition. We introduce 4 dataset variants: 1) PartNet only, 2) PartNet+ScanNet (including ScanNet-Block), 3) PartNet+ShapeNet (pseudo labels), and 4) PartNet+ShapeNet+ScanNet. We train Point-SAM on these variants, resulting in different models. Table 4 shows the comparison of these models in zero-shot prompt segmentation on PartNet-Mobility (held-out categories). The model trained on PartNet+ScanNet surpasses the one trained solely on PartNet, although the evaluation dataset (part-level labels) has a markedly different distribution from the added ScanNet (object-level labels). Moreover, the model trained on PartNet+ShapeNet achieves even better performance, particularly with a single prompt. Note that the IoU@1 metric assesses whether the model captures sufficient mask diversity, since a single prompt is inherently ambiguous and the ground-truth label depends on dataset bias. It suggests that our pseudo labels effectively incorporate part-level knowledge distilled from SAM. Furthermore, it is observed that the zero-shot performance on out-of-distribution data consistently improves, as we utilize increasingly larger and more diverse data.
+
+Trained on ScanNet only To ensure a fair comparison with AGILE3D, we conduct an ablation study, where Point-SAM is trained exclusively on ScanNet, following the same settings as AGILE3D. This resulting model is denoted as Point-SAM*. Table 5a presents the comparison among AGILE3D, Point-SAM* and the original Point-SAM. Provided similar training data, Point-SAM* significantly
+
+![](images/9fd67124add433e80d7ad56d0025e9466c8fa867c65009c339692e82ce5794ad.jpg)  
+Figure 4: Qualitative results of prompt segmentation are presented for three different settings: KITTI360 for zero-shot outdoor scene segmentation, S3DIS for indoor scene segmentation, and PartNet-Mobility for zero-shot part segmentation. We compare our results with AGILE3D on KITTI360 and S3DIS, and with MVSAM on PartNet-Mobility. Point-SAM demonstrates superior segmentation results with fewer prompt points across all three datasets. Red points represent positive prompt points, while blue points indicate negative prompt points.
+
+outperforms AGILE3D on the OOD datasets like KITTI360 and PartNet-Mobility, although performing worse on the in-domain dataset S3DIS. We hypothesize that AGILE3D is highly optimized for ScanNet (the only dataset it was trained on), and some of its design choices may lead to overfitting to this dataset, like its objective of simultaneously generating exclusive multiple objects.
+
+Data Engine To demonstrate the effectiveness of distilling knowledge from SAM, we conduct an ablation study on our design of the data engine. We compare our current pipeline with a baseline that uses a pre-trained Point-SAM to generate instance proposals as pseudo labels. Specifically, we sample
+
+(a) Quantitative results for Point-SAM trained on ScanNet. Point-SAM* is trained on ScanNet following the setting of AGILE3D. Point-SAM refers to our original setting.
+
+<table><tr><td>Dataset</td><td>Method</td><td>IoU@1</td><td>IoU@3</td><td>IoU@5</td><td>IoU@7</td><td>IoU@10</td></tr><tr><td rowspan="3">PartNet-Mobility</td><td>AGILE3D</td><td>26.4</td><td>40.8</td><td>50.8</td><td>57.4</td><td>61.9</td></tr><tr><td>Point-SAM*</td><td>33.5</td><td>48.5</td><td>57.0</td><td>61.2</td><td>66.8</td></tr><tr><td>Point-SAM</td><td>47.9</td><td>67.7</td><td>74.2</td><td>77.0</td><td>78.6</td></tr><tr><td rowspan="3">ScanObjNN</td><td>AGILE3D</td><td>34.8</td><td>52.0</td><td>61.6</td><td>67.2</td><td>72.3</td></tr><tr><td>Point-SAM*</td><td>32.0</td><td>56.7</td><td>63.2</td><td>68.8</td><td>70.5</td></tr><tr><td>Point-SAM</td><td>49.4</td><td>75.3</td><td>82.0</td><td>84.8</td><td>86.3</td></tr><tr><td rowspan="3">S3DIS</td><td>AGILE3D</td><td>58.7</td><td>77.4</td><td>83.6</td><td>86.4</td><td>88.5</td></tr><tr><td>Point-SAM*</td><td>38.8</td><td>67.1</td><td>72.2</td><td>78.9</td><td>80.6</td></tr><tr><td>Point-SAM</td><td>47.6</td><td>78.4</td><td>86.2</td><td>89.2</td><td>90.4</td></tr><tr><td rowspan="3">KITTI360</td><td>AGILE3D</td><td>34.8</td><td>42.7</td><td>44.4</td><td>45.8</td><td>49.6</td></tr><tr><td>Point-SAM*</td><td>44.0</td><td>67.1</td><td>72.2</td><td>78.9</td><td>80.8</td></tr><tr><td>Point-SAM</td><td>49.4</td><td>74.4</td><td>87.1</td><td>84.3</td><td>85.8</td></tr></table>
+
+(b) Ablation study on the data engine. Different models are trained on different pseudo label datasets, and evaluated on PartNet-Mobility. * means using pseudo labels generated by Point-SAM ("segment everything") without SAM.
+
+Table 5: Ablation studies on model design and data engine.  
+
+<table><tr><td>PartNet Mobility</td><td>PartNet</td><td>PartNet +ShapeNet</td><td>PartNet +ShapeNet*</td></tr><tr><td>IoU@1</td><td>38.2</td><td>39.6</td><td>44.5</td></tr><tr><td>IoU@3</td><td>56.3</td><td>65.2</td><td>56.7</td></tr><tr><td>IoU@5</td><td>60.6</td><td>71.8</td><td>64.6</td></tr><tr><td>IoU@10</td><td>63.5</td><td>76.2</td><td>67.8</td></tr></table>
+
+Table 6: Sensitivity to point count. We report the IoU@k metrics for zero-shot prompt-segmentation on S3DIS, with varying the number of patches and patch size.  
+
+<table><tr><td>(#patches, patch size)</td><td>(512,64)</td><td>(512,256)</td><td>(2048,64)</td><td>(2048,256)</td></tr><tr><td>IoU@1</td><td>41.9</td><td>49.0</td><td>47.4</td><td>47.6</td></tr><tr><td>IoU@3</td><td>64.2</td><td>69.9</td><td>74.3</td><td>78.4</td></tr><tr><td>IoU@5</td><td>72.2</td><td>76.4</td><td>81.7</td><td>86.2</td></tr><tr><td>IoU@10</td><td>76.7</td><td>80.2</td><td>85.8</td><td>90.5</td></tr></table>
+
+1,024 prompt points from the point cloud, using each point to prompt Point-SAM. Non-Maximum Suppression (NMS) is applied to filter out duplicate instances. Table 5b compares the models trained on pseudo labels generated by our pipeline and the baseline. Interestingly, Point-SAM even benefits from pseudo labels generated by itself. Moreover, incorporating 2D SAM plays a crucial role in improving the quality of the pseudo labels, leading to a substantial boost in overall performance.
+
+Sensitivity to Point Count Point clouds are typically irregular. When handling point clouds with more points than those used in our training, we have to adjust the number of patches and the patch size accordingly. Thus, we conduct experiments to study the effect of these two hyperparameters. Table 6 shows the qualitative results of zero-shot prompt-segmentation on S3DIS (Armeni et al., 2016). We select S3DIS, because the average number of points for S3DIS is about 500K, 50 times larger than that of our training datasets. Our results indicate that it is important to increase the number of patches to accommodate larger point clouds. Enlarging the patch size is also crucial due to the different neighborhood densities compared to our training distribution.
+
+# 6 CONCLUSION
+
+In conclusion, our work presents significant strides towards developing a foundation model for 3D promptable segmentation using point clouds. By adopting a transformer-based architecture, we have successfully implemented Point-SAM, which effectively and efficiently responds to 3D point and mask prompts. Our model leverages a robust training strategy across mixed datasets like PartNet and ScanNet, which has proven beneficial, especially when enhanced with pseudo labels generated through our novel pipeline that distills knowledge from SAM.
+
+However, there are inherent limitations and challenges in our approach. The diversity and scale of the 3D datasets used still lag behind those available in 2D, posing a challenge for training models that can generalize well across different 3D environments and tasks. Furthermore, the computational demands of processing large-scale 3D data and the complexity of developing efficient 3D-specific operations remain significant hurdles. Our reliance on pseudo labels, while beneficial for expanding label diversity, also introduces dependencies on the quality and variability of the 2D labels provided by SAM, which may not always capture the complex nuances of 3D structures.
+
+# REFERENCES
+
+Iro Armeni, Ozan Sener, Amir R Zamir, Helen Jiang, Ioannis Brilakis, Martin Fischer, and Silvio Savarese. 3d semantic parsing of large-scale indoor spaces. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1534-1543, 2016.  
+Jiazhong Cen, Jiemin Fang, Chen Yang, Lingxi Xie, Xiaopeng Zhang, Wei Shen, and Qi Tian. Segment any 3d gaussians. arXiv preprint arXiv:2312.00860, 2023a.  
+Jiazhong Cen, Zanwei Zhou, Jiemin Fang, Wei Shen, Lingxi Xie, Dongsheng Jiang, Xiaopeng Zhang, Qi Tian, et al. Segment anything in 3d with nerfs. Advances in Neural Information Processing Systems, 36:25971-25990, 2023b.  
+Jiazhong Cen, Jiemin Fang, Zanwei Zhou, Chen Yang, Lingxi Xie, Xiaopeng Zhang, Wei Shen, and Qi Tian. Segment anything in 3d with radiance fields, 2024. URL https://arxiv.org/abs/2304.12308.  
+Angel X. Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, Jianxiong Xiao, Li Yi, and Fisher Yu. Shapenet: An information-rich 3d model repository, 2015.  
+Linghao Chen, Yuzhe Qin, Xiaowei Zhou, and Hao Su. Easyhec: Accurate and automatic hand-eye calibration via differentiable rendering and space exploration. IEEE Robotics and Automation Letters, 2023.  
+Angela Dai, Angel X Chang, Manolis Savva, Maciej Halber, Thomas Funkhouser, and Matthias Nießner. Scannet: Richly-annotated 3d reconstructions of indoor scenes. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 5828-5839, 2017.  
+Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale. ICLR, 2021.  
+Benjamin Graham, Martin Engelcke, and Laurens Van Der Maaten. 3d semantic segmentation with submanifold sparse convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 9224-9232, 2018.  
+Yining Hong, Haoyu Zhen, Peihao Chen, Shuhong Zheng, Yilun Du, Zhenfang Chen, and Chuang Gan. 3d-llm: Injecting the 3d world into large language models. Advances in Neural Information Processing Systems, 36:20482-20494, 2023.  
+Binh-Son Hua, Quang-Hieu Pham, Duc Thanh Nguyen, Minh-Khoi Tran, Lap-Fai Yu, and Sai-Kit Yeung. Scenenn: A scene meshes dataset with annotations. In 2016 fourth international conference on 3D vision (3DV), pp. 92-101. IEEE, 2016.  
+Jiangyong Huang, Silong Yong, Xiaojian Ma, Xiongkun Linghu, Puhao Li, Yan Wang, Qing Li, Song-Chun Zhu, Baoxiong Jia, and Siyuan Huang. An embodied generalist agent in 3d world. In Proceedings of the International Conference on Machine Learning (ICML), 2024.  
+Rui Huang, Songyou Peng, Ayca Takmaz, Federico Tombari, Marc Pollefeys, Shiji Song, Gao Huang, and Francis Engelmann. Segment3d: Learning fine-grained class-agnostic 3d segmentation without manual labels. arXiv preprint arXiv:2312.17232, 2023.  
+Alexander Kirillov, Eric Mintun, Nikhila Ravi, Hanzi Mao, Chloe Rolland, Laura Gustafson, Tete Xiao, Spencer Whitehead, Alexander C Berg, Wan-Yen Lo, et al. Segment anything. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4015-4026, 2023.  
+Theodora Kontogianni, Ekin Celikkan, Siyu Tang, and Konrad Schindler. Interactive object segmentation in 3d point clouds. In 2023 IEEE International Conference on Robotics and Automation (ICRA), pp. 2891-2897. IEEE, 2023.
+
+Joseph G. Lambourne, Karl D.D. Willis, Pradeep Kumar Jayaraman, Aditya Sanghi, Peter Meltzer, and Hooman Shayani. Brepnet: A topological message passing system for solid models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 12773-12782, June 2021.  
+Lianian Harold Li, Pengchuan Zhang, Haotian Zhang, Jianwei Yang, Chunyuan Li, Yiwu Zhong, Lijuan Wang, Lu Yuan, Lei Zhang, Jenq-Neng Hwang, et al. Grounded language-image pre-training. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 10965-10975, 2022.  
+Yiyi Liao, Jun Xie, and Andreas Geiger. KITTI-360: A novel dataset and benchmarks for urban scene understanding in 2d and 3d. Pattern Analysis and Machine Intelligence (PAMI), 2022.  
+Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dólar. Focal loss for dense object detection. In Proceedings of the IEEE international conference on computer vision, pp. 2980-2988, 2017.  
+Minghua Liu, Ruoxi Shi, Linghao Chen, Zhuoyang Zhang, Chao Xu, Xinyue Wei, Hansheng Chen, Chong Zeng, Jiayuan Gu, and Hao Su. One-2-3-45++: Fast single image to 3d objects with consistent multi-view generation and 3d diffusion. arXiv preprint arXiv:2311.07885, 2023a.  
+Minghua Liu, Yinhao Zhu, Hong Cai, Shizhong Han, Zhan Ling, Fatih Porikli, and Hao Su. Partslip: Low-shot part segmentation for 3d point clouds via pretrained image-language models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 21736-21746, 2023b.  
+Minghua Liu, Ruoxi Shi, Kaiming Kuang, Yinhao Zhu, Xuanlin Li, Shizhong Han, Hong Cai, Fatih Porikli, and Hao Su. Openshape: Scaling up 3d shape representation towards open-world understanding. Advances in Neural Information Processing Systems, 36, 2024.  
+Ruoshi Liu, Rundi Wu, Basile Van Hoorick, Pavel Tokmakov, Sergey Zakharov, and Carl Vondrick. Zero-1-to-3: Zero-shot one image to 3d object. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 9298–9309, 2023c.  
+Yichen Liu, Benran Hu, Chi-Keung Tang, and Yu-Wing Tai. Sanerf-hq: Segment anything for nerf in high quality. arXiv preprint arXiv:2312.01531, 2023d.  
+Fausto Miletari, Nassir Navab, and Seyed-Ahmad Ahmadi. V-net: Fully convolutional neural networks for volumetric medical image segmentation. In 2016 fourth international conference on 3D vision (3DV), pp. 565-571. IEEE, 2016.  
+Kaichun Mo, Shilin Zhu, Angel X Chang, Li Yi, Subarna Tripathi, Leonidas J Guibas, and Hao Su. Partnet: A large-scale benchmark for fine-grained and hierarchical part-level 3d object understanding. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 909-918, 2019.  
+Aljosa Osep, Tim Meinhardt, Francesco Ferroni, Neehar Peri, Deva Ramanan, and Laura Leal-Taixe. Better call sal: Towards learning to segment anything in lidar. arXiv preprint arXiv:2403.13129, 2024.  
+Songyou Peng, Kyle Genova, Chiyu Jiang, Andrea Tagliasacchi, Marc Pollefeys, Thomas Funkhouser, et al. Openscene: 3d scene understanding with open vocabularies. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 815-824, 2023.  
+Charles R Qi, Hao Su, Kaichun Mo, and Leonidas J Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 652-660, 2017a.  
+Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J Guibas. Pointnet++: Deep hierarchical feature learning on point sets in a metric space. Advances in neural information processing systems, 30, 2017b.
+
+Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pp. 8748-8763. PMLR, 2021.  
+Jonas Schult, Francis Engelmann, Alexander Hermans, Or Litany, Siyu Tang, and Bastian Leibe. Mask3D: Mask Transformer for 3D Semantic Instance Segmentation. 2023.  
+Julian Straub, Thomas Whelan, Lingni Ma, Yufan Chen, Erik Wijmans, Simon Green, Jakob J. Engel, Raul Mur-Artal, Carl Ren, Shobhit Verma, Anton Clarkson, Mingfei Yan, Brian Budge, Yajie Yan, Xiaqing Pan, June Yon, Yuyang Zou, Kimberly Leon, Nigel Carter, Jesus Braiales, Tyler Gillingham, Elias Mueggler, Luis Pesqueira, Manolis Savva, Dhruv Batra, Hauke M. Strasdat, Renzo De Nardi, Michael Goesele, Steven Lovegrove, and Richard Newcombe. The Replica dataset: A digital replica of indoor spaces. arXiv preprint arXiv:1906.05797, 2019.  
+Ayca Takmaz, Elisabetta Fedele, Robert Sumner, Marc Pollefeys, Federico Tombari, and Francis Engelmann. Openmask3d: Open-vocabulary 3d instance segmentation. Advances in Neural Information Processing Systems, 36, 2024.  
+Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. Advances in neural information processing systems, 33:7537-7547, 2020.  
+Mikaela Angelina Uy, Quang-Hieu Pham, Binh-Son Hua, Thanh Nguyen, and Sai-Kit Yeung. Revisiting point cloud classification: A new benchmark dataset and classification model on real-world data. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 1588-1597, 2019.  
+Ziyu Wang, Yanjie Ze, Yifei Sun, Zhecheng Yuan, and Huazhe Xu. Generalizable visual reinforcement learning with segment anything model. arXiv preprint arXiv:2312.17116, 2023.  
+Xiaoyang Wu, Yixing Lao, Li Jiang, Xihui Liu, and Hengshuang Zhao. Point transformer v2: Grouped vector attention and partition-based pooling. Advances in Neural Information Processing Systems, 35:33330-33342, 2022.  
+Fanbo Xiang, Yuzhe Qin, Kaichun Mo, Yikuan Xia, Hao Zhu, Fangchen Liu, Minghua Liu, Hanxiao Jiang, Yifu Yuan, He Wang, 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.  
+Mutian Xu, Xingyilang Yin, Lingteng Qiu, Yang Liu, Xin Tong, and Xiaoguang Han. Sampro3d: Locating sam prompts in 3d for zero-shot scene segmentation. arXiv preprint arXiv:2311.17707, 2023.  
+Yunhan Yang, Xiaoyang Wu, Tong He, Hengshuang Zhao, and Xihui Liu. Sam3d: Segment anything in 3d scenes. arXiv preprint arXiv:2306.03908, 2023.  
+Li Yi, Vladimir G Kim, Duygu Ceylan, I-Chao Shen, Mengyan Yan, Hao Su, Cewu Lu, Qixing Huang, Alla Sheffer, and Leonidas Guibas. A scalable active framework for region annotation in 3d shape collections. ACM Transactions on Graphics (ToG), 35(6):1-12, 2016.  
+Xumin Yu, Lulu Tang, Yongming Rao, Tiejun Huang, Jie Zhou, and Jiwen Lu. Point-bert: Pretraining 3d point cloud transformers with masked point modeling. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 19313-19322, 2022a.  
+Xumin Yu, Lulu Tang, Yongming Rao, Tiejun Huang, Jie Zhou, and Jiwen Lu. Point-bert: Pre-training 3d point cloud transformers with masked point modeling, 2022b.  
+Yuanwen Yue, Sabarinath Mahadevan, Jonas Schult, Francis Engelmann, Bastian Leibe, Konrad Schindler, and Theodora Kontogianni. Agile3d: Attention guided interactive multi-object 3d segmentation. arXiv preprint arXiv:2306.00977, 2023.
+
+Zhuoyang Zhang, Han Cai, and Song Han. Efficientvit-sam: Accelerated segment anything model without performance loss. arXiv preprint arXiv:2402.05008, 2024.  
+Hengshuang Zhao, Li Jiang, Jiaya Jia, Philip HS Torr, and Vladlen Koltun. Point transformer. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 16259-16268, 2021.  
+Junsheng Zhou, Jinsheng Wang, Baorui Ma, Yu-Shen Liu, Tiejun Huang, and Xinlong Wang. Uni3d: Exploring unified 3d representation at scale, 2023a.  
+Kaichen Zhou, Lanqing Hong, Enze Xie, Yongxin Yang, Zhenguo Li, and Wei Zhang. Serf: Fine-grained interactive 3d segmentation and editing with radiance fields, 2024. URL https:// arxiv.org/abs/2312.15856.  
+Yuchen Zhou, Jiayuan Gu, Xuanlin Li, Minghua Liu, Yunhao Fang, and Hao Su. Partslip++: Enhancing low-shot 3d part segmentation via multi-view instance segmentation and maximum likelihood estimation. arXiv preprint arXiv:2312.03015, 2023b.
+
+# A TRAINING DETAILS
+
+Training recipe. Point-SAM is trained with the AdamW optimizer. We train Point-SAM for 100k iterations. The learning rate  $(lr)$  is set to 5e-5 after learning rate warmup. Initially, the  $lr$  is warmed up for 3k iterations, starting at 5e-8. A step-wise  $lr$  scheduler with a decay factor of 0.1 is then used, with  $lr$  reductions at 60k and 90k iterations. The weight decay is set to 0.1. The training batch size for Point-SAM, utilizing ViT-g as the encoder, is set to 4 per GPU with a gradient accumulation of 4, and it is trained on 8 NVIDIA H100 GPUs with a total batch size of 128. The ViT-1 version can be trained across 2 NVIDIA A100 GPUs, with a batch size of 16 per GPU and gradient accumulation of 4, for 50k iterations. For Point-SAM utilizing ViT-1 as the backbone, the step-wise learning rate decay milestones are set at 30k and 40k iterations.
+
+Data augmentation. We apply several data augmentation techniques during training. For each object, we pre-sample 32,768 points before training and then perform online random sampling of 10,000 points from these 32,768 points for actual training. We apply a random scale for the normalized points with a scale factor of [0.8, 1.0] and a random rotation along y-axis from  $-180^{\circ}$  to  $180^{\circ}$ . For object point clouds we also apply a random rotation perturbation to x- and z-axis. The perturbation angles are sampled from a normal distribution with a standard deviation (sigma) of 0.06, and then these angles are clipped to the range [0, 0.18].
+
+# B ADDITIONAL EXPERIMENTS
+
+# B.1 ZERO-SHOT OBJECT PROPOSALS
+
+In this section, we evaluate Point-SAM on zero-shot object proposal generation. The ability to automatically generating masks for all possible instances is known as "segment everything" in SAM. SAM samples a 64x64 point grid on the image as prompts, and uses Non-Maximum Suppression (NMS) based on bounding boxes to remove duplicate instances. We adapt this approach for 3D point clouds with some modifications. First, we sample prompts using FPS, and then prompt Point-SAM to generate 3 masks per prompt. For post-processing, a modified version of NMS based on point-wise masks is applied.
+
+We compare with OpenMask3D (Takmaz et al., 2024) on Replica (Straub et al., 2019). OpenMask3D utilizes a class-agnostic version of Mask3D (Schult et al., 2023) trained on ScanNet200 to generate object proposals. For our Point-SAM, we sample 1024 prompts and set the NMS threshold to 0.3. In addition, to handle the extensive point counts in Replica, we downsample each scene to 100,000 points and later propagate the predictions to their nearest neighbors at the original resolution. We also adjust the number of patches and the patch size to 4096 and 64 respectively. For both methods, we truncate the proposals to the top 250.
+
+We use the average recall (AR) metric. We filter out "undefined" and "floor" categories from the ground truth labels. Table 7a shows the quantitative results. Point-SAM showcases strong performance compared to OpenMask3D, which is tailored for this task, even though our model is never trained on such a large number of points and is zero-shot evaluated on unseen data. It highlights the robust zero-shot capabilities of our method.
+
+Table 7: Qualitative results of zero-shot object proposal generation and few-shot part segmentation.  
+(a) Zero-shot object proposal generation on Replica.  
+
+<table><tr><td>Method</td><td>AR25</td><td>AR50</td></tr><tr><td>OpenMask3D</td><td>40.2</td><td>31.5</td></tr><tr><td>Ours</td><td>49.2</td><td>31.5</td></tr></table>
+
+(b) Few-shot part segmentation on ShapeNetPart. The numbers with * are reported by Uni3D (Zhou et al., 2023a).  
+
+<table><tr><td></td><td>PointBERT</td><td>Uni3D(close)</td><td>Uni3D(open)</td><td>Ours</td></tr><tr><td>1-shot</td><td>66.2*</td><td>71.5</td><td>75.9*</td><td>73.9</td></tr><tr><td>2-shot</td><td>71.9*</td><td>73.8</td><td>78.2*</td><td>76.1</td></tr></table>
+
+# B.2 FEW-SHOT PART SEGMENTATION
+
+Foundation models can be effectively fine-tuned for various tasks. In this section, we demonstrate that Point-SAM has captured good representations for part segmentation. We compare with
+
+PointBERT (Yu et al., 2022b) and Uni3D (Zhou et al., 2023a) on close-vocabulary, few-shot part segmentation. We use ShapeNetPart (Yi et al., 2016) and report the mIoU $_C$ , which is the mean IoU averaged across categories. Similar to Uni3D, we adapt Point-SAM for close-vocabulary part segmentation. Specifically, we extract features from the 4th, 8th, and last layers of the ViT in our encoder and use feature propagation (Qi et al., 2017b) to upscale them into point-wise features, followed by an MLP to predict point-wise multi-class logits. During few-shot training, we freeze our encoder and only optimize the feature propagation layer as well as the MLP using cross-entropy loss. Unlike PointBERT and our method, Uni3D originally aligns point-wise features with text features of ground truth part labels extracted by CLIP. We refer to it as Uni3D (open), since it is designed for open-vocabulary part segmentation. We also evaluate its variant sharing our modification for close-vocabulary part segmentation, denoted as Uni3D (close). Table 7b presents the results for both 1-shot and 2-shot settings. Point-SAM surpasses both PointBERT and Uni3D (close), which indicates that our approach has acquired versatile knowledge applicable to downstream tasks.
+
+# B.3 NORMALIZATION SCALE FOR AGILE3D
+
+We conduct a grid search to determine the optimal normalization scale for AGILE3D on PartNet-Mobility. Table 8 shows the effect of normalization scale for AGILE3D in zero-shot prompt segmentation on PartNet-Mobility (held-out categories). We find that AGILE3D achieves its best performance with a normalization scale of 5.
+
+Table 8: The effect of normalization scale for AGILE3D.  
+
+<table><tr><td>Normalization scale</td><td>IoU@3</td><td>IoU@5</td><td>IoU@7</td><td>IoU@9</td></tr><tr><td>1</td><td>29.0</td><td>36.4</td><td>41.0</td><td>44.7</td></tr><tr><td>3</td><td>35.7</td><td>43.1</td><td>48.4</td><td>51.7</td></tr><tr><td>5</td><td>40.8</td><td>50.8</td><td>57.4</td><td>61.9</td></tr><tr><td>7</td><td>36.8</td><td>45.6</td><td>51.3</td><td>57.5</td></tr><tr><td>10</td><td>35.1</td><td>42.2</td><td>48.6</td><td>53.1</td></tr></table>
+
+# C PROMPT SAMPLING
+
+Following AGILE3D (Yue et al., 2023), we sample two prompt points for each instance: one from false positive points and another from false negative points. The prompts are selected by identifying the foreground point that has the furthest distance to the nearest background point. Specifically, this involves computing pairwise distances from foreground points to background points, determining the minimum distance to background points for each foreground point, and selecting the foreground point with the maximum of these minimum distances.
+
+After computing the distances and selecting the candidates, we have two prompt point candidates: the point sampled from false positive points serves as a negative prompt, and the point sampled from false negative points serves as a positive prompt. We select the one with the furthest distance to the nearest background points as the final prompt point.
+
+# D VORONOITOKENIZER
+
+We show the whole table of the Voronoi tokenizer efficiency improvement in Table 9. Voronoi-based tokenizer surpasses the KNN-based tokenizer on all the datasets for both time and memory. Different with 2D SAM, the mask encoder of Point-SAM also need a point cloud tokenizer, which makes the voronoi tokenizer more important.
+
+# E TRAINING DATASETS ABLATION
+
+Table 10 presents the ablation study on training datasets evaluated across different datasets. This experiment highlights the scaling-up effect as more data is incorporated. The results demonstrate
+
+Table 9: This table shows the evaluation results of the Voronoi tokenizer. As the Voronoi tokenizer increases the time and memory efficiency significantly, it's very important for the real-world applications.  
+
+<table><tr><td></td><td>Method</td><td>FPS@1</td><td>FPS@5</td><td>FPS@10</td><td>Memory</td></tr><tr><td rowspan="2">PartNet-Mobility</td><td>KNN</td><td>22.3</td><td>11.7</td><td>7.3</td><td>2524</td></tr><tr><td>Voronoi</td><td>28.6 (+34.4%)</td><td>22.4 (+82.0%)</td><td>17.8 (+143.8%)</td><td>2078 (-21.5%)</td></tr><tr><td rowspan="2">ScanObjectNN</td><td>KNN</td><td>5.9</td><td>3.0</td><td>1.4</td><td>8548</td></tr><tr><td>Voronoi</td><td>7.3 (+23.7%)</td><td>4.9 (+63.3%)</td><td>2.3 (+62.0%)</td><td>6316 (-26.1%)</td></tr><tr><td rowspan="2">S3DIS</td><td>KNN</td><td>3.31</td><td>1.57</td><td>0.93</td><td>13680</td></tr><tr><td>Voronoi</td><td>3.52 (+34.4%)</td><td>1.72 (+8.7%)</td><td>1.04 (+11.8%)</td><td>10788 (-20.5%)</td></tr><tr><td rowspan="2">KITTI360</td><td>KNN</td><td>15.7</td><td>8.9</td><td>5.2</td><td>3890</td></tr><tr><td>Voronoi</td><td>21.1 (+34.4%)</td><td>16.2 (+82.0%)</td><td>13.7 (+163.4%)</td><td>3172 (-18.4%)</td></tr></table>
+
+Table 10: This table shows the datasets ablation results evaluated on all the evaluation datasets.  
+
+<table><tr><td>Eval Dataset</td><td>Point Number</td><td>PartNet</td><td>ScanNet</td><td>PartNet +ScanNet</td><td>PartNet +ShapeNet</td><td>PartNet +ScanNet+ShapeNet</td><td>Full</td></tr><tr><td rowspan="4">PartNet -Mobility</td><td>IoU@1</td><td>38.2</td><td>33.5</td><td>39.7</td><td>44.5</td><td>45.4</td><td>47.9</td></tr><tr><td>IoU@3</td><td>56.3</td><td>48.5</td><td>58.6</td><td>65.2</td><td>66.5</td><td>67.7</td></tr><tr><td>IoU@5</td><td>60.6</td><td>57.0</td><td>68.8</td><td>71.8</td><td>72.6</td><td>74.2</td></tr><tr><td>IoU@10</td><td>63.5</td><td>66.8</td><td>71.9</td><td>76.2</td><td>77.5</td><td>78.6</td></tr><tr><td rowspan="4">ScanObjectNN</td><td>IoU@1</td><td>38.4</td><td>32.0</td><td>44.8</td><td>45.2</td><td>48.2</td><td>49.4</td></tr><tr><td>IoU@3</td><td>63.9</td><td>56.7</td><td>71.8</td><td>72.4</td><td>73.9</td><td>75.3</td></tr><tr><td>IoU@5</td><td>66.8</td><td>63.2</td><td>78.9</td><td>80.2</td><td>81.4</td><td>82.0</td></tr><tr><td>IoU@10</td><td>67.7</td><td>70.5</td><td>80.8</td><td>81.4</td><td>84.8</td><td>86.3</td></tr><tr><td rowspan="4">S3DIS</td><td>IoU@1</td><td>25.6</td><td>38.8</td><td>43.5</td><td>30.5</td><td>46.2</td><td>47.6</td></tr><tr><td>IoU@3</td><td>48.7</td><td>65.3</td><td>73.6</td><td>54.1</td><td>75.9</td><td>78.4</td></tr><tr><td>IoU@5</td><td>63.2</td><td>71.4</td><td>83.9</td><td>67.9</td><td>85.8</td><td>86.2</td></tr><tr><td>IoU@10</td><td>67.9</td><td>80.6</td><td>88.3</td><td>76.3</td><td>90.1</td><td>90.4</td></tr><tr><td rowspan="4">KITTI360</td><td>IoU@1</td><td>36.2</td><td>44.0</td><td>46.3</td><td>39.0</td><td>47.3</td><td>49.4</td></tr><tr><td>IoU@3</td><td>62.9</td><td>67.1</td><td>70.5</td><td>67.8</td><td>72.1</td><td>74.4</td></tr><tr><td>IoU@5</td><td>68.9</td><td>72.2</td><td>77.9</td><td>73.5</td><td>80.4</td><td>81.7</td></tr><tr><td>IoU@10</td><td>74.8</td><td>80.8</td><td>83.6</td><td>80.9</td><td>85.0</td><td>85.8</td></tr></table>
+
+that adding data consistently enhances performance, although the extent of improvement varies depending on the type of data and the evaluation dataset. For example, adding ShapeNet leads to greater improvements on PartNet-Mobility, while adding ScanNet has a more pronounced effect on S3DIS. Furthermore, we observe that increasing the dataset size also improves performance on KITTI360, suggesting that the transferability of Point-SAM increases as the amount of training data grows.
+
+# F MORE VISUALIZATION
+
+We provide additional qualitative results in the appendix. Figure 5 presents qualitative results on the Waymo Open dataset. As a fully out-of-distribution experiment, this figure highlights the transferability of Point-SAM, demonstrating its ability to correctly segment outdoor objects such as cars and trees.
+
+Figure 4 shows the ground truth segmentation results alongside outcomes with varying numbers of prompt points. As Waymo is an OOD dataset, this figure demonstrates the superior transferability of Point-SAM.
+
+Figure 6 illustrates the few-shot segmentation results on the ShapeNet-Part dataset. The pre-trained model is used as the embedding for linear probing. We randomly select one sample from each category
+
+![](images/ae2753cab2297fc9f753c1eb66c940dd56e0d477952412a39ce58c27959d7dd9.jpg)  
+Figure 5: This figure shows the segmentation results of Waymo.
+
+![](images/241604ca7ff609733bfd112c6f55ad064a57afcfeb0d22f339da04beab5384bf.jpg)
+
+![](images/439e6453fb5e6e2d7ac8cb9bd77794016808a75d79c44accee9ea1f5250d3943.jpg)
+
+![](images/30b6cd6df6e06c18324bc2d8a5c7ad168bb5862e380b2a4aec8c765a5508b312.jpg)
+
+![](images/bf9297d701a84cb2dd98e8680df489e424276a232cdd34c3ee2eeff604da87f9.jpg)
+
+![](images/c7e45c33de6699fbf3fd3489a4c4000baaeefbaa0201dece44404912e8a80a83.jpg)
+
+![](images/063af03201d4a4ddc49af6dc5d1a6a4b3750db0e7f6b70f3933ff55bb5d666fb.jpg)
+
+![](images/bdb25891ccf6113624652ed4193580b1cf4e0517c40ec2ca8f58db62290af2dd.jpg)
+
+![](images/f0efb370a2f12e56d0c3f3573bffa1a55b1fcba783dd25a3e7f2d3e3cc62bbb7.jpg)
+
+![](images/2befd378268ba0faa160f62700205e514501aed83efc3e7ecf1291a1f89c86d6.jpg)
+
+![](images/1af3928ea2f081174c83dc8d9102c1d35fb42489acfe94ac9b7366174976af36.jpg)  
+Figure 6: This figure shows the few-shot segmentation results on ShapeNet-Part.
+
+![](images/e6b3c360c23de1c5d3e21e63540cb3ff9257130b8dd5fdb2d0b9320b59b02139.jpg)
+
+![](images/90c643811289f5f020ab7a8828e985e738797a97ca84fcfd2b364ee7e027bf93.jpg)
+
+![](images/5f126919af9c61855e2e9d512ce93e9312cfbf88cca2482c0f757ae5867786a8.jpg)
+
+![](images/19b80aff1baaed48398ce6dd94ffb577f7c7e364c436d8971e8935e05891da33.jpg)
+
+![](images/dbb12554c0c081f80ca0ee0e55b7ab569dd606d4fe05d03562053c56708c4044.jpg)
+
+as the training set, then perform inference on the evaluation set to obtain the one-shot probing results. These results demonstrate the superior pre-training quality of Point-SAM for segmentation tasks.
+
+# G EVALUATION FOR INTERIOR SEGMENTATION
+
+Table 11 shows the segmentation results for the StorageFurniture from PartNet-Mobility. We use the Point-SAM trained only on PartNet, and ScanNet. As shown in Table 11, MV-SAM achieves worse performance on StorageFurniture than the other 3 categories shown in Table 4. Point-SAM achieves better performance than MV-SAM with interior segmentation masks.
+
+Table 11: We present the evaluation results for the StorageFurniture category from PartNet-Mobility. To ensure the evaluation data is not in the training dataset, we used Point-SAM trained on PartNet, ShapeNet and ScanNet.
+
+<table><tr><td></td><td>IoU@1</td><td>IoU@3</td><td>IoU@5</td><td>IoU@7</td><td>IoU@10</td></tr><tr><td>MV-SAM</td><td>26.9</td><td>51.0</td><td>63.3</td><td>65.3</td><td>67.3</td></tr><tr><td>Point-SAM</td><td>42.3</td><td>63.4</td><td>68.6</td><td>71.8</td><td>74.5</td></tr></table>
+
+Table 12: This table presents the evaluation results on Replica. Following the evaluation procedure of AGILE3D, we crop the point clouds centered on the segmentation mask.  
+
+<table><tr><td></td><td>IoU@1</td><td>IoU@3</td><td>IoU@5</td><td>IoU@7</td><td>IoU@10</td></tr><tr><td>AGILE3D</td><td>55.9</td><td>74.8</td><td>81.7</td><td>85.4</td><td>87.9</td></tr><tr><td>Point-SAM</td><td>58.3</td><td>79.5</td><td>86.2</td><td>90.1</td><td>91.4</td></tr></table>
+
+# H EVALUATION ON REPLICA
+
+We present the quantitative results on Replica in Table 12. We still follow the progress of evaluating S3DIS and KITTI360 to split the scene into blocks centering on the segmentation target. In this experiment, we use the Point-SAM trained on the whole training dataset with Voronoi-based tokenizer. Point-SAM outperforms AGILE3D on the evaluation of Replica.
+
+# I MORE INTERACTIVE SEGMENTATION RESULTS
+
+Figure 7 shows more visualization results for complicated scenes and objects. We show both the segmentation results projected to meshes and the raw point cloud results. Figure 7 shows the superior transferability of Point-SAM.
+
+![](images/9b1f42defd1009d3a11e24dca15a09862c1e468cc937cd4985d373624322e4ee.jpg)
+
+![](images/8f0a8074ecebae0c9d4a55ecfce148d79fd3f58953e67e600fe14036032411b5.jpg)
+
+![](images/c077e53e517e13a4abdde0d21f9fcebea6629127afedbce9cf9999df7cefff28.jpg)
+
+![](images/4d221dc89b44772ee85475657ba73c48b5516a88753aebafd6f7086ac40b49e9.jpg)  
+mesh results
+
+![](images/d5cdb00ecca5fd156f6884e8710d19fe694e84b5adca9e5f0734b68b7c9f2a1b.jpg)
+
+![](images/49f8cd935228103f757cd1f57a9f5125ac1bf67975d446819ccf48e5dfd49e3d.jpg)
+
+![](images/2d4acb6bdc44823e2637f71b69114c57a2a4858c2ea2c84fcd293b2207badfeb.jpg)
+
+![](images/38123ee9be424f02a03b2fc5be00e4315a7150cf56f71adedba9aaf27e40e4bc.jpg)  
+point cloud results
+
+![](images/efb5732cbe2bb699d0f7410885d782a186cfbdde6d532a6b8cda5150ee0bc658.jpg)
+
+![](images/50f84a54f08a85d1a51d8c5025106eea9b3379dd09683f17b8b9675c9cabe847.jpg)
+
+![](images/285e36b3e89f0ec9012db58a9983ab2e13c90cec124971135ccdc615e7690c1f.jpg)
+
+![](images/bfa50fd044866976f92f90daa41e74db3013604f707be2af2bc6cc84cfff55d2.jpg)  
+mesh results
+
+![](images/30735831c95dd3dbf45da738b8938fb643382d6a1635da7072a34cd4a9de63c8.jpg)
+
+![](images/6426f5b8e8cb34f1fa025b8df6de24c03a6ea9784f8086922353fd8ed850f43f.jpg)
+
+![](images/986fa2708c336b74e4c33dc61c8bfca59c47ddbc7f5c4cc712cdfffbb04db7f2.jpg)
+
+![](images/3f66748a18c4547a05c545f01e37806ebb1ec1e49c1373bb2d35507dde395af5.jpg)  
+point cloud results  
+Figure 7: This figure presents additional visualization results of the interactive promptable segmentation. All objects were sourced from Polycam and Objaverse. We show both the projection results on meshes and the raw results on point clouds.

2025/Point-SAM_ Promptable 3D Segmentation Model for Point Clouds/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:32411cb5e28f0f6c15b1b46ab571a9325f7eefc9ea6c3bc54d0766982c3f5040
+size 1002544

2025/Point-based Instance Completion with Scene Constraints/62e2f7f2-4d65-4c81-bc65-22dc127b6093_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:dbd5db7d3489f0f873f88b062b83863989c031a47ef4d432005089eed814e783
+size 15121625

2025/Point-based Instance Completion with Scene Constraints/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b55b893073ac9f2d8d8269dd01df41b9c54c151c4c5c93e2ad3456c49940c75d
+size 1792425

2025/PointOBB-v2_ Towards Simpler, Faster, and Stronger Single Point Supervised Oriented Object Detection/860cef88-93f3-4f16-92f7-c5ac52feb051_content_list.json

@@ -0,0 +1,1988 @@
+[
+    {
+        "type": "text",
+        "text": "POINTOBB-v2: TOWARDS SIMPLER, FASTER, AND STRONGER SINGLE POINT SUPERVISED ORIENTED OBJECT DETECTION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            98,
+            823,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Botao Ren $^{1*}$ , Xue Yang $^{2*}$ , Yi Yu $^{3*}$ , Junwei Luo $^{4}$ , Zhidong Deng $^{1\\dagger}$",
+        "bbox": [
+            179,
+            193,
+            643,
+            210
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$^{1}$ BNRist, THUAI, Department of Computer Science, Tsinghua University",
+        "bbox": [
+            179,
+            210,
+            666,
+            224
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$^{2}$ Department of Automation, Shanghai Jiao Tong University",
+        "bbox": [
+            179,
+            224,
+            580,
+            238
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$^{3}$ Southeast University  $^{4}$ Wuhan University",
+        "bbox": [
+            179,
+            238,
+            475,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Code: https://github.com/taugeren/PointOBB-v2",
+        "bbox": [
+            179,
+            253,
+            635,
+            266
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
+        "bbox": [
+            450,
+            303,
+            547,
+            318
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Single point supervised oriented object detection has gained attention and made initial progress within the community. Diverse from those approaches relying on one-shot samples or powerful pretrained models (e.g. SAM), PointOBB has shown promise due to its prior-free feature. In this paper, we propose PointOBBv2, a simpler, faster, and stronger method to generate pseudo rotated boxes from points without relying on any other prior. Specifically, we first generate a Class Probability Map (CPM) by training the network with non-uniform positive and negative sampling. We show that the CPM is able to learn the approximate object regions and their contours. Then, Principal Component Analysis (PCA) is applied to accurately estimate the orientation and the boundary of objects. By further incorporating a separation mechanism, we resolve the confusion caused by the overlapping on the CPM, enabling its operation in high-density scenarios. Extensive comparisons demonstrate that our method achieves a training speed  $15.58 \\times$  faster and an accuracy improvement of  $11.60\\% / 25.15\\% / 21.19\\%$  on the DOTAv1.0/v1.5/v2.0 datasets compared to the previous state-of-the-art, PointOBB. This significantly advances the cutting edge of single point supervised oriented detection in the modular track.",
+        "bbox": [
+            228,
+            335,
+            767,
+            570
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            173,
+            597,
+            336,
+            612
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Oriented object detection is essential for accurately labeling small and densely packed objects, especially in scenarios like remote sensing imagery, retail analysis, and scene text detection, where Oriented Bounding Boxes (OBBs) provide precise annotations. However, annotating OBBs is labor-intensive and costly. Therefore, numerous weakly supervised methods have emerged in recent years, including horizontal bounding box supervision and point supervision. Representative methods for horizontal bounding box supervision include H2Box (Yang et al., 2023a) and H2Box-v2 (Yu et al., 2023). In addition, point supervision, which only requires labeling the point and category for each object, significantly reduces the annotation cost. Notable point-supervised methods include P2Box (Cao et al., 2024), Point2Box (Yu et al., 2024), and PointOBB (Luo et al., 2024).",
+        "bbox": [
+            169,
+            628,
+            826,
+            755
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Fig. 1, existing point-supervised oriented object detection methods can be broadly categorized into three paradigms: (a) SAM-based methods (Cao et al., 2024; Zhang et al., 2024) rely on the powerful SAM (Kirillov et al., 2023) model, which, although effective in natural images, struggles with cross-domain tasks like aerial imagery, particularly in small object and densely packed scenarios. Additionally, SAM-based methods are slow and memory-intensive due to post-processing; (b) Prior-based Weakly-supervised Oriented Object Detection (WOOD) methods, such as Point2RBox (Yu et al., 2024), integrate human priors which reduce generalizability since different datasets require distinct prior knowledge. Further, the end-to-end setup limits flexibility, preventing these methods from leveraging more powerful detectors and benefiting from their performance improvements; (c) Modular WOOD methods (Luo et al., 2024) do not rely on manually designed",
+        "bbox": [
+            169,
+            760,
+            826,
+            902
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "*Equal contribution. †Corresponding author",
+        "bbox": [
+            199,
+            909,
+            462,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/0a1eb4a75f3f9107bee695a4cf484ee986041d336a1270efd534a1dd44f5a560.jpg",
+        "image_caption": [
+            "Figure 1: Compare with existing point supervised methods, including (a) Prompt OOD (i.e. SAM based); (b-c) Weakly OOD: Prior-based and Modular. OOD means Oriented Object Detection."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            102,
+            290,
+            292
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/8da11f1dcd12aa2cfec3b83c555d6f06caa1f0d75e921f80115adfc68c681b87.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            321,
+            101,
+            444,
+            291
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/673777d9a5a05a77fd41764d70c61b48847fdf411ab5486545253475426ca2ea.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            452,
+            101,
+            812,
+            291
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "prriors and offer greater flexibility by decoupling pseudo-label generation from the detector, making them more suitable for efficient and scalable detection tasks.",
+        "bbox": [
+            169,
+            361,
+            823,
+            390
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As a previous state-of-the-art method, PointOBB falls under the modular WOOD category and offers a feasible solution for point-supervised detection. However, it has several practical limitations: the pseudo-label generation process is very slow, taking approximately 7-8 times longer than the subsequent detector training. Additionally, its training requires significant GPU memory due to multiple view transformations. Moreover, the variability in the number of Region of Interest (RoI) proposals can lead to out-of-memory issues, particularly in dense object scenarios. Although limiting the number of RoI proposals can mitigate this issue, it results in degraded performance.",
+        "bbox": [
+            169,
+            397,
+            823,
+            494
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Considering the aforementioned issues, our motivation is to design a simpler, faster, and stronger method, which leads to the development of PointOBB-v2. Our approach aims to retain the strengths of the modular WOOD paradigm while addressing the inefficiencies of PointOBB, particularly in terms of speed and memory consumption, making it more suitable for real-world applications.",
+        "bbox": [
+            169,
+            501,
+            823,
+            559
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "PointOBB-v2 introduces a novel and concise pipeline that discards the teacher-student structure, achieving significant improvements in the accuracy and generation speed of the pseudo-label, and improving memory efficiency, especially in scenarios with small and dense objects. Specifically, we generate Class Probability Maps (CPM) from point annotations and design a novel sample assignment strategy to capture object contours and orientations from the CPM. Next, we apply non-uniform sampling based on the probability distribution and use Principal Component Analysis (PCA) to determine object boundaries and directions. To address dense object distributions, we design a separation mechanism to reduce confusion in pseudo-label generation caused by connected CPMs.",
+        "bbox": [
+            169,
+            564,
+            825,
+            676
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Experimental results demonstrate that our method consistently improves accuracy, speed, and memory efficiency across various datasets compared to PointOBB, achieving several state-of-the-art results. Specifically, on the DOTA-v1.0 dataset, our method, when using pseudo-labels for training with Rotated FCOS, improves the mAP from  $30.08\\%$  (PointOBB) to  $41.68\\%$ , a gain of  $11.60\\%$  mAP. In more challenging datasets like DOTA-v1.5 and DOTA-v2.0, which contain a higher density of small objects, our method achieves mAP of  $36.39\\%$  and  $27.22\\%$ , with respective gains of  $25.15\\%$  and  $21.19\\%$  over PointOBB, demonstrating its robustness in handling small and densely packed objects. Furthermore, our pseudo-label generation process is 15.58 times faster, reducing the time from 22.28 hours to 1.43 hours. On the DOTA-v1.5 and DOTA-v2.0 datasets, PointOBB requires limiting the number of RoI proposals due to high memory consumption, while our method operates without such restrictions, with a memory usage of approximately 8GB.",
+        "bbox": [
+            169,
+            681,
+            825,
+            837
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are summarized as follows:",
+        "bbox": [
+            171,
+            842,
+            473,
+            856
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "- We propose a novel and efficient pipeline for point-supervised oriented object detection, which eliminates the time- and memory-consuming teacher-student structure, significantly improving pseudo-label generation speed and reducing memory usage.",
+        "bbox": [
+            171,
+            881,
+            823,
+            925
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Without any additional deep network design, our method relies solely on class probability maps to generate accurate object contours, using efficient PCA to determine object directions and boundaries. We also design a vector constraint method to distinguish small objects in dense scenarios, improving detection performance.",
+            "- Experimental results show that our method consistently outperforms PointOBB across multiple datasets, achieving  $11.60\\% / 25.15\\% / 21.19\\%$  gain on the DOTA-v1.0/v1.5/v2.0 datasets, with a  $15.58 \\times$  speedup in pseudo-label generation and memory usage reduced to approximately 8GB without limiting RoI proposals."
+        ],
+        "bbox": [
+            171,
+            103,
+            826,
+            220
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK",
+        "text_level": 1,
+        "bbox": [
+            171,
+            242,
+            346,
+            258
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In addition to horizontal detection (Zhao et al., 2019; Liu et al., 2020), oriented object detection (Yang et al., 2018; Wen et al., 2023) has received extensive attention. In this section, we first introduce oriented detection supervised by rotated boxes. Then, approaches to point-supervised oriented detection and other weakly-supervised settings are discussed.",
+        "bbox": [
+            169,
+            273,
+            826,
+            330
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 RBOX-SUPERVISED ORIENTED DETECTION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            348,
+            516,
+            363
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Representative works include anchor-based detector Rotated RetinaNet (Lin et al., 2020), anchor-free detector Rotated FCOS (Tian et al., 2019), and two-stage solutions, e.g. RoI Transformer (Ding et al., 2019), Oriented R-CNN (Xie et al., 2021), and ReDet (Han et al., 2021). Some research enhances the detector by exploiting alignment features, e.g.  $\\mathbf{R}^3\\mathrm{Det}$  (Yang et al., 2021b) and  $\\mathbf{S}^2\\mathbf{A}$ -Net (Han et al., 2022). The angle regression may face boundary discontinuity and remedies are developed, including modulated losses (Yang et al., 2019a; 2022; Qian et al., 2021) that alleviate loss jumps, angle coders (Yang & Yan, 2020; Yang et al., 2021a; Yang & Yan, 2022; Yu & Da, 2023) that convert the angle into boundary-free coded data, and Gaussian-based losses (Yang et al., 2021c;d; 2023b;c; Murragarra-Llerena et al., 2024) transforming rotated bounding boxes into Gaussian distributions. RepPoint-based methods (Yang et al., 2019b; Hou et al., 2023; Li et al., 2022) provide alternatives that predict a set of sample points that bounds the spatial extent of an object.",
+        "bbox": [
+            169,
+            375,
+            826,
+            529
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 POINT-SUPERVISED ORIENTED DETECTION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            546,
+            516,
+            559
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recently, several methods for point-supervised oriented detection have been proposed: 1) P2RBox (Cao et al., 2024), PMHO (Zhang et al., 2024) and PointSAM (Liu et al., 2024) propose oriented object detection with point prompts by employing the zero-shot Point-to-Mask ability of SAM (Kirillov et al., 2023). 2) Point2RBox (Yu et al., 2024) has introduced a novel end-to-end approach based on knowledge combination in this domain. 3) PointOBB (Luo et al., 2024) achieves point annotation based RBox generation method for oriented object detection through scale-sensitive consistency and multiple instance learning.",
+        "bbox": [
+            169,
+            571,
+            823,
+            671
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Among these methods, P2RBox, PMHO and PointSAM require SAM model pre-trained on massive amounts of labeled data, whereas Point2RBox requires one-shot samples (i.e. human priors) for each category. Although achieving better accuracy, they are not as general as PointOBB. Hence, we choose PointOBB as our baseline to develop a simpler, faster, and stronger method, PointOBB-v2.",
+        "bbox": [
+            169,
+            676,
+            823,
+            734
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 OTHER WEAKLY-SUPERVISED SETTINGS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            751,
+            496,
+            763
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Compared to the Point-to-RBox, some other weakly-supervised settings have been better studied. These methods are potentially applicable to our Point-to-RBox task setting by using a cascade pipeline, such as Point-to-HBox-to-RBox and Point-to-Mask-to-RBox. In our experiment, cascade pipelines powered by state-of-the-art weakly-supervised approaches are also adopted for comparison. Here, several representative work are introduced.",
+        "bbox": [
+            169,
+            777,
+            823,
+            847
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "HBox-to-RBox. The seminal work H2RBox (Yang et al., 2023a) circumvents the segmentation step and achieves RBox detection directly from HBox annotation. With HBox annotations for the same object in various orientations, the geometric constraint limits the object to a few candidate angles. Supplemented with a self-supervised branch eliminating the undesired results, an HBox-to-RBox paradigm is established. An enhanced version H2RBox-v2 (Yu et al., 2023) is proposed to",
+        "bbox": [
+            169,
+            854,
+            826,
+            925
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/5301b9c0d962d71bd8e1c39c40f190778058bc02e68accb99ea91eff31f4ccb5.jpg",
+        "image_caption": [
+            "Figure 2: Our PointOBB-v2 first generates a Class Probability Map (CPM) with a positive and negative sample assignment strategy during training. It then applies Principal Component Analysis (PCA) to infer object orientation and boundaries to generate pseudo labels."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            99,
+            808,
+            385
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "leverage the reflection symmetry of objects to estimate their angle, further boosting the HBox-to-RBox performance. EIE-Det (Wang et al., 2024) uses an explicit equivariance branch for learning rotation consistency, and an implicit equivariance branch for learning position, aspect ratio, and scale consistency. Some studies (Iqbal et al., 2021; Sun et al., 2021; Zhu et al., 2023) use additional annotated data for training, which are also attractive but less general.",
+        "bbox": [
+            169,
+            457,
+            823,
+            529
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Point-to-HBox. Several related approaches have been developed, including: 1) P2BNet (Chen et al., 2022) samples box proposals of different sizes around the labeled point and classify them to achieve point-supervised horizontal object detection. 2) PSOD (Gao et al., 2022) achieves point-supervised salient object detection using an edge detector and adaptive masked flood fill.",
+        "bbox": [
+            169,
+            534,
+            823,
+            592
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Point-to-Mask. Point2Mask (Li et al., 2023) is proposed to achieve panoptic segmentation using only a single point annotation per target for training. SAM (Segment Anything Model) (Kirillov et al., 2023) produces object masks from input point/HBox prompts. Though RBoxes can be obtained from the segmentation mask by finding the minimum circumscribed rectangle, such a complex pipeline can be less cost-efficient and perform worse (Yang et al., 2023a; Yu et al., 2023).",
+        "bbox": [
+            169,
+            597,
+            823,
+            669
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD",
+        "text_level": 1,
+        "bbox": [
+            171,
+            689,
+            282,
+            704
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our task focuses on oriented object detection with single point supervision. We first utilize point annotations for each object in the training dataset to generate pseudo labels, which are then used to train an existing detector. As shown in Fig. 2, the model first generates a Class Probability Map (CPM) based on the point annotations. Specifically, during training, we devise a positive and negative sample assignment strategy, where the resulting CPM outlines the rough object contours, with higher probabilities concentrated around the point and along the object axes.",
+        "bbox": [
+            169,
+            720,
+            823,
+            806
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We generate pseudo oriented bounding boxes according to CPM. We perform non-uniform sampling around the point annotation of each object, guided by the probability distribution within the CPM. We convert the sampling process into a weighted probability approach, which maintains the same expected result while eliminating the variance introduced by random sampling. By applying Principal Component Analysis (PCA) to the weighted grid points, we can infer the object's orientation. We then determine the object boundaries by combining the thresholded CPM with the inferred orientation. Furthermore, to address densely populated object scenarios, we introduce a mechanism for differentiating between closely situated objects, ensuring effective separation and accurate detection.",
+        "bbox": [
+            169,
+            811,
+            826,
+            925
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/363db275a12fa093b7a98a9b6b195c6e43e167aef7865bc1808c9a33e080fec9.jpg",
+        "image_caption": [
+            "Figure 3: Visualization of class probability map (CPM)."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            102,
+            491,
+            220
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/de55cbc82d801ac4684b76d258aaa9bde3e438dc2ead318a1dbe3b052a6f1553.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            504,
+            101,
+            635,
+            220
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/6e1ccfc9e13d3527b1f6685390f1c77904ce9bfb52b5928190edf054b0734d9c.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            635,
+            101,
+            761,
+            220
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 CLASS PROBABILITY MAP GENERATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            260,
+            496,
+            273
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The Class Probability Map (CPM) represents the per-class probability for each point on the feature map, with values ranging between [0, 1]. To generate the CPM, our model first takes an image  $I$  of dimension  $(C, H, W)$  as input and processes it through a ResNet-50 (He et al., 2016) backbone with an FPN (Lin et al., 2017) structure. The final class probability map is derived from the highest-resolution feature map of the FPN, which is then projected through a projection layer. The output is a map of size  $(N_{\\mathrm{class}}, H_0, W_0)$ . Formally, it is defined as:",
+        "bbox": [
+            169,
+            286,
+            823,
+            371
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {C P M} = \\operatorname {P r o j} (f (I) _ {0}), \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            424,
+            378,
+            823,
+            395
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where CPM is the class probability map,  $\\mathrm{Proj}(\\cdot)$  represents the projection layer, and  $f(I)_0$  is the highest-resolution feature map from the ResNet-50 + FPN.",
+        "bbox": [
+            169,
+            402,
+            823,
+            431
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 LABEL ASSIGNMENT",
+        "text_level": 1,
+        "bbox": [
+            171,
+            449,
+            362,
+            462
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A key component of our approach is the design of a robust sample assignment strategy for both positive and negative samples. This strategy is essential for building an accurate CPM, which outlines rough object contours, concentrating higher probabilities around object centers and along their axes. To ensure reliable separation of objects, especially in densely populated scenarios, our method addresses the challenge of closely situated objects by introducing additional mechanisms for effective differentiation. We illustrate this label assignment in the upper-right part of Fig. 2. The specific details of the sample assignment process are outlined below:",
+        "bbox": [
+            169,
+            476,
+            823,
+            573
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Positive Label Assignment. For positive samples, we select all points within a fixed radius  $b_{1}$  (set to 6 in our model) around each point. If a point lies within multiple such radii, it is assigned to the closest center. The condition for positive samples is as follows:",
+        "bbox": [
+            169,
+            580,
+            823,
+            622
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\exists \\mathbf {G T} _ {i} \\in \\mathbf {G T} _ {1 \\sim N}, (d (p, \\mathbf {G T} _ {i}) <   b _ {1}) \\wedge (d (p, \\mathbf {G T} _ {i}) = \\min  (d (p, \\mathbf {G T} _ {1 \\sim N}))) \\\\ \\Rightarrow p \\text {i s p o s i t i v e , c l s} (p) = \\operatorname {c l s} \\left(\\mathrm {G T} _ {i}\\right). \\tag {2} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            628,
+            823,
+            664
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Negative Label Assignment. Given  $N$  ground truth objects (GT), for each  $\\mathrm{GT}_i$ , we identify its nearest neighboring object  $\\mathrm{GT}_j$  based on the Euclidean distance. This gives us a vector  $\\mathrm{dist}_{min}$  with dimension  $[N]$ , where each element  $\\mathrm{dist}_i$  represents the minimum distance between  $\\mathrm{GT}_i$  and its closest neighbor. We then draw a circle with radius  $\\alpha \\times \\mathrm{dist}_i$  around  $\\mathrm{GT}_i$ , where  $\\alpha$  (set to 1 in our model) is a fixed proportional constant. Points outside all such circles are designated as negative samples. The negative sample condition is formulated as:",
+        "bbox": [
+            169,
+            679,
+            823,
+            763
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\forall \\mathrm {G T} _ {i} \\in \\mathrm {G T} _ {1 \\sim N}, d (p, \\mathrm {G T} _ {i}) > \\alpha \\times \\operatorname {d i s t} _ {i} \\Rightarrow p \\text {i s n e g a t i v e}. \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            772,
+            823,
+            787
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In addition to the above defined negative labels, we also set the middle region between objects as negative to make the boundaries clearer between densely packed objects (denoted as \"Neg./M\" in ablation Tab. 4). For each  $\\mathrm{GT}_i$ , we identify its nearest neighbor  $\\mathrm{GT}_j$  that belongs to the same class. A circle is drawn with a radius  $b_2$  (set to 4 in our model) and centered at the midpoint of the line connecting  $\\mathrm{GT}_i$  and  $\\mathrm{GT}_j$ , and points within this circle are assigned as negative samples. The condition is defined as:",
+        "bbox": [
+            169,
+            803,
+            823,
+            886
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\forall \\mathrm {G T} _ {i} \\in \\mathrm {G T} _ {1 \\sim N}, \\exists \\mathrm {G T} _ {j} \\in \\mathrm {G T} _ {1 \\sim N}, d (\\mathrm {G T} _ {i}, \\mathrm {G T} _ {j}) = \\min  (d (\\mathrm {G T} _ {i}, \\mathrm {G T} _ {1 \\sim N})) \\\\ \\wedge \\operatorname {c l s} \\left(\\mathrm {G T} _ {i}\\right) = \\operatorname {c l s} \\left(\\mathrm {G T} _ {j}\\right) \\wedge d (p, \\left(\\mathrm {G T} _ {i} + \\mathrm {G T} _ {j}\\right) / 2) <   b _ {2} \\Rightarrow p \\text {i s n e g a t i v e}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            893,
+            740,
+            926
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Robustness. While we do not explicitly define positive and negative samples based on precise object contours or oriented bounding boxes, which may result in some inaccuracies during label assignment (i.e. incorrectly assigning a small portion of positive or negative samples during training), this does not significantly hinder our method's ability to learn accurate object contours. These minor label assignment inaccuracies, particularly in densely populated regions or for objects with extreme aspect ratios, do not affect the overall robustness and effectiveness of the approach. As demonstrated in Fig. 3, our strategy is capable of learning the correct contours, even for objects with large aspect ratios and in densely packed scenarios.",
+        "bbox": [
+            169,
+            103,
+            826,
+            218
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 ORIENTATION AND BOUNDARY ESTIMATION VIA PCA",
+        "text_level": 1,
+        "bbox": [
+            171,
+            232,
+            591,
+            247
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "After obtaining the CPM, we sample points around each ground truth based on the class probabilities, and then apply Principal Component Analysis (PCA) on the sampled points to determine the object's orientation. As shown in bottom part of Fig. 2, we sample points around the GT based on the probabilities in the CPM for the corresponding object class. We choose a  $7 \\times 7$  grid centered at the GT with the coordinates of the 49 integer points  $z_{1 \\sim 49}$  as:",
+        "bbox": [
+            169,
+            257,
+            823,
+            329
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n(x, y), x \\in [ - 3, 3 ], y \\in [ - 3, 3 ]. \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            334,
+            823,
+            351
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For each grid point  $z_{i}$ , we can compute the CPM probability  $p_{i}$  and decide whether to sample the point based on this probability. Once we have the sampled points, we apply PCA to find the primary direction of the point set, which represents the object's orientation. While PCA provides the correct primary direction in expectation, the randomness introduced by sampling can cause variance in the result from a single pass. Although averaging over multiple sampling runs can mitigate this variance, it also increases computational cost.",
+        "bbox": [
+            169,
+            363,
+            823,
+            448
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To address this, we propose an equivalent method that transforms probabilistic sampling into a weighted coordinate transformation. Instead of sampling points probabilistically, we assign a weight of  $p_i$  to each point  $z_i$ , ensuring the same expected outcome while eliminating the variance caused by random sampling. The covariance matrix is then defined as:",
+        "bbox": [
+            169,
+            454,
+            826,
+            510
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nC _ {z} = \\sum_ {i = 1} ^ {N} p _ {i} \\left(z _ {i} - \\mu_ {z}\\right) ^ {\\mathrm {T}} \\left(z _ {i} - \\mu_ {z}\\right). \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            517,
+            823,
+            558
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We then perform eigenvalue decomposition on  $C_z$ :",
+        "bbox": [
+            171,
+            571,
+            509,
+            585
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nC _ {z} v _ {i} = \\lambda_ {i} v _ {i}. \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            450,
+            593,
+            823,
+            608
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The eigenvector  $v_{1}$  corresponding to the largest eigenvalue  $\\lambda_{1}$  is chosen as the primary direction. Since  $C_z$  is a real symmetric matrix, the secondary direction is guaranteed to be orthogonal to the primary direction. This orthogonality corresponds to the perpendicular relationship between the two adjacent sides of an oriented bounding box.",
+        "bbox": [
+            169,
+            622,
+            823,
+            678
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "After identifying the primary and secondary directions, we determine the object boundaries along these directions. Starting from the center, we move along each direction and stop when the value at a position falls below a threshold, indicating the object boundary.",
+        "bbox": [
+            169,
+            684,
+            823,
+            728
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 OBJECT DIFFERENTIATION IN DENSE SCENARIOS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            743,
+            558,
+            757
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In dense scenarios, objects can be difficult to distinguish on the CPM. This can affect the PCA's ability to determine the object orientation and the boundary identification. To address this, we design a \"Vector Constraint Suppression\" method to resolve boundary ambiguity.",
+        "bbox": [
+            169,
+            768,
+            823,
+            813
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Vector Constraint Suppression. Even after determining the correct orientation in dense scenarios, the object boundaries may still be unclear, making it difficult to precisely locate them using the probability threshold described in Sec. 3.3. In most cases, simply distinguishing between two closely positioned objects is sufficient to define the object's boundary.",
+        "bbox": [
+            169,
+            818,
+            823,
+            876
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We propose a simple constraint: For each  $\\mathrm{GT}_i$ , we first find its nearest same-class neighbor  $\\mathrm{GT}_j$  and compute the vector  $u = \\langle \\mathrm{GT}_i, \\mathrm{GT}_j \\rangle$  between  $\\mathrm{GT}_i$  and  $\\mathrm{GT}_j$ . If the angle between this vector and the primary or secondary direction is smaller than a threshold  $\\alpha$  (set to  $\\pi/6$  in our model),",
+        "bbox": [
+            169,
+            881,
+            825,
+            925
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/82f92b532300f3ed0bcf0501b0a64ee745fa1c25b7d0aaa606abf4c291a48243.jpg",
+        "table_caption": [
+            "Table 1: Results of each category on the DOTA-v1.0 dataset. FCOSR and ORCNN refer to Rotated FCOS and Oriented R-CNN. * indicates using additional human knowledge priors."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>PL</td><td>BD</td><td>BR</td><td>GTF</td><td>SV</td><td>LV</td><td>SH</td><td>TC</td><td>BC</td><td>ST</td><td>SBF</td><td>RA</td><td>HA</td><td>SP</td><td>HC</td><td>\\( \\mathbf{{mAP}}{}_{50} \\)</td></tr><tr><td>Point2Mask-RBox</td><td>4.0</td><td>23.1</td><td>3.8</td><td>1.3</td><td>15.1</td><td>1.0</td><td>3.3</td><td>19.0</td><td>1.0</td><td>29.1</td><td>0.0</td><td>9.5</td><td>7.4</td><td>21.1</td><td>7.1</td><td>9.72</td></tr><tr><td>P2BNet+H2RBox</td><td>24.7</td><td>35.9</td><td>7.0</td><td>27.9</td><td>3.3</td><td>12.1</td><td>17.5</td><td>17.5</td><td>0.8</td><td>34.0</td><td>6.3</td><td>49.6</td><td>11.6</td><td>27.2</td><td>18.8</td><td>19.63</td></tr><tr><td>P2BNet+H2RBox-v2</td><td>11.0</td><td>44.8</td><td>14.9</td><td>15.4</td><td>36.8</td><td>16.7</td><td>27.8</td><td>12.1</td><td>1.8</td><td>31.2</td><td>3.4</td><td>50.6</td><td>12.6</td><td>36.7</td><td>12.5</td><td>21.87</td></tr><tr><td>Point2RBox-RC</td><td>62.9</td><td>64.3</td><td>14.4</td><td>35.0</td><td>28.2</td><td>38.9</td><td>33.3</td><td>25.2</td><td>2.2</td><td>44.5</td><td>3.4</td><td>48.1</td><td>25.9</td><td>45.0</td><td>22.6</td><td>34.07</td></tr><tr><td>Point2RBox-SK*</td><td>53.3</td><td>63.9</td><td>3.7</td><td>50.9</td><td>40.0</td><td>39.2</td><td>45.7</td><td>76.7</td><td>10.5</td><td>56.1</td><td>5.4</td><td>49.5</td><td>24.2</td><td>51.2</td><td>33.8</td><td>40.27</td></tr><tr><td>PointOBB (FCOSR)</td><td>26.1</td><td>65.7</td><td>9.1</td><td>59.4</td><td>65.8</td><td>34.9</td><td>29.8</td><td>0.5</td><td>2.3</td><td>16.7</td><td>0.6</td><td>49.0</td><td>21.8</td><td>41.0</td><td>36.7</td><td>30.08</td></tr><tr><td>PointOBB-v2 (FCOSR)</td><td>64.5</td><td>27.8</td><td>1.9</td><td>36.2</td><td>58.8</td><td>47.2</td><td>53.4</td><td>90.5</td><td>62.2</td><td>45.3</td><td>12.1</td><td>41.7</td><td>8.1</td><td>43.7</td><td>32.0</td><td>41.68</td></tr><tr><td>PointOBB (ORCNN)</td><td>28.3</td><td>70.7</td><td>1.5</td><td>64.9</td><td>68.8</td><td>46.8</td><td>33.9</td><td>9.1</td><td>10.0</td><td>20.1</td><td>0.2</td><td>47.0</td><td>29.7</td><td>38.2</td><td>30.6</td><td>33.31</td></tr><tr><td>PointOBB-v2 (ORCNN)</td><td>63.7</td><td>45.6</td><td>2.0</td><td>39.5</td><td>50.5</td><td>49.6</td><td>45.4</td><td>89.8</td><td>62.9</td><td>41.3</td><td>13.6</td><td>42.8</td><td>8.9</td><td>39.5</td><td>29.5</td><td>41.64</td></tr><tr><td>PointOBB (ReDet)</td><td>24.2</td><td>75.0</td><td>0.5</td><td>60.6</td><td>59.3</td><td>46.1</td><td>45.7</td><td>6.1</td><td>10.1</td><td>25.0</td><td>0.2</td><td>50.4</td><td>30.0</td><td>45.0</td><td>31.1</td><td>33.95</td></tr><tr><td>PointOBB-v2 (ReDet)</td><td>65.4</td><td>52.1</td><td>2.2</td><td>44.4</td><td>55.0</td><td>49.3</td><td>51.8</td><td>89.0</td><td>70.2</td><td>47.0</td><td>16.2</td><td>43.9</td><td>13.0</td><td>43.8</td><td>29.4</td><td>44.85</td></tr></table>",
+        "bbox": [
+            173,
+            136,
+            823,
+            286
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "we consider this direction valid for boundary definition. The boundary is then constrained by the following condition:",
+        "bbox": [
+            169,
+            306,
+            823,
+            335
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nu \\cdot v _ {k} <   \\frac {1}{2} \\times d \\left(\\mathrm {G T} _ {i}, \\mathrm {G T} _ {j}\\right) \\quad \\text {i f a n g l e} (u, v _ {k}) <   \\alpha \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            342,
+            825,
+            372
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where  $v_{k}$  is the primary or secondary direction,  $\\mathrm{GT}_j$  is the nearest same-class object to  $\\mathrm{GT}_i$ , and  $\\frac{1}{2}$  means that the boundary should be closer to  $\\mathrm{GT}_i$  than to  $\\mathrm{GT}_j$ .",
+        "bbox": [
+            169,
+            381,
+            823,
+            411
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENT",
+        "text_level": 1,
+        "bbox": [
+            171,
+            431,
+            318,
+            448
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 DATASETS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            465,
+            290,
+            479
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "DOTA (Xia et al., 2018) is a large-scale dataset designed for object detection in aerial images, covering various object categories and complexities. DOTA has three versions:",
+        "bbox": [
+            169,
+            491,
+            823,
+            521
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "DOTA-v1.0 has 2,806 images with 188,282 instances across 15 categories, annotated as: Plane (PL), Baseball Diamond (BD), Bridge (BR), Ground Track Field (GTF), Small Vehicle (SV), Large Vehicle (LV), Ship (SH), Tennis Court (TC), Basketball Court (BC), Storage Tank (ST), Soccer Ball Field (SBF), Roundabout (RA), Harbor (HA), Swimming Pool (SP), Helicopter (HC). The images range from  $800 \\times 800$  to  $4,000 \\times 4,000$  pixels and exhibit significant variation in scale and orientation.",
+        "bbox": [
+            169,
+            526,
+            825,
+            611
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "DOTA-v1.5 extends DOTA-v1.0 by adding annotations for extremely small objects (less than 10 pixels) and introducing a new category, Container Crane (CC). It includes a total of 403,318 instances while retaining the same image count and dataset split as DOTA-v1.0.",
+        "bbox": [
+            169,
+            617,
+            825,
+            659
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "DOTA-v2.0 further expands the dataset to 11,268 images and 1,793,658 instances, covering 18 categories. Two additional categories, Airport (AP) and Helipad (HP), are introduced, providing a more diverse and challenging set of aerial images.",
+        "bbox": [
+            169,
+            666,
+            823,
+            709
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "RSAR (Zhang et al., 2025) is a large-scale dataset for rotated object detection in SAR imagery, comprising 95,842 images and 183,534 annotated instances across 6 categories.",
+        "bbox": [
+            169,
+            715,
+            823,
+            744
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "SKU110K-R (Pan et al., 2020) is a dataset focused on detecting densely packed retail scenes, consisting of 11,762 images and 110,712 annotated objects.",
+        "bbox": [
+            169,
+            750,
+            823,
+            780
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTAL SETTINGS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            797,
+            401,
+            811
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our implementation is based on the MMRotate library (Zhou et al., 2022). In the pseudo-label generation stage, we train the model for 6 epochs using momentum SGD as the optimizer. We set the weight decay to 1e-4, with an initial learning rate of 0.005, which decays by a factor of 10 after the  $4^{\\text{th}}$  epoch. The batch size for training is set to 2. For the detector training phase using pseudolabels, we use the same detector configurations as the default settings in MMRotate. Throughout the entire training process, random flipping is employed as the only data augmentation technique. Our experiments were accelerated using two GeForce RTX 3090 GPUs.",
+        "bbox": [
+            169,
+            824,
+            825,
+            925
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/df096b1a274569f6bed076c48c4653008d102ee20fb302363f2c60a5708b7d98.jpg",
+        "table_caption": [
+            "Table 2: Results on DOTA-v1.0/v1.5/v2.0, RSAR and SKU110K-R, reporting the  $\\mathbf{mAP}_{50}$  metric. FCOSR and ORCNN refer to Rotated FCOS and Oriented R-CNN. * indicates using additional human knowledge priors."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>DOTA-v1.0</td><td>DOTA-v1.5</td><td>DOTA-v2.0</td><td>RSAR</td><td>SKU110K-R</td></tr><tr><td>Point2Mask-RBox</td><td>9.72</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>P2BNet+H2RBox</td><td>19.63</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>P2BNet+H2RBox-v2</td><td>21.87</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Point2RBox-RC</td><td>34.07</td><td>24.31</td><td>14.69</td><td>-</td><td>3.41</td></tr><tr><td>Point2RBox-SK*</td><td>40.27</td><td>30.51</td><td>23.43</td><td>-</td><td>-</td></tr><tr><td>PointOBB (FCOSR)</td><td>30.08</td><td>10.66</td><td>5.53</td><td>13.80</td><td>-</td></tr><tr><td>PointOBB-v2 (FCOSR)</td><td>41.68 (+11.60)</td><td>30.59 (+19.93)</td><td>20.64 (+15.11)</td><td>18.99 (+5.19)</td><td>56.63</td></tr><tr><td>PointOBB (ORCNN)</td><td>33.31</td><td>10.92</td><td>6.29</td><td>12.30</td><td>-</td></tr><tr><td>PointOBB-v2 (ORCNN)</td><td>41.64 (+8.33)</td><td>32.01 (+21.09)</td><td>23.40 (+17.11)</td><td>22.61 (+10.31)</td><td>62.76</td></tr><tr><td>PointOBB (ReDet)</td><td>33.95</td><td>11.24</td><td>6.03</td><td>-</td><td>-</td></tr><tr><td>PointOBB-v2 (ReDet)</td><td>44.85 (+10.90)</td><td>36.39 (+25.15)</td><td>27.22 (+21.19)</td><td>-</td><td>-</td></tr></table>",
+        "bbox": [
+            173,
+            148,
+            823,
+            311
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/ebc121913fe89461ff168d209f7d25918611020923f1675c1dc3490abb7b6d79.jpg",
+        "table_caption": [
+            "Table 3: Comparison of the training time in pseudolabel generation phase and the accuracy between PointOBB and PointOBB-v2. The reported  $\\mathbf{mAP}_{50}$  is trained with Rotated FCOS."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Epochs</td><td>Train Hours</td><td>mAP50</td></tr><tr><td>PointOBB</td><td>24</td><td>22.28</td><td>30.08</td></tr><tr><td>PointOBB-v2</td><td>6</td><td>1.43</td><td>41.68</td></tr></table>",
+        "bbox": [
+            179,
+            388,
+            509,
+            439
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/98bd9121cd2099195f97e635a7560d9e33cc98c39522ae0689818777466a6750.jpg",
+        "table_caption": [
+            "Table 4: Ablation study with different label assignment strategies."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Pos.</td><td>Neg.</td><td>Neg./M</td><td>mAP50</td></tr><tr><td>✓</td><td></td><td></td><td>23.62</td></tr><tr><td>✓</td><td>✓</td><td></td><td>44.75</td></tr><tr><td>✓</td><td></td><td>✓</td><td>18.21</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>44.85</td></tr></table>",
+        "bbox": [
+            535,
+            364,
+            815,
+            439
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 MAIN RESULTS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            458,
+            323,
+            472
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on DOTA-v1.0. As shown in Tab. 1, our method achieves state-of-the-art performance compared to the previous leading approaches, i.e. PointOBB and Point2RBox. Specifically, under three different detectors, our method attains  $\\mathrm{mAP}_{50}$  scores of  $41.68\\%$ ,  $41.64\\%$ , and  $44.85\\%$ , representing improvements of  $11.60\\%$ ,  $8.33\\%$ , and  $10.90\\%$  over PointOBB, respectively. Additionally, when compared to Point2RBox-RC, which does not incorporate human prior knowledge, our approach achieves a substantial gain of  $10.78\\%$ . Even when compared with Point2RBox-SK, which leverages manual sketches to assist in boundary determination, our method still outperforms it by  $4.58\\%$ . These results demonstrate the robustness and effectiveness of our approach, even without the need for manual prior knowledge.",
+        "bbox": [
+            169,
+            484,
+            823,
+            611
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on DOTA-v1.5/v2.0. Both DOTA-v1.5 and DOTA-v2.0 present a higher level of difficulty due to the increased number of densely packed and smaller objects. As Tab. 2 shows, our method demonstrates significant improvements over other approaches on these more challenging datasets, indicating its strength in handling small and densely distributed objects, attributed to the separation mechanism we designed. In comparison to PointOBB, our method achieves substantial gains on both DOTA-v1.5 and DOTA-v2.0, with greater absolute improvements and higher percentages. For instance, when trained with ReDet, our approach improves by  $36.39\\%$  on DOTA-v1.5 and  $27.22\\%$  on DOTA-v2.0, corresponding to  $25.15\\%$  and  $21.19\\%$  increases, which surpass the  $10.90\\%$  improvement on DOTA-v1.0. Furthermore, our method consistently outperforms Point2RBox. Even when compared to Point2RBox-SK, which incorporates human prior knowledge, our method achieves improvements of  $5.88\\% / 3.79\\%$  on DOTA-v1.5/v2.0, respectively.",
+        "bbox": [
+            169,
+            617,
+            826,
+            772
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Computational Cost. Our method is highly lightweight, primarily due to its single-branch structure, which eliminates the need for the traditional teacher-student framework. Unlike other methods, we do not require multiple image transformations or consistency constraints within the model. As shown in Tab. 3, the pseudo-label training process for our model takes only 1.43 hours, which is 15.58 times faster than the 22.28 hours required by PointOBB.",
+        "bbox": [
+            169,
+            777,
+            823,
+            848
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In terms of memory consumption, our approach is also more efficient. For dense object scenarios such as DOTA-v2.0, our method uses approximately 8GB of memory, making it suitable for most GPUs. In contrast, PointOBB faces out-of-memory issues when handling such dense scenes, requiring restrictions on the number of RoIs to run properly. However, this limitation severely impacts the detector's performance, resulting in numerous small objects being undetected.",
+        "bbox": [
+            169,
+            854,
+            823,
+            925
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "page_number",
+        "text": "8",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/781c18aa38f622afac46318bb7a71fe09a6b779970e6a2a1135b306dc606e0e0.jpg",
+        "table_caption": [
+            "Table 5: Ablation study of sample methods. \"Probabilistic\" means sample according to the CPM probability."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Sample</td><td>Probabilistic</td><td>Weighted</td></tr><tr><td>mAP50</td><td>41.40</td><td>44.85</td></tr></table>",
+        "bbox": [
+            181,
+            148,
+            485,
+            186
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/d7d9fec4f67f3630545bef8a8200f27bbe88bed350d38aa79b8654cdff936eae.jpg",
+        "table_caption": [
+            "Table 6: Ablation of vector constraint module to handle dense objects. \"w\" and \"w/o\" indicate the \"with\" and \"without\" the module."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Vector Constraint</td><td>w/o</td><td>w</td></tr><tr><td>mAP50</td><td>27.88</td><td>44.85</td></tr></table>",
+        "bbox": [
+            506,
+            148,
+            812,
+            186
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/fc54c2007856382dc4d16c9dbe057ebc01cd25ae469842e42e3dd09bde8e7e1e.jpg",
+        "table_caption": [
+            "Table 7: Ablation study of the sampling size of PCA."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Size</td><td>mAP50</td></tr><tr><td>5</td><td>43.35</td></tr><tr><td>7</td><td>44.85</td></tr><tr><td>9</td><td>44.17</td></tr><tr><td>11</td><td>43.63</td></tr></table>",
+        "bbox": [
+            183,
+            241,
+            357,
+            321
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/837e3c30a37b9930a5feb9473e7195d62100402847915f9fcf361eabd6e0b3bc.jpg",
+        "table_caption": [
+            "Table 8: Ablation study of point annotations inaccuracy. We report the pseudo-label generation quality (measured by mIoU) and detection performance (measured by  $\\mathbf{mAP}_{50}$ )"
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Point Range</td><td colspan=\"2\">DOTA-v1.0</td><td colspan=\"2\">DOTA-v1.5</td><td colspan=\"2\">DOTA-v2.0</td></tr><tr><td>mIoU</td><td>\\( \\mathbf{mAP}_{50} \\)</td><td>mIoU</td><td>\\( \\mathbf{mAP}_{50} \\)</td><td>mIoU</td><td>\\( \\mathbf{mAP}_{50} \\)</td></tr><tr><td>0%</td><td>45.40</td><td>44.85</td><td>43.65</td><td>36.39</td><td>42.91</td><td>27.22</td></tr><tr><td>10%</td><td>42.89</td><td>42.30</td><td>41.85</td><td>34.20</td><td>41.46</td><td>25.14</td></tr><tr><td>20%</td><td>40.22</td><td>38.46</td><td>39.47</td><td>30.95</td><td>39.30</td><td>23.45</td></tr></table>",
+        "bbox": [
+            379,
+            247,
+            815,
+            321
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/e73ea38bba6b193c6f512c44af6ef2f469fe86326fc89c52b29b103eefdcde98.jpg",
+        "table_caption": [
+            "Table 9: Comparison of pseudo-label generation quality (measured by mIoU) and detection performance (measured by  $\\mathbf{mAP}_{50}$ ) across different datasets. Additionally, we selected three representative categories—Small Vehicle (SV), Large Vehicle (LV), and Ship (SH)—which are abundant and exhibit characteristics of small objects and densely packed distributions. mean represents the average value across all classes in the dataset."
+        ],
+        "table_footnote": [
+            "* The training stage requires constraining to around 70 RoIs; otherwise it results in out-of-memory errors."
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td rowspan=\"2\">Method</td><td rowspan=\"2\">Memory (GB)</td><td colspan=\"4\">mIoU</td><td colspan=\"4\">mAP50</td></tr><tr><td>SV</td><td>LV</td><td>SH</td><td>mean</td><td>SV</td><td>LV</td><td>SH</td><td>mean</td></tr><tr><td rowspan=\"2\">DOTA-v1.0</td><td>PointOBB</td><td>&gt;24*</td><td>57.06</td><td>49.78</td><td>46.59</td><td>44.88</td><td>59.32</td><td>46.06</td><td>45.71</td><td>33.95</td></tr><tr><td>PointOBB-v2</td><td>5.99</td><td>52.74</td><td>47.65</td><td>48.82</td><td>45.40</td><td>55.00</td><td>49.27</td><td>51.77</td><td>44.85</td></tr><tr><td rowspan=\"2\">DOTA-v1.5</td><td>PointOBB</td><td>&gt;24*</td><td>9.69</td><td>26.58</td><td>26.49</td><td>30.65</td><td>0.01</td><td>4.15</td><td>11.40</td><td>11.24</td></tr><tr><td>PointOBB-v2</td><td>7.35</td><td>40.42</td><td>45.62</td><td>49.74</td><td>43.65</td><td>20.52</td><td>42.97</td><td>48.53</td><td>36.39</td></tr><tr><td rowspan=\"2\">DOTA-v2.0</td><td>PointOBB</td><td>&gt;24*</td><td>10.01</td><td>24.45</td><td>21.33</td><td>26.63</td><td>0.49</td><td>1.22</td><td>0.36</td><td>6.03</td></tr><tr><td>PointOBB-v2</td><td>7.67</td><td>42.47</td><td>48.14</td><td>46.45</td><td>42.91</td><td>20.94</td><td>28.76</td><td>22.15</td><td>27.22</td></tr></table>",
+        "bbox": [
+            179,
+            414,
+            818,
+            525
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDIES",
+        "text_level": 1,
+        "bbox": [
+            171,
+            561,
+            354,
+            575
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Label Assignment. Tab. 4 demonstrates the impact of our three label assignment strategies on model performance. In these experiments, different label assignment strategies were used to train and generate the CPM. When a specific strategy was applied to define positive and negative samples, the remaining points were ignored during training. We observed that using a simple circular strategy to determine positive samples resulted in only  $23.62\\%$ . However, by incorporating a more comprehensive strategy to identify negative samples, the performance increased significantly to  $44.75\\%$ . Furthermore, assigning middle region between objects to negative (denoted as “Neg./M”) yielded a slight improvement, raising the mAP to  $44.85\\%$ . This underscores the crucial role of the negative label assignment strategy, which contributes greatly to the performance gains.",
+        "bbox": [
+            169,
+            589,
+            823,
+            715
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "PCA Sampling Strategy and Size. We conducted ablation experiments on the PCA sampling strategy and the range of sampling sizes. As shown in Tab. 5, our weighted method for PCA calculation improves accuracy by  $3.45\\%$  compared to the probabilistic method. We also found that this improvement primarily benefits classes with larger aspect ratios, such as large vehicles and harbors. This is because CPM in elongated objects exhibit significant probability variation along the short axis of their oriented bounding boxes, and probabilistic sample method introduces considerable instability. Additionally, we evaluated the impact of the PCA sampling size. As shown in Tab. 7, our method achieves the best performance when the sampling size is set to 7.",
+        "bbox": [
+            169,
+            720,
+            823,
+            834
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Vector Constraint. As shown in Tab. 6, applying the vector constraint significantly improves detection performance. Through further analysis, we found that the improvement is primarily concentrated in dense object categories, such as small vehicles, large vehicles, and ships. In contrast, sparse categories like harbors and swimming pools are almost unaffected. This observation aligns with the motivation behind the design of this module, which especially addresses densely packed object scenarios.",
+        "bbox": [
+            169,
+            839,
+            826,
+            925
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.5 ANALYSIS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            104,
+            287,
+            118
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Label Accuracy. Recognizing the potential inaccuracies in human annotations, where the center point might not be perfectly labeled, we conducted experiments by adding noise to the center points to evaluate the robustness of our model. We selected different thresholds  $\\sigma$  and calculated the object's scale as  $S = \\sqrt{wh}$ . The center points were randomly shifted along a uniformly sampled direction, with the offset distance drawn from a uniform distribution over the range  $[- \\sigma S, \\sigma S]$ . We observed a slight performance decrease as the center points were perturbed. As Tab. 8 shows, the average mAP dropped by only  $2.27\\%$  with a  $10\\%$  shift. Despite this decline, our method still significantly outperforms PointOBB, demonstrating the strong robustness of our model.",
+        "bbox": [
+            169,
+            128,
+            826,
+            242
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Quality of Pseudo Labels. As shown in Tab. 9, our method consistently outperforms PointOBB in generating pseudo labels, with performance gains increasing in more challenging datasets like DOTA-v1.5 and DOTA-v2.0. Specifically, we observe mIoU improvements of  $0.52\\%$ ,  $13.00\\%$ , and  $16.28\\%$  across DOTA-v1.0, DOTA-v1.5, and DOTA-v2.0, respectively. Notably, although the improvement in DOTA-v1.0 is only  $0.52\\%$ , training the same detector with our pseudo labels yields a nearly  $10\\%$  increase in mAP compared to PointOBB. As shown in Fig. 5, the first and second columns illustrate that our model learns more accurate object scales, while the third column demonstrates that, unlike PointOBB—which produces overlapping pseudo labels for small vehicles—our method effectively distinguishes between these densely packed objects.",
+        "bbox": [
+            169,
+            247,
+            826,
+            376
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "**Dense Object Scenarios.** As shown in Tab. 9, we selected three representative categories—Small Vehicle (SV), Large Vehicle (LV), and Ship (SH)—which are characterized by small and densely packed objects. Datasets like DOTA-v1.5 and DOTA-v2.0 introduce a much larger number of these densely packed objects compared to DOTA-v1.0. In these challenging scenarios, our method significantly outperforms PointOBB. For example, in DOTA-v2.0, our method achieves a mean mIoU of  $42.91\\%$  and mAP of  $27.22\\%$ , whereas PointOBB drops to  $26.63\\%$  and  $6.03\\%$ , respectively. Visualizations further confirm that our model generates better pseudo labels in dense scenes. In terms of detection results, in the last column of Fig. 5, we show a dense scene with 25 large vehicles, where our method detects all of them, while PointOBB identifies only 15.",
+        "bbox": [
+            169,
+            380,
+            826,
+            506
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Limitations. (a) Our method assigns negative samples based on the minimum distance between objects, requiring at least two point annotations per image. In scenarios with extremely sparse objects, it may degrade the performance. (b) Some hyperparameters (e.g. the radius in label assignment) are set based on the dataset. They may require adjustments when facing other scenarios.",
+        "bbox": [
+            169,
+            511,
+            826,
+            570
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            588,
+            320,
+            604
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduced PointOBB-v2, a simpler, faster, and stronger approach for single point-supervised oriented object detection. By employing class probability maps and Principal Component Analysis (PCA) for object orientation and boundary estimation, our method improves detection accuracy while discarding the traditional time- and memory-heavy teacher-student structure. Experimental results demonstrate that PointOBB-v2 consistently outperforms the previous state-of-the-art across multiple datasets, achieving a training speed  $15.58 \\times$  faster and accuracy improvements of  $11.60\\% / 25.15\\% / 21.19\\%$  on the DOTA-v1.0/v1.5/v2.0 datasets, with notable gains in small and densely packed object scenarios. Our method achieves a substantial speedup and accuracy boost while using less memory, showcasing its effectiveness for real-world applications.",
+        "bbox": [
+            169,
+            619,
+            826,
+            747
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 ACKNOWLEDGMENTS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            765,
+            387,
+            780
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant No. 62176134, by research and application on AI technologies for smart mobility funded by SAIC Motor, and by a grant from the Institute Guo Qiang (2019GQG0002), Tsinghua University.",
+        "bbox": [
+            169,
+            796,
+            825,
+            840
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was also supported in part by the National Natural Science Foundation of China (NSFC) under Grant No. 62306069.",
+        "bbox": [
+            169,
+            845,
+            823,
+            875
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "page_number",
+        "text": "10",
+        "bbox": [
+            490,
+            946,
+            509,
+            960
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            287,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Guangming Cao, Xuehui Yu, Wenwen Yu, Xumeng Han, Xue Yang, Guorong Li, Jianbin Jiao, and Zhenjun Han. P2rbox: Point prompt oriented object detection with SAM. arXiv preprint arXiv:2311.13128, 2024.",
+            "Pengfei Chen, Xuehui Yu, Xumeng Han, Najmul Hassan, Kai Wang, Jiachen Li, Jian Zhao, Humphrey Shi, Zhenjun Han, and Qixiang Ye. Point-to-box network for accurate object detection via single point supervision. In European Conference on Computer Vision, 2022.",
+            "Jian Ding, Nan Xue, Yang Long, Gui-Song Xia, and Qikai Lu. Learning roi transformer for oriented object detection in aerial images. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 2849-2858, 2019.",
+            "Shuyong Gao, Wei Zhang, Yan Wang, Qianyu Guo, Chenglong Zhang, Yangji He, and Wenqiang Zhang. Weakly-supervised salient object detection using point supervision. In AAAI Conference on Artificial Intelligence, volume 36, pp. 670-678, 2022.",
+            "Jiaming Han, Jian Ding, Nan Xue, and Gui-Song Xia. Redet: A rotation-equivariant detector for aerial object detection. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 2785-2794, 2021. doi: 10.1109/CVPR46437.2021.00281.",
+            "Jiaming Han, Jian Ding, Jie Li, and Gui-Song Xia. Align deep features for oriented object detection. IEEE Transactions on Geoscience and Remote Sensing, 60:1-11, 2022. doi: 10.1109/TGRS.2021.3062048.",
+            "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016.",
+            "Liping Hou, Ke Lu, Xue Yang, Yuqiu Li, and Jian Xue. G-rep: Gaussian representation for arbitrary-oriented object detection. Remote Sensing, 15(3):757, 2023.",
+            "Javed Iqbal, Muhammad Akhtar Munir, Arif Mahmood, Afsheen Rafaqat Ali, and Mohsen Ali. Leveraging orientation for weakly supervised object detection with application to firearm localization. Neurocomputing, 440:310-320, 2021. ISSN 0925-2312. doi: https://doi.org/10.1016/j.neucom.2021.01.075.",
+            "Alexander Kirillov, Eric Mintun, Nikhila Ravi, Hanzi Mao, Chloe Rolland, Laura Gustafson, Tete Xiao, Spencer Whitehead, Alexander C Berg, Wan-Yen Lo, et al. Segment anything. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4015-4026, 2023.",
+            "Wentong Li, Yijie Chen, Kaixuan Hu, and Jianke Zhu. Oriented reppoints for aerial object detection. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 1829-1838, 2022.",
+            "Wentong Li, Yuqian Yuan, Song Wang, Jianke Zhu, Jianshu Li, Jian Liu, and Lei Zhang. Point2mask: Point-supervised panoptic segmentation via optimal transport. In IEEE International Conference on Computer Vision, 2023.",
+            "Tsung-Yi Lin, Piotr Dollar, Ross Girshick, Kaiming He, Bharath Hariharan, and Serge Belongie. Feature pyramid networks for object detection. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2117-2125, 2017.",
+            "Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollar. Focal loss for dense object detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(2):318-327, 2020. doi: 10.1109/TPAMI.2018.2858826.",
+            "Li Liu, Wanli Ouyang, Xiaogang Wang, Paul Fieguth, Jie Chen, Xinwang Liu, and Matti Pietikainen. Deep learning for generic object detection: A survey. International Journal of Computer Vision, 128(2):261-318, 2020.",
+            "Nanqing Liu, Xun Xu, Yongyi Su, Haojie Zhang, and Heng-Chao Li. Pointsam: Pointly-supervised segment anything model for remote sensing images. arXiv preprint arXiv:2409.13401, 2024."
+        ],
+        "bbox": [
+            171,
+            126,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "page_number",
+        "text": "11",
+        "bbox": [
+            490,
+            948,
+            506,
+            959
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Junwei Luo, Xue Yang, Yi Yu, Qingyun Li, Junchi Yan, and Yansheng Li. Pointobb: Learning oriented object detection via single point supervision. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024.",
+            "Jeffri Murragarra-Llerena, Lucas N Kirsten, Luis Felipe Zeni, and Claudio R Jung. Probabilistic intersection-over-union for training and evaluation of oriented object detectors. IEEE Transactions on Image Processing, 2024.",
+            "Xingjia Pan, Yuqiang Ren, Kekai Sheng, Weiming Dong, Haolei Yuan, Xiaowei Guo, Chongyang Ma, and Changsheng Xu. Dynamic refinement network for oriented and densely packed object detection. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 11207-11216, 2020.",
+            "Wen Qian, Xue Yang, Silong Peng, Junchi Yan, and Yue Guo. Learning modulated loss for rotated object detection. In AAAI Conference on Artificial Intelligence, volume 35, pp. 2458-2466, 2021.",
+            "Yongqing Sun, Jie Ran, Feng Yang, Chenqiang Gao, Takayuki Kurozumi, Hideaki Kimata, and Ziqi Ye. Oriented object detection for remote sensing images based on weakly supervised learning. In IEEE International Conference on Multimedia & Expo Workshops, pp. 1-6, 2021.",
+            "Zhi Tian, Chunhua Shen, Hao Chen, and Tong He. Fcos: Fully convolutional one-stage object detection. In IEEE/CVF International Conference on Computer Vision, pp. 9626-9635, 2019. doi: 10.1109/ICCV.2019.00972.",
+            "Linfei Wang, Yibing Zhan, Xu Lin, Baosheng Yu, Liang Ding, Jianqing Zhu, and Dapeng Tao. Explicit and implicit box equivariance learning for weakly-supervised rotated object detection. IEEE Transactions on Emerging Topics in Computational Intelligence, 2024.",
+            "Long Wen, Yu Cheng, Yi Fang, and Xinyu Li. A comprehensive survey of oriented object detection in remote sensing images. Expert Systems with Applications, pp. 119960, 2023.",
+            "Gui-Song Xia, Xiang Bai, Jian Ding, Zhen Zhu, Serge Belongie, Jiebo Luo, Mihai Datcu, Marcello Pelillo, and Liangpei Zhang. Dota: A large-scale dataset for object detection in aerial images. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 3974-3983, 2018.",
+            "Xingxing Xie, Gong Cheng, Jiabao Wang, Xiwen Yao, and Junwei Han. Oriented r-cnn for object detection. In IEEE/CVF International Conference on Computer Vision, pp. 3520-3529, 2021.",
+            "Xue Yang and Junchi Yan. Arbitrary-oriented object detection with circular smooth label. In European Conference on Computer Vision, pp. 677-694, 2020.",
+            "Xue Yang and Junchi Yan. On the arbitrary-oriented object detection: Classification based approaches revisited. International Journal of Computer Vision, 130(5):1340-1365, 2022.",
+            "Xue Yang, Hao Sun, Kun Fu, Jirui Yang, Xian Sun, Menglong Yan, and Zhi Guo. Automatic ship detection in remote sensing images from google earth of complex scenes based on multiscale rotation dense feature pyramid networks. Remote sensing, 10(1):132, 2018.",
+            "Xue Yang, Jirui Yang, Junchi Yan, Yue Zhang, Tengfei Zhang, Zhi Guo, Xian Sun, and Kun Fu. Scdet: Towards more robust detection for small, cluttered and rotated objects. In IEEE/CVF International Conference on Computer Vision, pp. 8231-8240, 2019a. doi: 10.1109/ICCV.2019.00832.",
+            "Xue Yang, Liping Hou, Yue Zhou, Wentao Wang, and Junchi Yan. Dense label encoding for boundary discontinuity free rotation detection. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 15814-15824, 2021a. doi: 10.1109/CVPR46437.2021.01556.",
+            "Xue Yang, Junchi Yan, Ziming Feng, and Tao He. R3det: Refined single-stage detector with feature refinement for rotating object. In AAAI Conference on Artificial Intelligence, volume 35, pp. 3163-3171, 2021b."
+        ],
+        "bbox": [
+            171,
+            102,
+            825,
+            922
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "page_number",
+        "text": "12",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Xue Yang, Junchi Yan, Ming Qi, Wentao Wang, Xiaopeng Zhang, and Tian Qi. Rethinking rotated object detection with gaussian Wasserstein distance loss. In 38th International Conference on Machine Learning, volume 139, pp. 11830-11841, 2021c.",
+            "Xue Yang, Xiaojiang Yang, Jirui Yang, Qi Ming, Wentao Wang, Qi Tian, and Junchi Yan. Learning high-precision bounding box for rotated object detection via kullback-leibler divergence. In Advances in Neural Information Processing Systems, volume 34, pp. 18381-18394, 2021d.",
+            "Xue Yang, Junchi Yan, Wenlong Liao, Xiaokang Yang, Jin Tang, and Tao He. Scdet++: Detecting small, cluttered and rotated objects via instance-level feature denoising and rotation loss smoothing. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(2):2384-2399, 2022.",
+            "Xue Yang, Gefan Zhang, Wentong Li, Xuehui Wang, Yue Zhou, and Junchi Yan. H2rbox: Horizontal box annotation is all you need for oriented object detection. International Conference on Learning Representations, 2023a.",
+            "Xue Yang, Gefan Zhang, Xiaojiang Yang, Yue Zhou, Wentao Wang, Jin Tang, Tao He, and Junchi Yan. Detecting rotated objects as gaussian distributions and its 3-d generalization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(4):4335-4354, 2023b.",
+            "Xue Yang, Yue Zhou, Gefan Zhang, Jirui Yang, Wentao Wang, Junchi Yan, Xiaopeng Zhang, and Qi Tian. The kfiou loss for rotated object detection. In International Conference on Learning Representations, 2023c.",
+            "Ze Yang, Shaohui Liu, Han Hu, Liwei Wang, and Stephen Lin. Repoints: Point set representation for object detection. In IEEE/CVF International Conference on Computer Vision, pp. 9656-9665, 2019b. doi: 10.1109/ICCV.2019.00975.",
+            "Yi Yu and Feipeng Da. Phase-shifting coder: Predicting accurate orientation in oriented object detection. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023.",
+            "Yi Yu, Xue Yang, Qingyun Li, Yue Zhou, Feipeng Da, and Junchi Yan. H2rbox-v2: Incorporating symmetry for boosting horizontal box supervised oriented object detection. In Advances in Neural Information Processing Systems, 2023.",
+            "Yi Yu, Xue Yang, Qingyun Li, Feipeng Da, Jifeng Dai, Yu Qiao, and Junchi Yan. Point2rbox: Combine knowledge from synthetic visual patterns for end-to-end oriented object detection with single point supervision. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024.",
+            "Shun Zhang, Jihui Long, Yaohui Xu, and Shaohui Mei. PMho: Point-supervised oriented object detection based on segmentation-driven proposal generation. IEEE Transactions on Geoscience and Remote Sensing, 2024.",
+            "Xin Zhang, Xue Yang, Yuxuan Li, Jian Yang, Ming-Ming Cheng, and Xiang Li. Rsar: Restricted state angle resolver and rotated sar benchmark. arXiv preprint arXiv:2501.04440, 2025.",
+            "Zhong-Qiu Zhao, Peng Zheng, Shou-Tao Xu, and Xindong Wu. Object detection with deep learning: A review. IEEE Transactions on Neural Networks and Learning Systems, 30(11):3212-3232, 2019. doi: 10.1109/TNNLS.2018.2876865.",
+            "Yue Zhou, Xue Yang, Gefan Zhang, Jiabao Wang, Yanyi Liu, Liping Hou, Xue Jiang, Xingzhao Liu, Junchi Yan, Chengqi Lyu, et al. Mmrotate: A rotated object detection benchmark using pytorch. In Proceedings of the 30th ACM International Conference on Multimedia, pp. 7331-7334, 2022.",
+            "Tianyu Zhu, Bryce Ferenczi, Pulak Purkait, Tom Drummond, Hamid Rezatofighi, and Anton van den Hengel. Knowledge combination to learn rotated detection without rotated annotation. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023."
+        ],
+        "bbox": [
+            171,
+            102,
+            825,
+            854
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "page_number",
+        "text": "13",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "7 APPENDIX",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            295,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "7.1 NEGATIVE LABEL ASSIGNMENT",
+        "text_level": 1,
+        "bbox": [
+            171,
+            138,
+            439,
+            152
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To further compare our dynamic minimum radius circle-based strategy for defining negative samples with the fixed radius circle-based strategy, we trained the CPM using both strategies and visualized the results. As shown in Fig. 4, our method better captures the contours of an object, whereas the fixed radius approach produces a CPM that resembles a circle centered on the object, with less distinct contours. This lack of clarity in the contours makes it difficult for PCA to accurately determine the orientation and boundaries based on the CPM.",
+        "bbox": [
+            169,
+            167,
+            826,
+            252
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/99e6829d43c42c45da2880ae5dee4c2345bb2baf88a8748ea6d6422ffe8c6009.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            270,
+            411,
+            404
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/83978d23cd67ea8ceeb86818b6a7978a86f1290a153e710c8868c89f9c1cd74e.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            413,
+            270,
+            586,
+            404
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1553ca21e1e68d1fb655c66a8be1ac330864c938a7596bd3aa0ac66db05add85.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            586,
+            270,
+            759,
+            404
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1ef42fbb827abe984de0b565a43ea90953c6d5a4aa7deb3ab3ddad9e7e2921db.jpg",
+        "image_caption": [
+            "Figure 4: Visualization of the class probability map (CPM). Our dynamic radius negative label assignment vs. fixed radius negative label assignment. Our method produces clearer object contours, whereas the fixed radius approach generates contours that are approximately circular, with the resulting CPM primarily concentrated around the object's center."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            406,
+            411,
+            537
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/9961ac097fe8d63bfcefee6a1246669d76b1b114fefc6c41d63a40027d765355.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            413,
+            406,
+            584,
+            537
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/0b12e12e5bfdb7d572de13e126a68598d7266422a66697c822ce6dcb24057013.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            586,
+            406,
+            759,
+            539
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "7.2 ROBUSTNESS OF CPM'S PARAMETER SELECTION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            638,
+            555,
+            651
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We conducted experiments on the DOTA dataset, focusing on parameters that could potentially impact the results  $(\\alpha, b1, b2)$ , and observed minimal fluctuations in the final results. Additionally, the performance consistently exceeded that of PointOBB  $(33.31\\%)$  by a significant margin (minimum  $7\\%$  gain on Oriented RCNN). Therefore, we conclude that the introduced thresholds do not require precise tuning to achieve strong performance, indicating that the model is not highly sensitive to threshold settings.",
+        "bbox": [
+            169,
+            667,
+            823,
+            752
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/2d34520a0f9c49d543d890c3b49c16f6320061e644e36ed4c50cb2fe19f1abb7.jpg",
+        "table_caption": [
+            "Table 10: The performance of our method under different parameter choices."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Parameters Choice</td><td>mAP50 (Oriented RCNN)</td></tr><tr><td>α,b1,b2= 1,6,4 (Result in Tab. 1)</td><td>41.64</td></tr><tr><td>α,b1,b2= 0.67,8,4</td><td>41.33</td></tr><tr><td>α,b1,b2= 1,8,4</td><td>40.68</td></tr><tr><td>α,b1,b2= 1.5,8,4</td><td>40.22</td></tr><tr><td>α,b1,b2= 1.5,6,4</td><td>41.95</td></tr><tr><td>α,b1,b2= 0.67,12,6</td><td>41.74</td></tr></table>",
+        "bbox": [
+            276,
+            797,
+            720,
+            906
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "page_number",
+        "text": "14",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/33247ff99bee7b938ce4d4c1082e7395dcb3a294c9ed8e71df8c036e4dce7ecb.jpg",
+        "image_caption": [
+            "Figure 5: Visualization of pseudo-labels and detection results from our model compared to PointOBB. For clarity, label texts for dense objects are hidden."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            104,
+            816,
+            338
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "7.3 VISUALIZATION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            392,
+            328,
+            405
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We visualize the quality of pseudo labels and detection results generated by our method compared to PointOBB. As shown in Figure 5, our method generates more precise object scales for pseudo labels and performs better in dense object scenarios, where PointOBB often produces overlapping pseudo labels. In terms of detection results, as seen in the fourth column, our approach produces bounding boxes with more accurate scales and fewer false detections. In the last column, we demonstrate a dense scenario with 25 large vehicles, where our method detects all of them, while PointOBB identifies only 15.",
+        "bbox": [
+            169,
+            417,
+            823,
+            516
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "7.4 DISCUSSION ON ANNOTATION COST VS PERFORMANCE GAP TO FULL-SUPERVISION ACCURACY",
+        "text_level": 1,
+        "bbox": [
+            171,
+            532,
+            790,
+            560
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In the field of object detection, the cost of point annotations is about  $36.5\\%$  lower than HBB and  $104.8\\%$  lower than OBB, and the efficiency of labeling point is significantly higher than both HBB and OBB. This cost analysis has also been adopted and validated in prior works, such as H2RBox (Yang et al., 2023a), PointOBB (Luo et al., 2024), and Point2RBox (Yu et al., 2024), further demonstrating its reliability and relevance.",
+        "bbox": [
+            169,
+            571,
+            823,
+            643
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "7.5 SOME IMPLEMENTATION DETAILS IN CPM GENERATION MODULE",
+        "text_level": 1,
+        "bbox": [
+            171,
+            659,
+            669,
+            672
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The activation function used in the middle layers is LeakyReLU, while the last layer employs a sigmoid function. For each point, we first generate its target using a label assignment strategy. Subsequently, we train the CPM using focal loss (Lin et al., 2020).",
+        "bbox": [
+            169,
+            685,
+            823,
+            728
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "page_number",
+        "text": "15",
+        "bbox": [
+            490,
+            946,
+            508,
+            959
+        ],
+        "page_idx": 14
+    }
+]

2025/PointOBB-v2_ Towards Simpler, Faster, and Stronger Single Point Supervised Oriented Object Detection/860cef88-93f3-4f16-92f7-c5ac52feb051_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:429e7215a23eed3995e15ae4620103308e43e00611466a2ce688f5365f8c586c
+size 2448444

2025/PointOBB-v2_ Towards Simpler, Faster, and Stronger Single Point Supervised Oriented Object Detection/full.md

@@ -0,0 +1,353 @@
+# POINTOBB-v2: TOWARDS SIMPLER, FASTER, AND STRONGER SINGLE POINT SUPERVISED ORIENTED OBJECT DETECTION
+
+Botao Ren $^{1*}$ , Xue Yang $^{2*}$ , Yi Yu $^{3*}$ , Junwei Luo $^{4}$ , Zhidong Deng $^{1\dagger}$
+
+$^{1}$ BNRist, THUAI, Department of Computer Science, Tsinghua University
+
+$^{2}$ Department of Automation, Shanghai Jiao Tong University
+
+$^{3}$ Southeast University  $^{4}$ Wuhan University
+
+Code: https://github.com/taugeren/PointOBB-v2
+
+# ABSTRACT
+
+Single point supervised oriented object detection has gained attention and made initial progress within the community. Diverse from those approaches relying on one-shot samples or powerful pretrained models (e.g. SAM), PointOBB has shown promise due to its prior-free feature. In this paper, we propose PointOBBv2, a simpler, faster, and stronger method to generate pseudo rotated boxes from points without relying on any other prior. Specifically, we first generate a Class Probability Map (CPM) by training the network with non-uniform positive and negative sampling. We show that the CPM is able to learn the approximate object regions and their contours. Then, Principal Component Analysis (PCA) is applied to accurately estimate the orientation and the boundary of objects. By further incorporating a separation mechanism, we resolve the confusion caused by the overlapping on the CPM, enabling its operation in high-density scenarios. Extensive comparisons demonstrate that our method achieves a training speed  $15.58 \times$  faster and an accuracy improvement of  $11.60\% / 25.15\% / 21.19\%$  on the DOTAv1.0/v1.5/v2.0 datasets compared to the previous state-of-the-art, PointOBB. This significantly advances the cutting edge of single point supervised oriented detection in the modular track.
+
+# 1 INTRODUCTION
+
+Oriented object detection is essential for accurately labeling small and densely packed objects, especially in scenarios like remote sensing imagery, retail analysis, and scene text detection, where Oriented Bounding Boxes (OBBs) provide precise annotations. However, annotating OBBs is labor-intensive and costly. Therefore, numerous weakly supervised methods have emerged in recent years, including horizontal bounding box supervision and point supervision. Representative methods for horizontal bounding box supervision include H2Box (Yang et al., 2023a) and H2Box-v2 (Yu et al., 2023). In addition, point supervision, which only requires labeling the point and category for each object, significantly reduces the annotation cost. Notable point-supervised methods include P2Box (Cao et al., 2024), Point2Box (Yu et al., 2024), and PointOBB (Luo et al., 2024).
+
+As illustrated in Fig. 1, existing point-supervised oriented object detection methods can be broadly categorized into three paradigms: (a) SAM-based methods (Cao et al., 2024; Zhang et al., 2024) rely on the powerful SAM (Kirillov et al., 2023) model, which, although effective in natural images, struggles with cross-domain tasks like aerial imagery, particularly in small object and densely packed scenarios. Additionally, SAM-based methods are slow and memory-intensive due to post-processing; (b) Prior-based Weakly-supervised Oriented Object Detection (WOOD) methods, such as Point2RBox (Yu et al., 2024), integrate human priors which reduce generalizability since different datasets require distinct prior knowledge. Further, the end-to-end setup limits flexibility, preventing these methods from leveraging more powerful detectors and benefiting from their performance improvements; (c) Modular WOOD methods (Luo et al., 2024) do not rely on manually designed
+
+![](images/0a1eb4a75f3f9107bee695a4cf484ee986041d336a1270efd534a1dd44f5a560.jpg)  
+Figure 1: Compare with existing point supervised methods, including (a) Prompt OOD (i.e. SAM based); (b-c) Weakly OOD: Prior-based and Modular. OOD means Oriented Object Detection.
+
+![](images/8da11f1dcd12aa2cfec3b83c555d6f06caa1f0d75e921f80115adfc68c681b87.jpg)
+
+![](images/673777d9a5a05a77fd41764d70c61b48847fdf411ab5486545253475426ca2ea.jpg)
+
+prriors and offer greater flexibility by decoupling pseudo-label generation from the detector, making them more suitable for efficient and scalable detection tasks.
+
+As a previous state-of-the-art method, PointOBB falls under the modular WOOD category and offers a feasible solution for point-supervised detection. However, it has several practical limitations: the pseudo-label generation process is very slow, taking approximately 7-8 times longer than the subsequent detector training. Additionally, its training requires significant GPU memory due to multiple view transformations. Moreover, the variability in the number of Region of Interest (RoI) proposals can lead to out-of-memory issues, particularly in dense object scenarios. Although limiting the number of RoI proposals can mitigate this issue, it results in degraded performance.
+
+Considering the aforementioned issues, our motivation is to design a simpler, faster, and stronger method, which leads to the development of PointOBB-v2. Our approach aims to retain the strengths of the modular WOOD paradigm while addressing the inefficiencies of PointOBB, particularly in terms of speed and memory consumption, making it more suitable for real-world applications.
+
+PointOBB-v2 introduces a novel and concise pipeline that discards the teacher-student structure, achieving significant improvements in the accuracy and generation speed of the pseudo-label, and improving memory efficiency, especially in scenarios with small and dense objects. Specifically, we generate Class Probability Maps (CPM) from point annotations and design a novel sample assignment strategy to capture object contours and orientations from the CPM. Next, we apply non-uniform sampling based on the probability distribution and use Principal Component Analysis (PCA) to determine object boundaries and directions. To address dense object distributions, we design a separation mechanism to reduce confusion in pseudo-label generation caused by connected CPMs.
+
+Experimental results demonstrate that our method consistently improves accuracy, speed, and memory efficiency across various datasets compared to PointOBB, achieving several state-of-the-art results. Specifically, on the DOTA-v1.0 dataset, our method, when using pseudo-labels for training with Rotated FCOS, improves the mAP from  $30.08\%$  (PointOBB) to  $41.68\%$ , a gain of  $11.60\%$  mAP. In more challenging datasets like DOTA-v1.5 and DOTA-v2.0, which contain a higher density of small objects, our method achieves mAP of  $36.39\%$  and  $27.22\%$ , with respective gains of  $25.15\%$  and  $21.19\%$  over PointOBB, demonstrating its robustness in handling small and densely packed objects. Furthermore, our pseudo-label generation process is 15.58 times faster, reducing the time from 22.28 hours to 1.43 hours. On the DOTA-v1.5 and DOTA-v2.0 datasets, PointOBB requires limiting the number of RoI proposals due to high memory consumption, while our method operates without such restrictions, with a memory usage of approximately 8GB.
+
+Our contributions are summarized as follows:
+
+- We propose a novel and efficient pipeline for point-supervised oriented object detection, which eliminates the time- and memory-consuming teacher-student structure, significantly improving pseudo-label generation speed and reducing memory usage.
+
+- Without any additional deep network design, our method relies solely on class probability maps to generate accurate object contours, using efficient PCA to determine object directions and boundaries. We also design a vector constraint method to distinguish small objects in dense scenarios, improving detection performance.  
+- Experimental results show that our method consistently outperforms PointOBB across multiple datasets, achieving  $11.60\% / 25.15\% / 21.19\%$  gain on the DOTA-v1.0/v1.5/v2.0 datasets, with a  $15.58 \times$  speedup in pseudo-label generation and memory usage reduced to approximately 8GB without limiting RoI proposals.
+
+# 2 RELATED WORK
+
+In addition to horizontal detection (Zhao et al., 2019; Liu et al., 2020), oriented object detection (Yang et al., 2018; Wen et al., 2023) has received extensive attention. In this section, we first introduce oriented detection supervised by rotated boxes. Then, approaches to point-supervised oriented detection and other weakly-supervised settings are discussed.
+
+# 2.1 RBOX-SUPERVISED ORIENTED DETECTION
+
+Representative works include anchor-based detector Rotated RetinaNet (Lin et al., 2020), anchor-free detector Rotated FCOS (Tian et al., 2019), and two-stage solutions, e.g. RoI Transformer (Ding et al., 2019), Oriented R-CNN (Xie et al., 2021), and ReDet (Han et al., 2021). Some research enhances the detector by exploiting alignment features, e.g.  $\mathbf{R}^3\mathrm{Det}$  (Yang et al., 2021b) and  $\mathbf{S}^2\mathbf{A}$ -Net (Han et al., 2022). The angle regression may face boundary discontinuity and remedies are developed, including modulated losses (Yang et al., 2019a; 2022; Qian et al., 2021) that alleviate loss jumps, angle coders (Yang & Yan, 2020; Yang et al., 2021a; Yang & Yan, 2022; Yu & Da, 2023) that convert the angle into boundary-free coded data, and Gaussian-based losses (Yang et al., 2021c;d; 2023b;c; Murragarra-Llerena et al., 2024) transforming rotated bounding boxes into Gaussian distributions. RepPoint-based methods (Yang et al., 2019b; Hou et al., 2023; Li et al., 2022) provide alternatives that predict a set of sample points that bounds the spatial extent of an object.
+
+# 2.2 POINT-SUPERVISED ORIENTED DETECTION
+
+Recently, several methods for point-supervised oriented detection have been proposed: 1) P2RBox (Cao et al., 2024), PMHO (Zhang et al., 2024) and PointSAM (Liu et al., 2024) propose oriented object detection with point prompts by employing the zero-shot Point-to-Mask ability of SAM (Kirillov et al., 2023). 2) Point2RBox (Yu et al., 2024) has introduced a novel end-to-end approach based on knowledge combination in this domain. 3) PointOBB (Luo et al., 2024) achieves point annotation based RBox generation method for oriented object detection through scale-sensitive consistency and multiple instance learning.
+
+Among these methods, P2RBox, PMHO and PointSAM require SAM model pre-trained on massive amounts of labeled data, whereas Point2RBox requires one-shot samples (i.e. human priors) for each category. Although achieving better accuracy, they are not as general as PointOBB. Hence, we choose PointOBB as our baseline to develop a simpler, faster, and stronger method, PointOBB-v2.
+
+# 2.3 OTHER WEAKLY-SUPERVISED SETTINGS
+
+Compared to the Point-to-RBox, some other weakly-supervised settings have been better studied. These methods are potentially applicable to our Point-to-RBox task setting by using a cascade pipeline, such as Point-to-HBox-to-RBox and Point-to-Mask-to-RBox. In our experiment, cascade pipelines powered by state-of-the-art weakly-supervised approaches are also adopted for comparison. Here, several representative work are introduced.
+
+HBox-to-RBox. The seminal work H2RBox (Yang et al., 2023a) circumvents the segmentation step and achieves RBox detection directly from HBox annotation. With HBox annotations for the same object in various orientations, the geometric constraint limits the object to a few candidate angles. Supplemented with a self-supervised branch eliminating the undesired results, an HBox-to-RBox paradigm is established. An enhanced version H2RBox-v2 (Yu et al., 2023) is proposed to
+
+![](images/5301b9c0d962d71bd8e1c39c40f190778058bc02e68accb99ea91eff31f4ccb5.jpg)  
+Figure 2: Our PointOBB-v2 first generates a Class Probability Map (CPM) with a positive and negative sample assignment strategy during training. It then applies Principal Component Analysis (PCA) to infer object orientation and boundaries to generate pseudo labels.
+
+leverage the reflection symmetry of objects to estimate their angle, further boosting the HBox-to-RBox performance. EIE-Det (Wang et al., 2024) uses an explicit equivariance branch for learning rotation consistency, and an implicit equivariance branch for learning position, aspect ratio, and scale consistency. Some studies (Iqbal et al., 2021; Sun et al., 2021; Zhu et al., 2023) use additional annotated data for training, which are also attractive but less general.
+
+Point-to-HBox. Several related approaches have been developed, including: 1) P2BNet (Chen et al., 2022) samples box proposals of different sizes around the labeled point and classify them to achieve point-supervised horizontal object detection. 2) PSOD (Gao et al., 2022) achieves point-supervised salient object detection using an edge detector and adaptive masked flood fill.
+
+Point-to-Mask. Point2Mask (Li et al., 2023) is proposed to achieve panoptic segmentation using only a single point annotation per target for training. SAM (Segment Anything Model) (Kirillov et al., 2023) produces object masks from input point/HBox prompts. Though RBoxes can be obtained from the segmentation mask by finding the minimum circumscribed rectangle, such a complex pipeline can be less cost-efficient and perform worse (Yang et al., 2023a; Yu et al., 2023).
+
+# 3 METHOD
+
+Our task focuses on oriented object detection with single point supervision. We first utilize point annotations for each object in the training dataset to generate pseudo labels, which are then used to train an existing detector. As shown in Fig. 2, the model first generates a Class Probability Map (CPM) based on the point annotations. Specifically, during training, we devise a positive and negative sample assignment strategy, where the resulting CPM outlines the rough object contours, with higher probabilities concentrated around the point and along the object axes.
+
+We generate pseudo oriented bounding boxes according to CPM. We perform non-uniform sampling around the point annotation of each object, guided by the probability distribution within the CPM. We convert the sampling process into a weighted probability approach, which maintains the same expected result while eliminating the variance introduced by random sampling. By applying Principal Component Analysis (PCA) to the weighted grid points, we can infer the object's orientation. We then determine the object boundaries by combining the thresholded CPM with the inferred orientation. Furthermore, to address densely populated object scenarios, we introduce a mechanism for differentiating between closely situated objects, ensuring effective separation and accurate detection.
+
+![](images/363db275a12fa093b7a98a9b6b195c6e43e167aef7865bc1808c9a33e080fec9.jpg)  
+Figure 3: Visualization of class probability map (CPM).
+
+![](images/de55cbc82d801ac4684b76d258aaa9bde3e438dc2ead318a1dbe3b052a6f1553.jpg)
+
+![](images/6e1ccfc9e13d3527b1f6685390f1c77904ce9bfb52b5928190edf054b0734d9c.jpg)
+
+# 3.1 CLASS PROBABILITY MAP GENERATION
+
+The Class Probability Map (CPM) represents the per-class probability for each point on the feature map, with values ranging between [0, 1]. To generate the CPM, our model first takes an image  $I$  of dimension  $(C, H, W)$  as input and processes it through a ResNet-50 (He et al., 2016) backbone with an FPN (Lin et al., 2017) structure. The final class probability map is derived from the highest-resolution feature map of the FPN, which is then projected through a projection layer. The output is a map of size  $(N_{\mathrm{class}}, H_0, W_0)$ . Formally, it is defined as:
+
+$$
+\mathrm {C P M} = \operatorname {P r o j} (f (I) _ {0}), \tag {1}
+$$
+
+where CPM is the class probability map,  $\mathrm{Proj}(\cdot)$  represents the projection layer, and  $f(I)_0$  is the highest-resolution feature map from the ResNet-50 + FPN.
+
+# 3.2 LABEL ASSIGNMENT
+
+A key component of our approach is the design of a robust sample assignment strategy for both positive and negative samples. This strategy is essential for building an accurate CPM, which outlines rough object contours, concentrating higher probabilities around object centers and along their axes. To ensure reliable separation of objects, especially in densely populated scenarios, our method addresses the challenge of closely situated objects by introducing additional mechanisms for effective differentiation. We illustrate this label assignment in the upper-right part of Fig. 2. The specific details of the sample assignment process are outlined below:
+
+Positive Label Assignment. For positive samples, we select all points within a fixed radius  $b_{1}$  (set to 6 in our model) around each point. If a point lies within multiple such radii, it is assigned to the closest center. The condition for positive samples is as follows:
+
+$$
+\begin{array}{l} \exists \mathbf {G T} _ {i} \in \mathbf {G T} _ {1 \sim N}, (d (p, \mathbf {G T} _ {i}) <   b _ {1}) \wedge (d (p, \mathbf {G T} _ {i}) = \min  (d (p, \mathbf {G T} _ {1 \sim N}))) \\ \Rightarrow p \text {i s p o s i t i v e , c l s} (p) = \operatorname {c l s} \left(\mathrm {G T} _ {i}\right). \tag {2} \\ \end{array}
+$$
+
+Negative Label Assignment. Given  $N$  ground truth objects (GT), for each  $\mathrm{GT}_i$ , we identify its nearest neighboring object  $\mathrm{GT}_j$  based on the Euclidean distance. This gives us a vector  $\mathrm{dist}_{min}$  with dimension  $[N]$ , where each element  $\mathrm{dist}_i$  represents the minimum distance between  $\mathrm{GT}_i$  and its closest neighbor. We then draw a circle with radius  $\alpha \times \mathrm{dist}_i$  around  $\mathrm{GT}_i$ , where  $\alpha$  (set to 1 in our model) is a fixed proportional constant. Points outside all such circles are designated as negative samples. The negative sample condition is formulated as:
+
+$$
+\forall \mathrm {G T} _ {i} \in \mathrm {G T} _ {1 \sim N}, d (p, \mathrm {G T} _ {i}) > \alpha \times \operatorname {d i s t} _ {i} \Rightarrow p \text {i s n e g a t i v e}. \tag {3}
+$$
+
+In addition to the above defined negative labels, we also set the middle region between objects as negative to make the boundaries clearer between densely packed objects (denoted as "Neg./M" in ablation Tab. 4). For each  $\mathrm{GT}_i$ , we identify its nearest neighbor  $\mathrm{GT}_j$  that belongs to the same class. A circle is drawn with a radius  $b_2$  (set to 4 in our model) and centered at the midpoint of the line connecting  $\mathrm{GT}_i$  and  $\mathrm{GT}_j$ , and points within this circle are assigned as negative samples. The condition is defined as:
+
+$$
+\begin{array}{l} \forall \mathrm {G T} _ {i} \in \mathrm {G T} _ {1 \sim N}, \exists \mathrm {G T} _ {j} \in \mathrm {G T} _ {1 \sim N}, d (\mathrm {G T} _ {i}, \mathrm {G T} _ {j}) = \min  (d (\mathrm {G T} _ {i}, \mathrm {G T} _ {1 \sim N})) \\ \wedge \operatorname {c l s} \left(\mathrm {G T} _ {i}\right) = \operatorname {c l s} \left(\mathrm {G T} _ {j}\right) \wedge d (p, \left(\mathrm {G T} _ {i} + \mathrm {G T} _ {j}\right) / 2) <   b _ {2} \Rightarrow p \text {i s n e g a t i v e}. \\ \end{array}
+$$
+
+Robustness. While we do not explicitly define positive and negative samples based on precise object contours or oriented bounding boxes, which may result in some inaccuracies during label assignment (i.e. incorrectly assigning a small portion of positive or negative samples during training), this does not significantly hinder our method's ability to learn accurate object contours. These minor label assignment inaccuracies, particularly in densely populated regions or for objects with extreme aspect ratios, do not affect the overall robustness and effectiveness of the approach. As demonstrated in Fig. 3, our strategy is capable of learning the correct contours, even for objects with large aspect ratios and in densely packed scenarios.
+
+# 3.3 ORIENTATION AND BOUNDARY ESTIMATION VIA PCA
+
+After obtaining the CPM, we sample points around each ground truth based on the class probabilities, and then apply Principal Component Analysis (PCA) on the sampled points to determine the object's orientation. As shown in bottom part of Fig. 2, we sample points around the GT based on the probabilities in the CPM for the corresponding object class. We choose a  $7 \times 7$  grid centered at the GT with the coordinates of the 49 integer points  $z_{1 \sim 49}$  as:
+
+$$
+(x, y), x \in [ - 3, 3 ], y \in [ - 3, 3 ]. \tag {5}
+$$
+
+For each grid point  $z_{i}$ , we can compute the CPM probability  $p_{i}$  and decide whether to sample the point based on this probability. Once we have the sampled points, we apply PCA to find the primary direction of the point set, which represents the object's orientation. While PCA provides the correct primary direction in expectation, the randomness introduced by sampling can cause variance in the result from a single pass. Although averaging over multiple sampling runs can mitigate this variance, it also increases computational cost.
+
+To address this, we propose an equivalent method that transforms probabilistic sampling into a weighted coordinate transformation. Instead of sampling points probabilistically, we assign a weight of  $p_i$  to each point  $z_i$ , ensuring the same expected outcome while eliminating the variance caused by random sampling. The covariance matrix is then defined as:
+
+$$
+C _ {z} = \sum_ {i = 1} ^ {N} p _ {i} \left(z _ {i} - \mu_ {z}\right) ^ {\mathrm {T}} \left(z _ {i} - \mu_ {z}\right). \tag {6}
+$$
+
+We then perform eigenvalue decomposition on  $C_z$ :
+
+$$
+C _ {z} v _ {i} = \lambda_ {i} v _ {i}. \tag {7}
+$$
+
+The eigenvector  $v_{1}$  corresponding to the largest eigenvalue  $\lambda_{1}$  is chosen as the primary direction. Since  $C_z$  is a real symmetric matrix, the secondary direction is guaranteed to be orthogonal to the primary direction. This orthogonality corresponds to the perpendicular relationship between the two adjacent sides of an oriented bounding box.
+
+After identifying the primary and secondary directions, we determine the object boundaries along these directions. Starting from the center, we move along each direction and stop when the value at a position falls below a threshold, indicating the object boundary.
+
+# 3.4 OBJECT DIFFERENTIATION IN DENSE SCENARIOS
+
+In dense scenarios, objects can be difficult to distinguish on the CPM. This can affect the PCA's ability to determine the object orientation and the boundary identification. To address this, we design a "Vector Constraint Suppression" method to resolve boundary ambiguity.
+
+Vector Constraint Suppression. Even after determining the correct orientation in dense scenarios, the object boundaries may still be unclear, making it difficult to precisely locate them using the probability threshold described in Sec. 3.3. In most cases, simply distinguishing between two closely positioned objects is sufficient to define the object's boundary.
+
+We propose a simple constraint: For each  $\mathrm{GT}_i$ , we first find its nearest same-class neighbor  $\mathrm{GT}_j$  and compute the vector  $u = \langle \mathrm{GT}_i, \mathrm{GT}_j \rangle$  between  $\mathrm{GT}_i$  and  $\mathrm{GT}_j$ . If the angle between this vector and the primary or secondary direction is smaller than a threshold  $\alpha$  (set to  $\pi/6$  in our model),
+
+Table 1: Results of each category on the DOTA-v1.0 dataset. FCOSR and ORCNN refer to Rotated FCOS and Oriented R-CNN. * indicates using additional human knowledge priors.  
+
+<table><tr><td>Method</td><td>PL</td><td>BD</td><td>BR</td><td>GTF</td><td>SV</td><td>LV</td><td>SH</td><td>TC</td><td>BC</td><td>ST</td><td>SBF</td><td>RA</td><td>HA</td><td>SP</td><td>HC</td><td>\( \mathbf{{mAP}}{}_{50} \)</td></tr><tr><td>Point2Mask-RBox</td><td>4.0</td><td>23.1</td><td>3.8</td><td>1.3</td><td>15.1</td><td>1.0</td><td>3.3</td><td>19.0</td><td>1.0</td><td>29.1</td><td>0.0</td><td>9.5</td><td>7.4</td><td>21.1</td><td>7.1</td><td>9.72</td></tr><tr><td>P2BNet+H2RBox</td><td>24.7</td><td>35.9</td><td>7.0</td><td>27.9</td><td>3.3</td><td>12.1</td><td>17.5</td><td>17.5</td><td>0.8</td><td>34.0</td><td>6.3</td><td>49.6</td><td>11.6</td><td>27.2</td><td>18.8</td><td>19.63</td></tr><tr><td>P2BNet+H2RBox-v2</td><td>11.0</td><td>44.8</td><td>14.9</td><td>15.4</td><td>36.8</td><td>16.7</td><td>27.8</td><td>12.1</td><td>1.8</td><td>31.2</td><td>3.4</td><td>50.6</td><td>12.6</td><td>36.7</td><td>12.5</td><td>21.87</td></tr><tr><td>Point2RBox-RC</td><td>62.9</td><td>64.3</td><td>14.4</td><td>35.0</td><td>28.2</td><td>38.9</td><td>33.3</td><td>25.2</td><td>2.2</td><td>44.5</td><td>3.4</td><td>48.1</td><td>25.9</td><td>45.0</td><td>22.6</td><td>34.07</td></tr><tr><td>Point2RBox-SK*</td><td>53.3</td><td>63.9</td><td>3.7</td><td>50.9</td><td>40.0</td><td>39.2</td><td>45.7</td><td>76.7</td><td>10.5</td><td>56.1</td><td>5.4</td><td>49.5</td><td>24.2</td><td>51.2</td><td>33.8</td><td>40.27</td></tr><tr><td>PointOBB (FCOSR)</td><td>26.1</td><td>65.7</td><td>9.1</td><td>59.4</td><td>65.8</td><td>34.9</td><td>29.8</td><td>0.5</td><td>2.3</td><td>16.7</td><td>0.6</td><td>49.0</td><td>21.8</td><td>41.0</td><td>36.7</td><td>30.08</td></tr><tr><td>PointOBB-v2 (FCOSR)</td><td>64.5</td><td>27.8</td><td>1.9</td><td>36.2</td><td>58.8</td><td>47.2</td><td>53.4</td><td>90.5</td><td>62.2</td><td>45.3</td><td>12.1</td><td>41.7</td><td>8.1</td><td>43.7</td><td>32.0</td><td>41.68</td></tr><tr><td>PointOBB (ORCNN)</td><td>28.3</td><td>70.7</td><td>1.5</td><td>64.9</td><td>68.8</td><td>46.8</td><td>33.9</td><td>9.1</td><td>10.0</td><td>20.1</td><td>0.2</td><td>47.0</td><td>29.7</td><td>38.2</td><td>30.6</td><td>33.31</td></tr><tr><td>PointOBB-v2 (ORCNN)</td><td>63.7</td><td>45.6</td><td>2.0</td><td>39.5</td><td>50.5</td><td>49.6</td><td>45.4</td><td>89.8</td><td>62.9</td><td>41.3</td><td>13.6</td><td>42.8</td><td>8.9</td><td>39.5</td><td>29.5</td><td>41.64</td></tr><tr><td>PointOBB (ReDet)</td><td>24.2</td><td>75.0</td><td>0.5</td><td>60.6</td><td>59.3</td><td>46.1</td><td>45.7</td><td>6.1</td><td>10.1</td><td>25.0</td><td>0.2</td><td>50.4</td><td>30.0</td><td>45.0</td><td>31.1</td><td>33.95</td></tr><tr><td>PointOBB-v2 (ReDet)</td><td>65.4</td><td>52.1</td><td>2.2</td><td>44.4</td><td>55.0</td><td>49.3</td><td>51.8</td><td>89.0</td><td>70.2</td><td>47.0</td><td>16.2</td><td>43.9</td><td>13.0</td><td>43.8</td><td>29.4</td><td>44.85</td></tr></table>
+
+we consider this direction valid for boundary definition. The boundary is then constrained by the following condition:
+
+$$
+u \cdot v _ {k} <   \frac {1}{2} \times d \left(\mathrm {G T} _ {i}, \mathrm {G T} _ {j}\right) \quad \text {i f a n g l e} (u, v _ {k}) <   \alpha \tag {8}
+$$
+
+where  $v_{k}$  is the primary or secondary direction,  $\mathrm{GT}_j$  is the nearest same-class object to  $\mathrm{GT}_i$ , and  $\frac{1}{2}$  means that the boundary should be closer to  $\mathrm{GT}_i$  than to  $\mathrm{GT}_j$ .
+
+# 4 EXPERIMENT
+
+# 4.1 DATASETS
+
+DOTA (Xia et al., 2018) is a large-scale dataset designed for object detection in aerial images, covering various object categories and complexities. DOTA has three versions:
+
+DOTA-v1.0 has 2,806 images with 188,282 instances across 15 categories, annotated as: Plane (PL), Baseball Diamond (BD), Bridge (BR), Ground Track Field (GTF), Small Vehicle (SV), Large Vehicle (LV), Ship (SH), Tennis Court (TC), Basketball Court (BC), Storage Tank (ST), Soccer Ball Field (SBF), Roundabout (RA), Harbor (HA), Swimming Pool (SP), Helicopter (HC). The images range from  $800 \times 800$  to  $4,000 \times 4,000$  pixels and exhibit significant variation in scale and orientation.
+
+DOTA-v1.5 extends DOTA-v1.0 by adding annotations for extremely small objects (less than 10 pixels) and introducing a new category, Container Crane (CC). It includes a total of 403,318 instances while retaining the same image count and dataset split as DOTA-v1.0.
+
+DOTA-v2.0 further expands the dataset to 11,268 images and 1,793,658 instances, covering 18 categories. Two additional categories, Airport (AP) and Helipad (HP), are introduced, providing a more diverse and challenging set of aerial images.
+
+RSAR (Zhang et al., 2025) is a large-scale dataset for rotated object detection in SAR imagery, comprising 95,842 images and 183,534 annotated instances across 6 categories.
+
+SKU110K-R (Pan et al., 2020) is a dataset focused on detecting densely packed retail scenes, consisting of 11,762 images and 110,712 annotated objects.
+
+# 4.2 EXPERIMENTAL SETTINGS
+
+Our implementation is based on the MMRotate library (Zhou et al., 2022). In the pseudo-label generation stage, we train the model for 6 epochs using momentum SGD as the optimizer. We set the weight decay to 1e-4, with an initial learning rate of 0.005, which decays by a factor of 10 after the  $4^{\text{th}}$  epoch. The batch size for training is set to 2. For the detector training phase using pseudolabels, we use the same detector configurations as the default settings in MMRotate. Throughout the entire training process, random flipping is employed as the only data augmentation technique. Our experiments were accelerated using two GeForce RTX 3090 GPUs.
+
+Table 2: Results on DOTA-v1.0/v1.5/v2.0, RSAR and SKU110K-R, reporting the  $\mathbf{mAP}_{50}$  metric. FCOSR and ORCNN refer to Rotated FCOS and Oriented R-CNN. * indicates using additional human knowledge priors.  
+
+<table><tr><td>Method</td><td>DOTA-v1.0</td><td>DOTA-v1.5</td><td>DOTA-v2.0</td><td>RSAR</td><td>SKU110K-R</td></tr><tr><td>Point2Mask-RBox</td><td>9.72</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>P2BNet+H2RBox</td><td>19.63</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>P2BNet+H2RBox-v2</td><td>21.87</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Point2RBox-RC</td><td>34.07</td><td>24.31</td><td>14.69</td><td>-</td><td>3.41</td></tr><tr><td>Point2RBox-SK*</td><td>40.27</td><td>30.51</td><td>23.43</td><td>-</td><td>-</td></tr><tr><td>PointOBB (FCOSR)</td><td>30.08</td><td>10.66</td><td>5.53</td><td>13.80</td><td>-</td></tr><tr><td>PointOBB-v2 (FCOSR)</td><td>41.68 (+11.60)</td><td>30.59 (+19.93)</td><td>20.64 (+15.11)</td><td>18.99 (+5.19)</td><td>56.63</td></tr><tr><td>PointOBB (ORCNN)</td><td>33.31</td><td>10.92</td><td>6.29</td><td>12.30</td><td>-</td></tr><tr><td>PointOBB-v2 (ORCNN)</td><td>41.64 (+8.33)</td><td>32.01 (+21.09)</td><td>23.40 (+17.11)</td><td>22.61 (+10.31)</td><td>62.76</td></tr><tr><td>PointOBB (ReDet)</td><td>33.95</td><td>11.24</td><td>6.03</td><td>-</td><td>-</td></tr><tr><td>PointOBB-v2 (ReDet)</td><td>44.85 (+10.90)</td><td>36.39 (+25.15)</td><td>27.22 (+21.19)</td><td>-</td><td>-</td></tr></table>
+
+Table 3: Comparison of the training time in pseudolabel generation phase and the accuracy between PointOBB and PointOBB-v2. The reported  $\mathbf{mAP}_{50}$  is trained with Rotated FCOS.  
+
+<table><tr><td>Method</td><td>Epochs</td><td>Train Hours</td><td>mAP50</td></tr><tr><td>PointOBB</td><td>24</td><td>22.28</td><td>30.08</td></tr><tr><td>PointOBB-v2</td><td>6</td><td>1.43</td><td>41.68</td></tr></table>
+
+Table 4: Ablation study with different label assignment strategies.  
+
+<table><tr><td>Pos.</td><td>Neg.</td><td>Neg./M</td><td>mAP50</td></tr><tr><td>✓</td><td></td><td></td><td>23.62</td></tr><tr><td>✓</td><td>✓</td><td></td><td>44.75</td></tr><tr><td>✓</td><td></td><td>✓</td><td>18.21</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>44.85</td></tr></table>
+
+# 4.3 MAIN RESULTS
+
+Results on DOTA-v1.0. As shown in Tab. 1, our method achieves state-of-the-art performance compared to the previous leading approaches, i.e. PointOBB and Point2RBox. Specifically, under three different detectors, our method attains  $\mathrm{mAP}_{50}$  scores of  $41.68\%$ ,  $41.64\%$ , and  $44.85\%$ , representing improvements of  $11.60\%$ ,  $8.33\%$ , and  $10.90\%$  over PointOBB, respectively. Additionally, when compared to Point2RBox-RC, which does not incorporate human prior knowledge, our approach achieves a substantial gain of  $10.78\%$ . Even when compared with Point2RBox-SK, which leverages manual sketches to assist in boundary determination, our method still outperforms it by  $4.58\%$ . These results demonstrate the robustness and effectiveness of our approach, even without the need for manual prior knowledge.
+
+Results on DOTA-v1.5/v2.0. Both DOTA-v1.5 and DOTA-v2.0 present a higher level of difficulty due to the increased number of densely packed and smaller objects. As Tab. 2 shows, our method demonstrates significant improvements over other approaches on these more challenging datasets, indicating its strength in handling small and densely distributed objects, attributed to the separation mechanism we designed. In comparison to PointOBB, our method achieves substantial gains on both DOTA-v1.5 and DOTA-v2.0, with greater absolute improvements and higher percentages. For instance, when trained with ReDet, our approach improves by  $36.39\%$  on DOTA-v1.5 and  $27.22\%$  on DOTA-v2.0, corresponding to  $25.15\%$  and  $21.19\%$  increases, which surpass the  $10.90\%$  improvement on DOTA-v1.0. Furthermore, our method consistently outperforms Point2RBox. Even when compared to Point2RBox-SK, which incorporates human prior knowledge, our method achieves improvements of  $5.88\% / 3.79\%$  on DOTA-v1.5/v2.0, respectively.
+
+Computational Cost. Our method is highly lightweight, primarily due to its single-branch structure, which eliminates the need for the traditional teacher-student framework. Unlike other methods, we do not require multiple image transformations or consistency constraints within the model. As shown in Tab. 3, the pseudo-label training process for our model takes only 1.43 hours, which is 15.58 times faster than the 22.28 hours required by PointOBB.
+
+In terms of memory consumption, our approach is also more efficient. For dense object scenarios such as DOTA-v2.0, our method uses approximately 8GB of memory, making it suitable for most GPUs. In contrast, PointOBB faces out-of-memory issues when handling such dense scenes, requiring restrictions on the number of RoIs to run properly. However, this limitation severely impacts the detector's performance, resulting in numerous small objects being undetected.
+
+Table 5: Ablation study of sample methods. "Probabilistic" means sample according to the CPM probability.  
+
+<table><tr><td>Sample</td><td>Probabilistic</td><td>Weighted</td></tr><tr><td>mAP50</td><td>41.40</td><td>44.85</td></tr></table>
+
+Table 6: Ablation of vector constraint module to handle dense objects. "w" and "w/o" indicate the "with" and "without" the module.  
+
+<table><tr><td>Vector Constraint</td><td>w/o</td><td>w</td></tr><tr><td>mAP50</td><td>27.88</td><td>44.85</td></tr></table>
+
+Table 7: Ablation study of the sampling size of PCA.  
+
+<table><tr><td>Size</td><td>mAP50</td></tr><tr><td>5</td><td>43.35</td></tr><tr><td>7</td><td>44.85</td></tr><tr><td>9</td><td>44.17</td></tr><tr><td>11</td><td>43.63</td></tr></table>
+
+Table 8: Ablation study of point annotations inaccuracy. We report the pseudo-label generation quality (measured by mIoU) and detection performance (measured by  $\mathbf{mAP}_{50}$ )  
+
+<table><tr><td rowspan="2">Point Range</td><td colspan="2">DOTA-v1.0</td><td colspan="2">DOTA-v1.5</td><td colspan="2">DOTA-v2.0</td></tr><tr><td>mIoU</td><td>\( \mathbf{mAP}_{50} \)</td><td>mIoU</td><td>\( \mathbf{mAP}_{50} \)</td><td>mIoU</td><td>\( \mathbf{mAP}_{50} \)</td></tr><tr><td>0%</td><td>45.40</td><td>44.85</td><td>43.65</td><td>36.39</td><td>42.91</td><td>27.22</td></tr><tr><td>10%</td><td>42.89</td><td>42.30</td><td>41.85</td><td>34.20</td><td>41.46</td><td>25.14</td></tr><tr><td>20%</td><td>40.22</td><td>38.46</td><td>39.47</td><td>30.95</td><td>39.30</td><td>23.45</td></tr></table>
+
+Table 9: Comparison of pseudo-label generation quality (measured by mIoU) and detection performance (measured by  $\mathbf{mAP}_{50}$ ) across different datasets. Additionally, we selected three representative categories—Small Vehicle (SV), Large Vehicle (LV), and Ship (SH)—which are abundant and exhibit characteristics of small objects and densely packed distributions. mean represents the average value across all classes in the dataset.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Method</td><td rowspan="2">Memory (GB)</td><td colspan="4">mIoU</td><td colspan="4">mAP50</td></tr><tr><td>SV</td><td>LV</td><td>SH</td><td>mean</td><td>SV</td><td>LV</td><td>SH</td><td>mean</td></tr><tr><td rowspan="2">DOTA-v1.0</td><td>PointOBB</td><td>&gt;24*</td><td>57.06</td><td>49.78</td><td>46.59</td><td>44.88</td><td>59.32</td><td>46.06</td><td>45.71</td><td>33.95</td></tr><tr><td>PointOBB-v2</td><td>5.99</td><td>52.74</td><td>47.65</td><td>48.82</td><td>45.40</td><td>55.00</td><td>49.27</td><td>51.77</td><td>44.85</td></tr><tr><td rowspan="2">DOTA-v1.5</td><td>PointOBB</td><td>&gt;24*</td><td>9.69</td><td>26.58</td><td>26.49</td><td>30.65</td><td>0.01</td><td>4.15</td><td>11.40</td><td>11.24</td></tr><tr><td>PointOBB-v2</td><td>7.35</td><td>40.42</td><td>45.62</td><td>49.74</td><td>43.65</td><td>20.52</td><td>42.97</td><td>48.53</td><td>36.39</td></tr><tr><td rowspan="2">DOTA-v2.0</td><td>PointOBB</td><td>&gt;24*</td><td>10.01</td><td>24.45</td><td>21.33</td><td>26.63</td><td>0.49</td><td>1.22</td><td>0.36</td><td>6.03</td></tr><tr><td>PointOBB-v2</td><td>7.67</td><td>42.47</td><td>48.14</td><td>46.45</td><td>42.91</td><td>20.94</td><td>28.76</td><td>22.15</td><td>27.22</td></tr></table>
+
+* The training stage requires constraining to around 70 RoIs; otherwise it results in out-of-memory errors.
+
+# 4.4 ABLATION STUDIES
+
+Label Assignment. Tab. 4 demonstrates the impact of our three label assignment strategies on model performance. In these experiments, different label assignment strategies were used to train and generate the CPM. When a specific strategy was applied to define positive and negative samples, the remaining points were ignored during training. We observed that using a simple circular strategy to determine positive samples resulted in only  $23.62\%$ . However, by incorporating a more comprehensive strategy to identify negative samples, the performance increased significantly to  $44.75\%$ . Furthermore, assigning middle region between objects to negative (denoted as “Neg./M”) yielded a slight improvement, raising the mAP to  $44.85\%$ . This underscores the crucial role of the negative label assignment strategy, which contributes greatly to the performance gains.
+
+PCA Sampling Strategy and Size. We conducted ablation experiments on the PCA sampling strategy and the range of sampling sizes. As shown in Tab. 5, our weighted method for PCA calculation improves accuracy by  $3.45\%$  compared to the probabilistic method. We also found that this improvement primarily benefits classes with larger aspect ratios, such as large vehicles and harbors. This is because CPM in elongated objects exhibit significant probability variation along the short axis of their oriented bounding boxes, and probabilistic sample method introduces considerable instability. Additionally, we evaluated the impact of the PCA sampling size. As shown in Tab. 7, our method achieves the best performance when the sampling size is set to 7.
+
+Vector Constraint. As shown in Tab. 6, applying the vector constraint significantly improves detection performance. Through further analysis, we found that the improvement is primarily concentrated in dense object categories, such as small vehicles, large vehicles, and ships. In contrast, sparse categories like harbors and swimming pools are almost unaffected. This observation aligns with the motivation behind the design of this module, which especially addresses densely packed object scenarios.
+
+# 4.5 ANALYSIS
+
+Label Accuracy. Recognizing the potential inaccuracies in human annotations, where the center point might not be perfectly labeled, we conducted experiments by adding noise to the center points to evaluate the robustness of our model. We selected different thresholds  $\sigma$  and calculated the object's scale as  $S = \sqrt{wh}$ . The center points were randomly shifted along a uniformly sampled direction, with the offset distance drawn from a uniform distribution over the range  $[- \sigma S, \sigma S]$ . We observed a slight performance decrease as the center points were perturbed. As Tab. 8 shows, the average mAP dropped by only  $2.27\%$  with a  $10\%$  shift. Despite this decline, our method still significantly outperforms PointOBB, demonstrating the strong robustness of our model.
+
+Quality of Pseudo Labels. As shown in Tab. 9, our method consistently outperforms PointOBB in generating pseudo labels, with performance gains increasing in more challenging datasets like DOTA-v1.5 and DOTA-v2.0. Specifically, we observe mIoU improvements of  $0.52\%$ ,  $13.00\%$ , and  $16.28\%$  across DOTA-v1.0, DOTA-v1.5, and DOTA-v2.0, respectively. Notably, although the improvement in DOTA-v1.0 is only  $0.52\%$ , training the same detector with our pseudo labels yields a nearly  $10\%$  increase in mAP compared to PointOBB. As shown in Fig. 5, the first and second columns illustrate that our model learns more accurate object scales, while the third column demonstrates that, unlike PointOBB—which produces overlapping pseudo labels for small vehicles—our method effectively distinguishes between these densely packed objects.
+
+**Dense Object Scenarios.** As shown in Tab. 9, we selected three representative categories—Small Vehicle (SV), Large Vehicle (LV), and Ship (SH)—which are characterized by small and densely packed objects. Datasets like DOTA-v1.5 and DOTA-v2.0 introduce a much larger number of these densely packed objects compared to DOTA-v1.0. In these challenging scenarios, our method significantly outperforms PointOBB. For example, in DOTA-v2.0, our method achieves a mean mIoU of  $42.91\%$  and mAP of  $27.22\%$ , whereas PointOBB drops to  $26.63\%$  and  $6.03\%$ , respectively. Visualizations further confirm that our model generates better pseudo labels in dense scenes. In terms of detection results, in the last column of Fig. 5, we show a dense scene with 25 large vehicles, where our method detects all of them, while PointOBB identifies only 15.
+
+Limitations. (a) Our method assigns negative samples based on the minimum distance between objects, requiring at least two point annotations per image. In scenarios with extremely sparse objects, it may degrade the performance. (b) Some hyperparameters (e.g. the radius in label assignment) are set based on the dataset. They may require adjustments when facing other scenarios.
+
+# 5 CONCLUSION
+
+In this paper, we introduced PointOBB-v2, a simpler, faster, and stronger approach for single point-supervised oriented object detection. By employing class probability maps and Principal Component Analysis (PCA) for object orientation and boundary estimation, our method improves detection accuracy while discarding the traditional time- and memory-heavy teacher-student structure. Experimental results demonstrate that PointOBB-v2 consistently outperforms the previous state-of-the-art across multiple datasets, achieving a training speed  $15.58 \times$  faster and accuracy improvements of  $11.60\% / 25.15\% / 21.19\%$  on the DOTA-v1.0/v1.5/v2.0 datasets, with notable gains in small and densely packed object scenarios. Our method achieves a substantial speedup and accuracy boost while using less memory, showcasing its effectiveness for real-world applications.
+
+# 6 ACKNOWLEDGMENTS
+
+This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant No. 62176134, by research and application on AI technologies for smart mobility funded by SAIC Motor, and by a grant from the Institute Guo Qiang (2019GQG0002), Tsinghua University.
+
+This work was also supported in part by the National Natural Science Foundation of China (NSFC) under Grant No. 62306069.
+
+# REFERENCES
+
+Guangming Cao, Xuehui Yu, Wenwen Yu, Xumeng Han, Xue Yang, Guorong Li, Jianbin Jiao, and Zhenjun Han. P2rbox: Point prompt oriented object detection with SAM. arXiv preprint arXiv:2311.13128, 2024.  
+Pengfei Chen, Xuehui Yu, Xumeng Han, Najmul Hassan, Kai Wang, Jiachen Li, Jian Zhao, Humphrey Shi, Zhenjun Han, and Qixiang Ye. Point-to-box network for accurate object detection via single point supervision. In European Conference on Computer Vision, 2022.  
+Jian Ding, Nan Xue, Yang Long, Gui-Song Xia, and Qikai Lu. Learning roi transformer for oriented object detection in aerial images. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 2849-2858, 2019.  
+Shuyong Gao, Wei Zhang, Yan Wang, Qianyu Guo, Chenglong Zhang, Yangji He, and Wenqiang Zhang. Weakly-supervised salient object detection using point supervision. In AAAI Conference on Artificial Intelligence, volume 36, pp. 670-678, 2022.  
+Jiaming Han, Jian Ding, Nan Xue, and Gui-Song Xia. Redet: A rotation-equivariant detector for aerial object detection. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 2785-2794, 2021. doi: 10.1109/CVPR46437.2021.00281.  
+Jiaming Han, Jian Ding, Jie Li, and Gui-Song Xia. Align deep features for oriented object detection. IEEE Transactions on Geoscience and Remote Sensing, 60:1-11, 2022. doi: 10.1109/TGRS.2021.3062048.  
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016.  
+Liping Hou, Ke Lu, Xue Yang, Yuqiu Li, and Jian Xue. G-rep: Gaussian representation for arbitrary-oriented object detection. Remote Sensing, 15(3):757, 2023.  
+Javed Iqbal, Muhammad Akhtar Munir, Arif Mahmood, Afsheen Rafaqat Ali, and Mohsen Ali. Leveraging orientation for weakly supervised object detection with application to firearm localization. Neurocomputing, 440:310-320, 2021. ISSN 0925-2312. doi: https://doi.org/10.1016/j.neucom.2021.01.075.  
+Alexander Kirillov, Eric Mintun, Nikhila Ravi, Hanzi Mao, Chloe Rolland, Laura Gustafson, Tete Xiao, Spencer Whitehead, Alexander C Berg, Wan-Yen Lo, et al. Segment anything. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4015-4026, 2023.  
+Wentong Li, Yijie Chen, Kaixuan Hu, and Jianke Zhu. Oriented reppoints for aerial object detection. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 1829-1838, 2022.  
+Wentong Li, Yuqian Yuan, Song Wang, Jianke Zhu, Jianshu Li, Jian Liu, and Lei Zhang. Point2mask: Point-supervised panoptic segmentation via optimal transport. In IEEE International Conference on Computer Vision, 2023.  
+Tsung-Yi Lin, Piotr Dollar, Ross Girshick, Kaiming He, Bharath Hariharan, and Serge Belongie. Feature pyramid networks for object detection. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2117-2125, 2017.  
+Tsung-Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollar. Focal loss for dense object detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(2):318-327, 2020. doi: 10.1109/TPAMI.2018.2858826.  
+Li Liu, Wanli Ouyang, Xiaogang Wang, Paul Fieguth, Jie Chen, Xinwang Liu, and Matti Pietikainen. Deep learning for generic object detection: A survey. International Journal of Computer Vision, 128(2):261-318, 2020.  
+Nanqing Liu, Xun Xu, Yongyi Su, Haojie Zhang, and Heng-Chao Li. Pointsam: Pointly-supervised segment anything model for remote sensing images. arXiv preprint arXiv:2409.13401, 2024.
+
+Junwei Luo, Xue Yang, Yi Yu, Qingyun Li, Junchi Yan, and Yansheng Li. Pointobb: Learning oriented object detection via single point supervision. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024.  
+Jeffri Murragarra-Llerena, Lucas N Kirsten, Luis Felipe Zeni, and Claudio R Jung. Probabilistic intersection-over-union for training and evaluation of oriented object detectors. IEEE Transactions on Image Processing, 2024.  
+Xingjia Pan, Yuqiang Ren, Kekai Sheng, Weiming Dong, Haolei Yuan, Xiaowei Guo, Chongyang Ma, and Changsheng Xu. Dynamic refinement network for oriented and densely packed object detection. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 11207-11216, 2020.  
+Wen Qian, Xue Yang, Silong Peng, Junchi Yan, and Yue Guo. Learning modulated loss for rotated object detection. In AAAI Conference on Artificial Intelligence, volume 35, pp. 2458-2466, 2021.  
+Yongqing Sun, Jie Ran, Feng Yang, Chenqiang Gao, Takayuki Kurozumi, Hideaki Kimata, and Ziqi Ye. Oriented object detection for remote sensing images based on weakly supervised learning. In IEEE International Conference on Multimedia & Expo Workshops, pp. 1-6, 2021.  
+Zhi Tian, Chunhua Shen, Hao Chen, and Tong He. Fcos: Fully convolutional one-stage object detection. In IEEE/CVF International Conference on Computer Vision, pp. 9626-9635, 2019. doi: 10.1109/ICCV.2019.00972.  
+Linfei Wang, Yibing Zhan, Xu Lin, Baosheng Yu, Liang Ding, Jianqing Zhu, and Dapeng Tao. Explicit and implicit box equivariance learning for weakly-supervised rotated object detection. IEEE Transactions on Emerging Topics in Computational Intelligence, 2024.  
+Long Wen, Yu Cheng, Yi Fang, and Xinyu Li. A comprehensive survey of oriented object detection in remote sensing images. Expert Systems with Applications, pp. 119960, 2023.  
+Gui-Song Xia, Xiang Bai, Jian Ding, Zhen Zhu, Serge Belongie, Jiebo Luo, Mihai Datcu, Marcello Pelillo, and Liangpei Zhang. Dota: A large-scale dataset for object detection in aerial images. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 3974-3983, 2018.  
+Xingxing Xie, Gong Cheng, Jiabao Wang, Xiwen Yao, and Junwei Han. Oriented r-cnn for object detection. In IEEE/CVF International Conference on Computer Vision, pp. 3520-3529, 2021.  
+Xue Yang and Junchi Yan. Arbitrary-oriented object detection with circular smooth label. In European Conference on Computer Vision, pp. 677-694, 2020.  
+Xue Yang and Junchi Yan. On the arbitrary-oriented object detection: Classification based approaches revisited. International Journal of Computer Vision, 130(5):1340-1365, 2022.  
+Xue Yang, Hao Sun, Kun Fu, Jirui Yang, Xian Sun, Menglong Yan, and Zhi Guo. Automatic ship detection in remote sensing images from google earth of complex scenes based on multiscale rotation dense feature pyramid networks. Remote sensing, 10(1):132, 2018.  
+Xue Yang, Jirui Yang, Junchi Yan, Yue Zhang, Tengfei Zhang, Zhi Guo, Xian Sun, and Kun Fu. Scdet: Towards more robust detection for small, cluttered and rotated objects. In IEEE/CVF International Conference on Computer Vision, pp. 8231-8240, 2019a. doi: 10.1109/ICCV.2019.00832.  
+Xue Yang, Liping Hou, Yue Zhou, Wentao Wang, and Junchi Yan. Dense label encoding for boundary discontinuity free rotation detection. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 15814-15824, 2021a. doi: 10.1109/CVPR46437.2021.01556.  
+Xue Yang, Junchi Yan, Ziming Feng, and Tao He. R3det: Refined single-stage detector with feature refinement for rotating object. In AAAI Conference on Artificial Intelligence, volume 35, pp. 3163-3171, 2021b.
+
+Xue Yang, Junchi Yan, Ming Qi, Wentao Wang, Xiaopeng Zhang, and Tian Qi. Rethinking rotated object detection with gaussian Wasserstein distance loss. In 38th International Conference on Machine Learning, volume 139, pp. 11830-11841, 2021c.  
+Xue Yang, Xiaojiang Yang, Jirui Yang, Qi Ming, Wentao Wang, Qi Tian, and Junchi Yan. Learning high-precision bounding box for rotated object detection via kullback-leibler divergence. In Advances in Neural Information Processing Systems, volume 34, pp. 18381-18394, 2021d.  
+Xue Yang, Junchi Yan, Wenlong Liao, Xiaokang Yang, Jin Tang, and Tao He. Scdet++: Detecting small, cluttered and rotated objects via instance-level feature denoising and rotation loss smoothing. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(2):2384-2399, 2022.  
+Xue Yang, Gefan Zhang, Wentong Li, Xuehui Wang, Yue Zhou, and Junchi Yan. H2rbox: Horizontal box annotation is all you need for oriented object detection. International Conference on Learning Representations, 2023a.  
+Xue Yang, Gefan Zhang, Xiaojiang Yang, Yue Zhou, Wentao Wang, Jin Tang, Tao He, and Junchi Yan. Detecting rotated objects as gaussian distributions and its 3-d generalization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(4):4335-4354, 2023b.  
+Xue Yang, Yue Zhou, Gefan Zhang, Jirui Yang, Wentao Wang, Junchi Yan, Xiaopeng Zhang, and Qi Tian. The kfiou loss for rotated object detection. In International Conference on Learning Representations, 2023c.  
+Ze Yang, Shaohui Liu, Han Hu, Liwei Wang, and Stephen Lin. Repoints: Point set representation for object detection. In IEEE/CVF International Conference on Computer Vision, pp. 9656-9665, 2019b. doi: 10.1109/ICCV.2019.00975.  
+Yi Yu and Feipeng Da. Phase-shifting coder: Predicting accurate orientation in oriented object detection. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023.  
+Yi Yu, Xue Yang, Qingyun Li, Yue Zhou, Feipeng Da, and Junchi Yan. H2rbox-v2: Incorporating symmetry for boosting horizontal box supervised oriented object detection. In Advances in Neural Information Processing Systems, 2023.  
+Yi Yu, Xue Yang, Qingyun Li, Feipeng Da, Jifeng Dai, Yu Qiao, and Junchi Yan. Point2rbox: Combine knowledge from synthetic visual patterns for end-to-end oriented object detection with single point supervision. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024.  
+Shun Zhang, Jihui Long, Yaohui Xu, and Shaohui Mei. PMho: Point-supervised oriented object detection based on segmentation-driven proposal generation. IEEE Transactions on Geoscience and Remote Sensing, 2024.  
+Xin Zhang, Xue Yang, Yuxuan Li, Jian Yang, Ming-Ming Cheng, and Xiang Li. Rsar: Restricted state angle resolver and rotated sar benchmark. arXiv preprint arXiv:2501.04440, 2025.  
+Zhong-Qiu Zhao, Peng Zheng, Shou-Tao Xu, and Xindong Wu. Object detection with deep learning: A review. IEEE Transactions on Neural Networks and Learning Systems, 30(11):3212-3232, 2019. doi: 10.1109/TNNLS.2018.2876865.  
+Yue Zhou, Xue Yang, Gefan Zhang, Jiabao Wang, Yanyi Liu, Liping Hou, Xue Jiang, Xingzhao Liu, Junchi Yan, Chengqi Lyu, et al. Mmrotate: A rotated object detection benchmark using pytorch. In Proceedings of the 30th ACM International Conference on Multimedia, pp. 7331-7334, 2022.  
+Tianyu Zhu, Bryce Ferenczi, Pulak Purkait, Tom Drummond, Hamid Rezatofighi, and Anton van den Hengel. Knowledge combination to learn rotated detection without rotated annotation. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023.
+
+# 7 APPENDIX
+
+# 7.1 NEGATIVE LABEL ASSIGNMENT
+
+To further compare our dynamic minimum radius circle-based strategy for defining negative samples with the fixed radius circle-based strategy, we trained the CPM using both strategies and visualized the results. As shown in Fig. 4, our method better captures the contours of an object, whereas the fixed radius approach produces a CPM that resembles a circle centered on the object, with less distinct contours. This lack of clarity in the contours makes it difficult for PCA to accurately determine the orientation and boundaries based on the CPM.
+
+![](images/99e6829d43c42c45da2880ae5dee4c2345bb2baf88a8748ea6d6422ffe8c6009.jpg)
+
+![](images/83978d23cd67ea8ceeb86818b6a7978a86f1290a153e710c8868c89f9c1cd74e.jpg)
+
+![](images/1553ca21e1e68d1fb655c66a8be1ac330864c938a7596bd3aa0ac66db05add85.jpg)
+
+![](images/1ef42fbb827abe984de0b565a43ea90953c6d5a4aa7deb3ab3ddad9e7e2921db.jpg)  
+Figure 4: Visualization of the class probability map (CPM). Our dynamic radius negative label assignment vs. fixed radius negative label assignment. Our method produces clearer object contours, whereas the fixed radius approach generates contours that are approximately circular, with the resulting CPM primarily concentrated around the object's center.
+
+![](images/9961ac097fe8d63bfcefee6a1246669d76b1b114fefc6c41d63a40027d765355.jpg)
+
+![](images/0b12e12e5bfdb7d572de13e126a68598d7266422a66697c822ce6dcb24057013.jpg)
+
+# 7.2 ROBUSTNESS OF CPM'S PARAMETER SELECTION
+
+We conducted experiments on the DOTA dataset, focusing on parameters that could potentially impact the results  $(\alpha, b1, b2)$ , and observed minimal fluctuations in the final results. Additionally, the performance consistently exceeded that of PointOBB  $(33.31\%)$  by a significant margin (minimum  $7\%$  gain on Oriented RCNN). Therefore, we conclude that the introduced thresholds do not require precise tuning to achieve strong performance, indicating that the model is not highly sensitive to threshold settings.
+
+Table 10: The performance of our method under different parameter choices.  
+
+<table><tr><td>Parameters Choice</td><td>mAP50 (Oriented RCNN)</td></tr><tr><td>α,b1,b2= 1,6,4 (Result in Tab. 1)</td><td>41.64</td></tr><tr><td>α,b1,b2= 0.67,8,4</td><td>41.33</td></tr><tr><td>α,b1,b2= 1,8,4</td><td>40.68</td></tr><tr><td>α,b1,b2= 1.5,8,4</td><td>40.22</td></tr><tr><td>α,b1,b2= 1.5,6,4</td><td>41.95</td></tr><tr><td>α,b1,b2= 0.67,12,6</td><td>41.74</td></tr></table>
+
+![](images/33247ff99bee7b938ce4d4c1082e7395dcb3a294c9ed8e71df8c036e4dce7ecb.jpg)  
+Figure 5: Visualization of pseudo-labels and detection results from our model compared to PointOBB. For clarity, label texts for dense objects are hidden.
+
+# 7.3 VISUALIZATION
+
+We visualize the quality of pseudo labels and detection results generated by our method compared to PointOBB. As shown in Figure 5, our method generates more precise object scales for pseudo labels and performs better in dense object scenarios, where PointOBB often produces overlapping pseudo labels. In terms of detection results, as seen in the fourth column, our approach produces bounding boxes with more accurate scales and fewer false detections. In the last column, we demonstrate a dense scenario with 25 large vehicles, where our method detects all of them, while PointOBB identifies only 15.
+
+# 7.4 DISCUSSION ON ANNOTATION COST VS PERFORMANCE GAP TO FULL-SUPERVISION ACCURACY
+
+In the field of object detection, the cost of point annotations is about  $36.5\%$  lower than HBB and  $104.8\%$  lower than OBB, and the efficiency of labeling point is significantly higher than both HBB and OBB. This cost analysis has also been adopted and validated in prior works, such as H2RBox (Yang et al., 2023a), PointOBB (Luo et al., 2024), and Point2RBox (Yu et al., 2024), further demonstrating its reliability and relevance.
+
+# 7.5 SOME IMPLEMENTATION DETAILS IN CPM GENERATION MODULE
+
+The activation function used in the middle layers is LeakyReLU, while the last layer employs a sigmoid function. For each point, we first generate its target using a label assignment strategy. Subsequently, we train the CPM using focal loss (Lin et al., 2020).

2025/PointOBB-v2_ Towards Simpler, Faster, and Stronger Single Point Supervised Oriented Object Detection/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:bf353a956a828f5bc3764315f0f8a8e513a6eb5c1d939a7425cbb85ac4e5adce
+size 800107

2025/Poisson-Dirac Neural Networks for Modeling Coupled Dynamical Systems across Domains/655be178-57e1-494e-92a1-8899e0bc0c03_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9f44c0f6fff843c044e582b97fc42ad34913ab8a699b22521ab123a0a91577e0
+size 2423205

2025/Poisson-Dirac Neural Networks for Modeling Coupled Dynamical Systems across Domains/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5d1b57b2a6fafd70ee0b39b11ef77fa47cf4df65dfeb9b1cabce2b35c39b4e32
+size 705493