Add Batch ddf34024-6614-4f5b-b2de-9e86383236ab data

.gitattributes

@@ -1961,3 +1961,67 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
 2025/Multi-Turn[[:space:]]Code[[:space:]]Generation[[:space:]]Through[[:space:]]Single-Step[[:space:]]Rewards/2a064cfb-d3b5-492f-8f90-b687f89dad03_origin.pdf filter=lfs diff=lfs merge=lfs -text
 2025/Multi-View[[:space:]]Graph[[:space:]]Clustering[[:space:]]via[[:space:]]Node-Guided[[:space:]]Contrastive[[:space:]]Encoding/50d310dc-c958-46c2-9536-788ea61cc810_origin.pdf filter=lfs diff=lfs merge=lfs -text
 2025/Multi-agent[[:space:]]Architecture[[:space:]]Search[[:space:]]via[[:space:]]Agentic[[:space:]]Supernet/25a746aa-2e14-4571-b770-111e7a05e017_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Multi-band[[:space:]]Frequency[[:space:]]Reconstruction[[:space:]]for[[:space:]]Neural[[:space:]]Psychoacoustic[[:space:]]Coding/acbe0cae-dd20-45e6-9725-8a41781631c7_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Multi-objective[[:space:]]Linear[[:space:]]Reinforcement[[:space:]]Learning[[:space:]]with[[:space:]]Lexicographic[[:space:]]Rewards/66599d63-c75d-4e63-a2bb-e5bb4f0dd0cd_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/MultiPDENet_[[:space:]]PDE-embedded[[:space:]]Learning[[:space:]]with[[:space:]]Multi-time-stepping[[:space:]]for[[:space:]]Accelerated[[:space:]]Flow[[:space:]]Simulation/fc6a1b12-5b25-4f65-b6e6-903ab0a4ef17_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Multiaccuracy[[:space:]]and[[:space:]]Multicalibration[[:space:]]via[[:space:]]Proxy[[:space:]]Groups/5d6e9d73-a3b3-40a7-bc11-bbcb3bfb308c_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Multidimensional[[:space:]]Adaptive[[:space:]]Coefficient[[:space:]]for[[:space:]]Inference[[:space:]]Trajectory[[:space:]]Optimization[[:space:]]in[[:space:]]Flow[[:space:]]and[[:space:]]Diffusion/a77e6807-0eb0-428d-bfb7-8ed224b510c2_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Multilayer[[:space:]]Matrix[[:space:]]Factorization[[:space:]]via[[:space:]]Dimension-Reducing[[:space:]]Diffusion[[:space:]]Variational[[:space:]]Inference/e19e76fc-b602-4624-9f13-cc4f80943df0_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Multimodal[[:space:]]Medical[[:space:]]Code[[:space:]]Tokenizer/932dc8e4-b1cb-4cd5-a163-6ce8fd1ac8f6_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Multinoulli[[:space:]]Extension_[[:space:]]A[[:space:]]Lossless[[:space:]]Yet[[:space:]]Effective[[:space:]]Probabilistic[[:space:]]Framework[[:space:]]for[[:space:]]Subset[[:space:]]Selection[[:space:]]over[[:space:]]Partition[[:space:]]Constraints/2f74a846-6168-4a9b-8587-04417ae152bc_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Multiobjective[[:space:]]distribution[[:space:]]matching/52ffe869-9827-40a0-89ab-c4311c335290_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Multiple-policy[[:space:]]Evaluation[[:space:]]via[[:space:]]Density[[:space:]]Estimation/99739d86-685b-4504-8ce0-347d4f4f250d_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Multivariate[[:space:]]Conformal[[:space:]]Selection/475a2517-c503-4ad7-9ac6-4e509938bf49_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/MuseControlLite_[[:space:]]Multifunctional[[:space:]]Music[[:space:]]Generation[[:space:]]with[[:space:]]Lightweight[[:space:]]Conditioners/c668bb86-80aa-40ec-a616-ae0beeb3dbde_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Mutual[[:space:]]Learning[[:space:]]for[[:space:]]SAM[[:space:]]Adaptation_[[:space:]]A[[:space:]]Dual[[:space:]]Collaborative[[:space:]]Network[[:space:]]Framework[[:space:]]for[[:space:]]Source-Free[[:space:]]Domain[[:space:]]Transfer/3064a7c3-baf6-419f-b1ff-2acc8e6835d4_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/MxMoE_[[:space:]]Mixed-precision[[:space:]]Quantization[[:space:]]for[[:space:]]MoE[[:space:]]with[[:space:]]Accuracy[[:space:]]and[[:space:]]Performance[[:space:]]Co-Design/5e40cf7f-a574-4d76-a637-8bccf4246552_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/N2GON_[[:space:]]Neural[[:space:]]Networks[[:space:]]for[[:space:]]Graph-of-Net[[:space:]]with[[:space:]]Position[[:space:]]Awareness/db6b7c24-641a-47f3-a682-e081848c2e62_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NBDI_[[:space:]]A[[:space:]]Simple[[:space:]]and[[:space:]]Effective[[:space:]]Termination[[:space:]]Condition[[:space:]]for[[:space:]]Skill[[:space:]]Extraction[[:space:]]from[[:space:]]Task-Agnostic[[:space:]]Demonstrations/0785fae8-6c78-42e9-b65b-4e8ef072ee3a_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NEAR_[[:space:]]Neural[[:space:]]Electromagnetic[[:space:]]Array[[:space:]]Response/027e85c3-91c5-401c-bc9b-6647cb361f96_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NETS_[[:space:]]A[[:space:]]Non-equilibrium[[:space:]]Transport[[:space:]]Sampler/d79e2199-89af-438a-8903-eb427b1d352f_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NExtLong_[[:space:]]Toward[[:space:]]Effective[[:space:]]Long-Context[[:space:]]Training[[:space:]]without[[:space:]]Long[[:space:]]Documents/c4d5ab8d-4e07-4d35-8895-9a16e9fc8d66_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NICE[[:space:]]Data[[:space:]]Selection[[:space:]]for[[:space:]]Instruction[[:space:]]Tuning[[:space:]]in[[:space:]]LLMs[[:space:]]with[[:space:]]Non-differentiable[[:space:]]Evaluation[[:space:]]Metric/03dcb7fd-8b61-433c-b711-b2a8b082a36f_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NMA-tune_[[:space:]]Generating[[:space:]]Highly[[:space:]]Designable[[:space:]]and[[:space:]]Dynamics[[:space:]]Aware[[:space:]]Protein[[:space:]]Backbones/2fb1a854-ba80-485e-a02b-349edd339412_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NTK-DFL_[[:space:]]Enhancing[[:space:]]Decentralized[[:space:]]Federated[[:space:]]Learning[[:space:]]in[[:space:]]Heterogeneous[[:space:]]Settings[[:space:]]via[[:space:]]Neural[[:space:]]Tangent[[:space:]]Kernel/9c21c15a-6cb1-4bd1-979d-715d8705555d_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NTPP_[[:space:]]Generative[[:space:]]Speech[[:space:]]Language[[:space:]]Modeling[[:space:]]for[[:space:]]Dual-Channel[[:space:]]Spoken[[:space:]]Dialogue[[:space:]]via[[:space:]]Next-Token-Pair[[:space:]]Prediction/be57bf67-f783-464c-a759-5f0c8e3c0ce8_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Natural[[:space:]]Perturbations[[:space:]]for[[:space:]]Black-box[[:space:]]Training[[:space:]]of[[:space:]]Neural[[:space:]]Networks[[:space:]]by[[:space:]]Zeroth-Order[[:space:]]Optimization/477467bd-19ce-49ae-91d3-2e4508aa3122_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Navigating[[:space:]]Conflicting[[:space:]]Views_[[:space:]]Harnessing[[:space:]]Trust[[:space:]]for[[:space:]]Learning/86a6d1fb-2666-41e6-84f5-98a16e115647_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Navigating[[:space:]]Semantic[[:space:]]Drift[[:space:]]in[[:space:]]Task-Agnostic[[:space:]]Class-Incremental[[:space:]]Learning/820f4c51-1089-4f0e-84b7-161631c663d8_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Navigating[[:space:]]the[[:space:]]Social[[:space:]]Welfare[[:space:]]Frontier_[[:space:]]Portfolios[[:space:]]for[[:space:]]Multi-objective[[:space:]]Reinforcement[[:space:]]Learning/28457464-b44c-4eec-aa91-b5a5b28eb0ba_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Near[[:space:]]Optimal[[:space:]]Best[[:space:]]Arm[[:space:]]Identification[[:space:]]for[[:space:]]Clustered[[:space:]]Bandits/cc98c2c7-66b2-4bad-802c-0a37909b19e2_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Near[[:space:]]Optimal[[:space:]]Non-asymptotic[[:space:]]Sample[[:space:]]Complexity[[:space:]]of[[:space:]]1-Identification/27e5a628-c996-4808-b633-16f555afdcba_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Near-Optimal[[:space:]]Consistency-Robustness[[:space:]]Trade-Offs[[:space:]]for[[:space:]]Learning-Augmented[[:space:]]Online[[:space:]]Knapsack[[:space:]]Problems/9b8c0b35-aae3-4272-bce5-35fda7a570f1_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Near-Optimal[[:space:]]Decision[[:space:]]Trees[[:space:]]in[[:space:]]a[[:space:]]SPLIT[[:space:]]Second/2f77ee14-4881-4981-b4e8-036afb4321cd_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Near-Optimal[[:space:]]Sample[[:space:]]Complexity[[:space:]]for[[:space:]]MDPs[[:space:]]via[[:space:]]Anchoring/96bb30d4-25f6-4b2c-aaa3-f8575eaefc9b_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Near-optimal[[:space:]]Regret[[:space:]]Using[[:space:]]Policy[[:space:]]Optimization[[:space:]]in[[:space:]]Online[[:space:]]MDPs[[:space:]]with[[:space:]]Aggregate[[:space:]]Bandit[[:space:]]Feedback/f99d1bf4-aabc-470b-82e2-9d7d63ad1ce4_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Near-optimal[[:space:]]Sketchy[[:space:]]Natural[[:space:]]Gradients[[:space:]]for[[:space:]]Physics-Informed[[:space:]]Neural[[:space:]]Networks/e682634b-8123-49ad-b591-e4956160ebb7_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Nearly[[:space:]]Optimal[[:space:]]Algorithms[[:space:]]for[[:space:]]Contextual[[:space:]]Dueling[[:space:]]Bandits[[:space:]]from[[:space:]]Adversarial[[:space:]]Feedback/5fa56811-b835-4b7f-9911-c3a3693a5a90_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Nearly[[:space:]]Optimal[[:space:]]Sample[[:space:]]Complexity[[:space:]]for[[:space:]]Learning[[:space:]]with[[:space:]]Label[[:space:]]Proportions/6e3f6d8a-2320-45a7-9409-b534eb4b3378_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NegMerge_[[:space:]]Sign-Consensual[[:space:]]Weight[[:space:]]Merging[[:space:]]for[[:space:]]Machine[[:space:]]Unlearning/44220826-a8a2-451d-aeee-749028c9f0db_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neighbour-Driven[[:space:]]Gaussian[[:space:]]Process[[:space:]]Variational[[:space:]]Autoencoders[[:space:]]for[[:space:]]Scalable[[:space:]]Structured[[:space:]]Latent[[:space:]]Modelling/4890c5bb-f9c7-4836-83be-4c673453d5fe_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Nemotron-CORTEXA_[[:space:]]Enhancing[[:space:]]LLM[[:space:]]Agents[[:space:]]for[[:space:]]Software[[:space:]]Engineering[[:space:]]Tasks[[:space:]]via[[:space:]]Improved[[:space:]]Localization[[:space:]]and[[:space:]]Solution[[:space:]]Diversity/798c6b9d-9afe-442c-9c20-e8804301dec2_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NestQuant_[[:space:]]nested[[:space:]]lattice[[:space:]]quantization[[:space:]]for[[:space:]]matrix[[:space:]]products[[:space:]]and[[:space:]]LLMs/781f2d98-9108-48a5-b562-cb1ec361c99c_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Nested[[:space:]]Expectations[[:space:]]with[[:space:]]Kernel[[:space:]]Quadrature/eb01350b-8455-44cc-9166-407e8b33829d_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Nesterov[[:space:]]Method[[:space:]]for[[:space:]]Asynchronous[[:space:]]Pipeline[[:space:]]Parallel[[:space:]]Optimization/6d30dbae-714d-493c-9844-08d6a7eaf6f9_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Network[[:space:]]Sparsity[[:space:]]Unlocks[[:space:]]the[[:space:]]Scaling[[:space:]]Potential[[:space:]]of[[:space:]]Deep[[:space:]]Reinforcement[[:space:]]Learning/dfb26fe7-e076-4889-8a8d-c7ad8cf8ba22_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neural[[:space:]]Collapse[[:space:]]Beyond[[:space:]]the[[:space:]]Unconstrained[[:space:]]Features[[:space:]]Model_[[:space:]]Landscape,[[:space:]]Dynamics,[[:space:]]and[[:space:]]Generalization[[:space:]]in[[:space:]]the[[:space:]]Mean-Field[[:space:]]Regime/ef2eab31-c7fb-4df3-96c6-5327ff822645_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neural[[:space:]]Discovery[[:space:]]in[[:space:]]Mathematics_[[:space:]]Do[[:space:]]Machines[[:space:]]Dream[[:space:]]of[[:space:]]Colored[[:space:]]Planes_/7b609452-05eb-44ae-8b20-9faa45d59c5f_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neural[[:space:]]Encoding[[:space:]]and[[:space:]]Decoding[[:space:]]at[[:space:]]Scale/e8c113e4-6386-44b8-8849-dfd9aa9344eb_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neural[[:space:]]Event-Triggered[[:space:]]Control[[:space:]]with[[:space:]]Optimal[[:space:]]Scheduling/4e7f5b88-3275-46b9-b124-590d4c7ebc33_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neural[[:space:]]Genetic[[:space:]]Search[[:space:]]in[[:space:]]Discrete[[:space:]]Spaces/c599d21f-6281-438e-bc09-f7928a9d4f10_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neural[[:space:]]Graph[[:space:]]Matching[[:space:]]Improves[[:space:]]Retrieval[[:space:]]Augmented[[:space:]]Generation[[:space:]]in[[:space:]]Molecular[[:space:]]Machine[[:space:]]Learning/a75c83ba-c090-452a-83b1-29f9b0621ec4_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neural[[:space:]]Guided[[:space:]]Diffusion[[:space:]]Bridges/66e9abae-5399-4468-b7be-6d72ddfa0010_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neural[[:space:]]Interpretable[[:space:]]PDEs_[[:space:]]Harmonizing[[:space:]]Fourier[[:space:]]Insights[[:space:]]with[[:space:]]Attention[[:space:]]for[[:space:]]Scalable[[:space:]]and[[:space:]]Interpretable[[:space:]]Physics[[:space:]]Discovery/3512346e-d058-426c-8187-3924f8011d36_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neural[[:space:]]Representational[[:space:]]Consistency[[:space:]]Emerges[[:space:]]from[[:space:]]Probabilistic[[:space:]]Neural-Behavioral[[:space:]]Representation[[:space:]]Alignment/c7d65a23-b1e7-4892-ad2c-61c802916925_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neural[[:space:]]Solver[[:space:]]Selection[[:space:]]for[[:space:]]Combinatorial[[:space:]]Optimization/e2e0e99a-1273-442d-b31b-f8806f355cbb_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NeuralCohort_[[:space:]]Cohort-aware[[:space:]]Neural[[:space:]]Representation[[:space:]]Learning[[:space:]]for[[:space:]]Healthcare[[:space:]]Analytics/060eaa63-1d00-4081-a598-1f578166504e_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NeuroTree_[[:space:]]Hierarchical[[:space:]]Functional[[:space:]]Brain[[:space:]]Pathway[[:space:]]Decoding[[:space:]]for[[:space:]]Mental[[:space:]]Health[[:space:]]Disorders/b9e5b9a6-a6bf-4d02-99d8-bf6d64017403_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NeuronTune_[[:space:]]Towards[[:space:]]Self-Guided[[:space:]]Spurious[[:space:]]Bias[[:space:]]Mitigation/5be0001a-fe4d-42bb-a5d8-890b8eccde70_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neurosymbolic[[:space:]]World[[:space:]]Models[[:space:]]for[[:space:]]Sequential[[:space:]]Decision[[:space:]]Making/faaef104-0a73-4f35-9929-d2c1764bba42_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/Neutral[[:space:]]residues_[[:space:]]revisiting[[:space:]]adapters[[:space:]]for[[:space:]]model[[:space:]]extension/be6d177e-f3bb-4e2e-aac2-39165a3d3694_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/New[[:space:]]Bounds[[:space:]]for[[:space:]]Sparse[[:space:]]Variational[[:space:]]Gaussian[[:space:]]Processes/3f70a0ee-8b17-44f2-920f-531128b42519_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/NextCoder_[[:space:]]Robust[[:space:]]Adaptation[[:space:]]of[[:space:]]Code[[:space:]]LMs[[:space:]]to[[:space:]]Diverse[[:space:]]Code[[:space:]]Edits/68d2b7d7-90b1-4321-8b18-d38f9a96eb9d_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/No[[:space:]]Free[[:space:]]Lunch[[:space:]]from[[:space:]]Random[[:space:]]Feature[[:space:]]Ensembles_[[:space:]]Scaling[[:space:]]Laws[[:space:]]and[[:space:]]Near-Optimality[[:space:]]Conditions/3a5c7cfa-419d-43ac-8624-60fac0303dbe_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/No[[:space:]]Metric[[:space:]]to[[:space:]]Rule[[:space:]]Them[[:space:]]All_[[:space:]]Toward[[:space:]]Principled[[:space:]]Evaluations[[:space:]]of[[:space:]]Graph-Learning[[:space:]]Datasets/3a5d31e4-4598-4fb5-b61f-693502bd4bfb_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/No[[:space:]]Soundness[[:space:]]in[[:space:]]the[[:space:]]Real[[:space:]]World_[[:space:]]On[[:space:]]the[[:space:]]Challenges[[:space:]]of[[:space:]]the[[:space:]]Verification[[:space:]]of[[:space:]]Deployed[[:space:]]Neural[[:space:]]Networks/5296027a-7bc0-48ca-90c8-2ab0e4a0c0c1_origin.pdf filter=lfs diff=lfs merge=lfs -text
+2025/No[[:space:]]Task[[:space:]]Left[[:space:]]Behind_[[:space:]]Isotropic[[:space:]]Model[[:space:]]Merging[[:space:]]with[[:space:]]Common[[:space:]]and[[:space:]]Task-Specific[[:space:]]Subspaces/a38669ab-8571-4ad6-acd6-a85d76bd0e81_origin.pdf filter=lfs diff=lfs merge=lfs -text

2025/Multi-band Frequency Reconstruction for Neural Psychoacoustic Coding/acbe0cae-dd20-45e6-9725-8a41781631c7_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7e282f52920fd8b64209d7668bf4a84f78ed1571708e6a84f2c8e74e8a701c26
+size 5968398

2025/Multi-band Frequency Reconstruction for Neural Psychoacoustic Coding/full.md

@@ -0,0 +1,484 @@
+# Multi-band Frequency Reconstruction for Neural Psychoacoustic Coding
+
+Dianwen Ng $^{12}$  Kun Zhou $^{3}$  Yi-Wen Chao $^{2}$  Zhiwei Xiong $^{2}$  Bin Ma $^{3}$  Eng Siong Chng $^{2}$
+
+# Abstract
+
+Achieving high-fidelity audio compression while preserving perceptual quality across diverse audio types remains a significant challenge in Neural Audio Coding (NAC). This paper introduces MUFFIN, a fully convolutional Neural Psychoacoustic Coding (NPC) framework that leverages psychoacoustically guided multi-band frequency reconstruction. Central to MUFFIN is the Multi-Band Spectral Residual Vector Quantization (MBS-RVQ) mechanism, which quantizes latent speech across different frequency bands. This approach optimizes bitrate allocation and enhances fidelity based on psychoacoustic studies, achieving efficient compression with unique perceptual features that separate content from speaker attributes through distinct codebooks. MUFFIN integrates a transformer-inspired convolutional architecture with proposed modified snake activation functions to capture fine frequency details with greater precision. Extensive evaluations on diverse datasets (LibriTTS, IEMOCAP, GTZAN, BBC) demonstrate MUFFIN's ability to consistently surpass existing performance in audio reconstruction across various domains. Notably, a high-compression variant achieves an impressive SOTA  $12.5\mathrm{Hz}$  rate while preserving reconstruction quality. Furthermore, MUFFIN excels in downstream generative tasks, demonstrating its potential as a robust token representation for integration with large language models. These results establish MUFFIN as a groundbreaking advancement in NAC and as the first NPC system. Speech demos and codes are available<sup>1</sup>
+
+<sup>1</sup>MiroMind, Singapore <sup>2</sup>College of Computing & Data Science, Nanyang Technological University, Singapore <sup>3</sup>Tongyi Speech Lab, Alibaba Group, Singapore. Correspondence to: Dianwen Ng <dianwen001@e.ntu.edu.sg>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+<sup>1</sup>https://demo46.github.io/muffin/
+
+$^{2}$ https://github.com/dianwen-ng/MUFFIN
+
+# 1. Introduction
+
+Neural Audio Coding (NAC) has emerged as a transformative technology in speech and audio processing, enabling high compression and high-fidelity reconstruction of audio signals with substantially reduced data representation sizes (Défossez et al., 2022; Du et al., 2024b; Wu et al., 2023; Kumar et al., 2024). This advancement not only optimizes storage and transmission efficiencies but also enhances the integration of semantic and acoustic details into speech-large language models (LLMs) (Zhang et al., 2023a; 2024; Défossez et al., 2024). Within NAC, neural autoencoders and quantization techniques have achieved significant success in building unit representations that effectively capture essential audio features and compress them into precise, compact tokenized forms. Among these quantization methods, Residual Vector Quantization (RVQ) has emerged as the choice of method to enhance information preservation by refining the quantization process through multiple boosting turns (Zeghidour et al., 2022). This progressive refinement reduces reconstruction errors, enhancing fidelity to the original signal, ensuring higher quality audio outputs.
+
+Psychoacoustics (Liu et al., 2017; Zhen et al., 2020; Byun et al., 2022), the study of how humans perceive sound, underpins traditional audio coding frameworks such as MP3 and OPUS (Herre & Dick, 2019), yet remains largely unexplored in NAC. This discipline offers crucial insights for designing perceptually oriented systems, particularly through principles like perceptual masking. Different frequency bands carry distinct types of information: low frequencies are critical for speech intelligibility, mid frequencies capture formant structures essential for content articulation, and high frequencies convey speaker identity, pitch, and timbre—attributes integral to naturalness and spatial realism (San Roman et al., 2024; Petermann et al., 2023). By segmenting and encoding these bands separately, we account only for perceptually relevant distortions, reducing effective entropy while allowing greater controllability of learned speech attributes. This psychoacoustic perspective has the potential to redefine NAC, guiding more efficient compression strategies and richer, more robust reconstructions.
+
+Leveraging psychoacoustic insights, we introduce MUFFIN, the first neural psychoacoustic codec (NPC) that utilizes the proposed multi-band spectral RVQ to achieve high-fidelity
+
+audio compression. MUFFIN aligns compression strategies with psychoacoustic principles, achieving an optimal balance between bitrate efficiency and perceptual quality. Additionally, we have enhanced frequency modeling through a novel modification of the snake activation function. Extensive experimental evaluations demonstrate MUFFIN's superior performance in audio reconstruction compared to existing NACs. Furthermore, our codec excels in important downstream tasks, such as zero-shot text-to-speech (TTS), showcasing its potential as a robust token representations for integrating with LLMs for more advanced applications.
+
+Contributions (1) We propose a novel multi-band spectral RVQ to optimize bitrate allocation, enhancing compression efficiency and perceptual audio quality. (2) We introduce MUFFIN, the first neural psychoacoustic codec that incorporates a modified snake activation function to achieve superior audio reconstruction at a SOTA  $12.5\mathrm{Hz}$  compression while preserving high fidelity. (3) Our empirical demonstrations and experiments analyze the characteristics of each novel perceptual codebook, offering a better understanding of their potential use in various audio processing tasks.
+
+# 2. Related Work
+
+Neural Audio Codec Neural audio compression frameworks such as Soundstream (Zeghidour et al., 2022) and EnCodec (Défossez et al., 2022) have advanced the field by implementing fully convolutional encoder-decoder architectures with RVQ bottlenecks, optimizing lossy tokenization while maintaining high fidelity. These models use HiFiGAN vocoder losses (Kong et al., 2020) to balance low bitrates with accurate reconstruction, ensuring robust audio quality even at reduced bandwidths (Ai et al., 2024). AudioDec (Wu et al., 2023) further enhances this approach by integrating grouped convolutions for real-time processing in a streamable format, while HiFi-Codec (Yang et al., 2023) introduces parallel group-RVQ layers to minimize redundancy. However, challenges persist. RVQ-based models often fail to preserve fine-grained details in complex acoustics, leading to perceptible artifacts (Langman et al., 2024). Transformer-based NAC models, like SpeechTokenizer (Zhang et al., 2023b), which disentangle semantic content from the remaining audio features, face limitations in generalizing to input sequences longer than those encountered during training (Varis & Bojar, 2021; Chen et al., 2023), resulting in inconsistent performance and diminished output fidelity. Moreover, the prevalent practice of applying quantization across full frequency bands overlooks potential efficiency gains that could be achieved by accounting for the perceptual importance of different spectral regions, which is effective for achieving high compression capability while maintaining high-fidelity audio (Petermann et al., 2023). These issues highlight the need for continued innovation in
+
+neural audio coding to better address the complexities of auditory perception.
+
+Multi-band Audio Processing Human auditory perception operates through a multi-band system, where the cochlea acts as a frequency analyzer, separating complex waveforms into bands that deliver crucial perceptual cues (Pulkki & Karjalainen, 2015). Psychoacoustic principles like critical bands and frequency masking are foundational to models such as the Bark scale (Zwicker & Terhardt, 1980) and Equivalent Rectangular Bandwidth (ERB) (Moore & Glasberg, 1983), reflecting human auditory sensitivity to frequency variations. Leveraging these principles, multi-band modeling has significantly advanced audio compression by enhancing perceptual efficiency (Zhen et al., 2020).
+
+Legacy CODECs like MPEG-1 Audio Layer III (MP3) and MPEG-2 Advanced Audio Coding (AAC) (Bosi et al., 1997) used these psychoacoustic models to dynamically allocate bits based on perceptual thresholds, achieving near-transparent sound quality. However, their static compression thresholds limited adaptability to complex audio signals. Modern neural audio CODEcs (Xiao et al., 2023; Luo et al., 2024; Langman et al., 2024; Nishimura et al., 2024; Chen et al., 2024) have overcome these limitations through flexible neural architectures that adapt more dynamically to signal complexities. For instance, the Penguins codec (Xiao et al., 2023) enhances perceptual audio quality by segmenting audio into frequency bands, using generative models for low frequencies and bandwidth extension for high frequencies to reduce bitrate without compromising quality. By quantizing mel-band features, spectral CODEcs (Langman et al., 2024) align more closely with human auditory perception, although this can sometimes impact the accuracy of spectral details and phase information. Additionally, Gull (Luo et al., 2024) uses the Band-Split RNN (BSRNN) (Luo & Yu, 2023) architecture to effectively capture temporal and inter-band dependencies through subband modeling and neural compression. However, its stacked BSRNN blocks and causal RNN layers increase computational latency, impacting real-time streaming applications.
+
+# 3. Methods: Multi-Band Frequency Coding
+
+# 3.1. Multi-band spectral residual vector quantization (MBS-RVQ)
+
+In traditional methods, a multi-band model divides the frequency domain of a discrete-time signal,  $x[n]$ , into different frequency bands using the Fast Fourier Transform (FFT). Mathematically, the FFT of the signal can be expressed as:
+
+$$
+X [ k ] = \sum_ {n = 0} ^ {N - 1} x [ n ] e ^ {- j \frac {2 \pi}{N} k n}, \quad k = 0, 1, \dots , N - 1 \tag {1}
+$$
+
+where  $X[k]$  represents the complex frequency components of the signal,  $N$  is the total number of samples, and  $k$  is the
+
+frequency index. We define  $K$  non-overlapping frequency bands, each corresponding to a specific range of frequencies, denoted as  $\mathbf{B}_k$  can be defined as:
+
+$$
+\mathbf {B} _ {k} = \left\{f: f _ {\min , k} \leq f <   f _ {\max , k} \right\}, \quad k = 1, 2,.., K \tag {2}
+$$
+
+where  $f_{\mathrm{min},k}$  and  $f_{\mathrm{max},k}$  denote the minimum and maximum frequencies of the  $k^{th}$  band, respectively. After splitting the signal into frequency bands, we are able to construct the multi-band neural tokenizer by feeding each band with a typical time-domain neural autoencoder. The spectral matrix of each band is converted back into a time-domain waveform using the inverse FFT. This waveform is then passed through the encoder for quantization. Although this approach is straightforward for processing the signal in separate bands, it increases latency, as each band-pass signal requires individual encoding steps through the autoencoder (i.e., scaling the computational burden in terms of FLOPs by the number of  $K$  bands) before being recombined to reconstruct the original signal. This added computational complexity limits its suitability for real-time streaming applications.
+
+![](images/0c0efe2a08f7388c860220be8d64551b712b6c5936a09bfff64e8a72e1e7e271.jpg)  
+Figure 1. Illustration of the MBS-RVQ process: Fast Fourier Transform (FFT) is applied to the encoded latent representation to isolate specific frequency bands, capturing targeted spectral information for each codebook. The filtered representation is reconstructed using inverse FFT before undergoing quantization. The quantization residuals are then passed to the next codebook
+
+Therefore, rather than splitting the bands at the input, we propose operating the multi-band processing within the encoder's latent space using its latent features. The audio is first encoded into a compressed representation  $z$ , capturing both spectral and temporal features. The latent representation  $z \in \mathbb{R}^{d \times T}$ , where  $d$  is the channel dimensionality and  $T$  is the temporal length, is then decomposed into multiple frequency bands. Spectral band splitting is performed using the FFT, as previously detailed. We specifically target frequency bands of 0 to  $18.75\mathrm{Hz}$ , 18.75 to  $37.5\mathrm{Hz}$ , and 37.5 to  $75\mathrm{Hz}$ , with corresponding scale factors of 4, 2, and 1, respectively. These scale factors are applied within the  $75\mathrm{Hz}$  bandwidth of the latent representation, which is derived from compressing  $24\mathrm{kHz}$  audio by a factor of 320 using the encoder. Each band is then quantized sequentially and continuously updated through an exponential moving average (Défossez et al., 2022). Note that the residuals from
+
+each subband quantizer will be accounted for in the subsequent subband feature input as residuals, following the RVQ approach, as we leverage the principle of successive refinement. This method enables each stage to quantize the residual error from the previous stage, thereby enhancing the preservation of detail in the audio representation. Figure 1 presents an illustration of the proposed MBS-RVQ.
+
+Besides, adopting a multi-band splitting approach enhances the expression of individual unit codebooks within their respective spectral bands during the quantization process. This method is informed by psychoacoustic research, which highlights how different frequency ranges convey distinct types of information (Appendix A). Specifically, low-frequency bands are essential for speech intelligibility due to their concentrated energy, while mid-frequency bands are crucial for articulating content through formant structures. High-frequency bands provide detailed acoustic features such as speaker identity, pitch, and timbre, contributing to the naturalness of speech (San Roman et al., 2024; Petermann et al., 2023). Furthermore, these higher frequencies enhance the spatial and ambient qualities of audio recordings.
+
+By applying this knowledge, the encoding process can be optimized to allocate resources more effectively across the audio spectrum, ensuring that each data segment is processed to maximize the preservation and clarity of key auditory cues. This strategy not only allows for the perceptually meaningful quantization of audio units but also improves the efficiency and effectiveness of data compression, thereby significantly improving the overall perceptual quality of the output at a much lower codebook bitrate.
+
+# 3.2. Analysis: Multi-band modeling improves generative quality by leveraging the perceptual entropy bound.
+
+To provide justification for MUFFIN, let  $\mathbf{x}(t)$  be a continuous-time audio signal defined over  $t\in \mathbb{R}$ , and let  $\hat{\mathbf{x}} (t)$  be its compressed approximation. Suppose the human auditory system is modeled by a set of perceptual filters that divide the frequency axis into  $K$  critical bands  $B_{1},B_{2},\ldots ,B_{K}$ . Each band  $B_{k}$  corresponds to a region where the ear has distinct sensitivity levels (e.g., Bark or Mel scales). For each band  $B_{k}$ , let  $P_{k}(\mathbf{x}(t))$  represent the perceptual threshold function indicating the maximum allowable distortion before artifacts become noticeable, and let  $D_{k}(\mathbf{x}(t),\hat{\mathbf{x}} (t))$  denote the band-specific distortion introduced by compression. The perceptual entropy  $E_{p}$  of  $\mathbf{x}(t)$  is defined as the minimal bit rate,  $R_{k}$ , needed so that  $D_{k}(\mathbf{x}(t),\hat{\mathbf{x}} (t))\leq P_{k}(\mathbf{x}(t))$  for all  $k$ , ensuring transparent audio compression:
+
+$$
+E _ {p} = \sum_ {k = 1} ^ {K} \min \{R _ {k}: D _ {k} (\mathbf {x} (t), \hat {\mathbf {x}} (t)) \leq P _ {k} (\mathbf {x} (t)) \}.
+$$
+
+Theorem 3.1 (Perceptual Entropy and Masking Bounds). (Cover, 1999) For the given audio signal  $\mathbf{x}(t)$  and com
+
+pressed representation  $\hat{\mathbf{x}}(t)$ , the perceptual entropy  $E_{p}$  satisfies the following lower bound when optimal multiband modeling is employed:
+
+$$
+E _ {p} \geq \sum_ {k = 1} ^ {K} H (B _ {k} | \mathbf {x} (t)) - \sum_ {k = 1} ^ {K} \Delta (B _ {k}, \mathbf {x} (t)),
+$$
+
+where  $H(B_k \mid \mathbf{x}(t))$  is the Shannon entropy of the signal components within band  $B_k$ , and  $\Delta(B_k, \mathbf{x}(t))$  is the perceptual masking effect that reduces the effective entropy by accounting for inaudible distortions in band  $B_k$ .
+
+Theorem 3.1 characterizes how multiband modeling leverages psychoacoustic properties to achieve lower bit rates without sacrificing perceived audio quality. The first term,  $\sum_{k=1}^{K} H(B_k \mid \mathbf{x}(t))$ , represents the total intrinsic entropy over the  $K$  critical bands, analogous to the information-theoretic bound one would calculate if there were no masking effects. However, human hearing does not require perfect fidelity in all frequency regions; many distortions remain hidden under the masking threshold (Zwicker & Fastl, 2013). This phenomenon allowsCodec's to allocate fewer bits to masked regions without degrading the subjective audio experience.
+
+In mathematical terms,  $\Delta (B_k,\mathbf{x}(t))$  represents the "masking offset" that effectively reduces the bits needed for band  $B_{k}$ . Due to this offset, the total perceptual entropy  $E_{p}$  can be substantially smaller than the naive entropy sum, reflecting how frequency regions with strong masking require fewer bits for transparent encoding. This principle underlies the design of many perceptual audio CODEs (Brandenburg & Bosi, 1997), where an FFT first decomposes the signal into subbands aligned with the human ear's critical bands, and then quantization is adapted based on psychoacoustic models (Johnston, 1988).
+
+# 3.3. Model Architecture
+
+MUFFIN features an autoencoder architecture inspired by HiFi-Codec (Yang et al., 2023), utilizing a fully convolutional encoder-decoder network for temporal downscaling. We adopt the same striding configuration (2, 4, 5, 8), optimized for  $24\mathrm{kHz}$  audio waveforms, achieving a total down sampling factor of 320 in the default configuration. A key component of our convolutional block is the multi-receptive field (MRF) fusion, adapted from Kong et al. (2020), which aggregates outputs from residual blocks with varying dilated kernel sizes. This allows the model to capture dependencies across multiple temporal scales, enhancing its capacity to handle long-range sequential information. To enhance the model's representational power, we integrate an inverted bottleneck layer with residual skip connections, inspired by ConvNeXt (Liu et al., 2022), which increases channel dimensions and adds neural complexity. This design aligns with SOTA architectures (Dao & Gu, 2024; Yu & Wang, 2024; Han et al., 2024), enabling the model to
+
+![](images/fdbd6c1d5d8541ce8f7cef414cec60d13410d3335cf9e938ee0b442ec21793f1.jpg)  
+Figure 2. Architecture of MUFFIN incorporating a fully convolutional structure.
+
+learn richer, more detailed representations, as shown in Figure 2. To reduce latency by improving the computational efficiency from the channel-upsampling layer, we employ grouped convolutions that upscale in groups of 32 channels, which significantly reduces the number of model parameters and computations. The model consists of 46.1M parameters, with 34.2M in the encoder and 11.9M in the decoder. The Multiply-Accumulate Operations (MACs) reach 31.6G per second of audio sampled at  $24\mathrm{kHz}$ , demonstrating enhanced performance efficiency compared to the HiFi-Codec's model, which has 61.5M parameters and taking 44.4G of MACs computational steps.
+
+It is noteworthy that our convolutional block mirrors transformer functionality. The communication block from MRF mimics self-attention, enabling global temporal interactions, while the complexity block from the inverted bottleneck parallels the transformer's feed-forward layer, adding depth and expressiveness. This transformer-inspired architecture efficiently captures hierarchical temporal patterns, ensuring robust audio signal reconstruction performance.
+
+Periodic Activation Function. To enhance periodic modeling for better spectral preservation and high-fidelity reconstruction, we draw inspiration from Kumar et al. (2024) by replacing all Leaky ReLU activations with the snake activation function (Lee et al., 2022),  $x + 1 / \alpha \sin^2\alpha x$ , which better preserves high-frequency information (Ziyin et al., 2020) and maintain Lipschitz continuity, since the derivative of snake activation function is bounded by a constant of 1. We further enhanced this function by introducing amplitude and bias adjustments to improve overall performance. In the original formulation, the parameter  $\alpha$  controls the frequency of the periodic component, while its reciprocal  $1 / \alpha$ , attenuates the amplitude. To overcome this limitation, Evans et al. (2024) introduced a learnable scaling factor  $\beta$  to adjust the amplitude independently. However, this adjustment may
+
+result in increased variance because of the amplified periodic magnitude relative to the input  $x$  (Appendix D, Figure 4). To address this, we propose adding a bias term  $\gamma$  that learns to shift the output, ensuring better adaptation to the new scale while preserving consistency with the input range. This approach mitigates variance issues and offers greater flexibility in fitting data, greatly enhancing high-fidelity reconstruction.
+
+# 3.4. Training Objectives
+
+Following the framework from HiFi-Codec, our training leverages two key components in the overall loss objectives.
+
+Reconstruction Loss We employ a multiscale mel spectrogram reconstruction loss, calculated as the L1 distance between predicted and target mel-spectrograms over multiple time scales (i.e., a 64-bins mel-spectrogram derived from an STFT, with a window size of  $2^{i}$  and a hop length of  $2^{i} / 4$  for  $i = 7,8,9,10,11$ ). Unlike Yang et al. (2023) that uses 80 mel-spectrogram bins, we opted for a lower bin count based on perceptual evaluations, which revealed improved naturalness in the reconstructed audio while preserving key spectral features. Although reducing the bin count compromises frequency resolution, our results indicate that stricter distance reconstruction criteria, while preserving more information, may slightly harm perceptual quality.
+
+Discriminative Loss We use three discriminators: a multiscale STFT discriminator (MS-STFT) (Zeghidour et al., 2022), a multi-period discriminator (MPD), and a multiscale discriminator (MSD) (Kong et al., 2020) to enhance perceptual quality through adversarial learning. We adopt the HingeGAN (Lim & Ye, 2017) adversarial loss formulation and L1 feature matching loss (Kumar et al., 2019).
+
+MBS-RVQ Commitment Loss We adopt quantization with commitment losses based on the VQ-VAE framework (Van Den Oord et al., 2017), using stop-gradients and the straight-through estimator (Bengio et al., 2013) for backpropagation through the codebook lookup. The input is split into  $K$  critical frequency bands  $(B_{i})$ , where  $B_{i}$  denotes the  $i$ -th band. For each band, the encoder output  $z_{e}^{(B_{i})}$  (encoded representation) is mapped to the nearest codebook vector  $z_{q}^{(B_{i})}$  (quantized representation), ensuring frequency-specific representation. The commitment loss is defined as  $\mathcal{L}_{\mathrm{commit}} = \sum_{i=1}^{K} \beta_{i} \| z_{e}^{(B_{i})} - z_{q}^{(B_{i})} \|^2$ , enabling precise mapping over the quantization of each spectral band.
+
+# 4. Experiments
+
+Data sources. We train our model on a modest collection of 1,600 hours of speech, music, and environmental sounds. For speech, we use LibriTTS (Zen et al., 2019) and EARS (Richter et al., 2024) datasets with expressive
+
+anechoic recordings of speech (585 and 100 hours, respectively). For music, we utilize Music4All (Santana et al., 2020) (910 hours). For environmental sounds, we use ESC-50 (Piczak, 2015) (3 hours, 50 classes with 40 examples per class, loosely arranged into 5 major categories: animal, human, natural sounds, interior, and exterior sounds). Music and environmental sounds are used in MUFFIN to learn broader audio expression, enhancing the foundational representations. All audio was resampled to  $24\mathrm{kHz}$ .
+
+Prior to training, all audio files are truncated to a maximum duration of 10 seconds. A text file containing the paths to the processed audio files is provided to the data loader. This ensures a balanced sample distribution of speech and music file samples, mitigating data imbalance and preventing under-performance skewing towards vocal or instrumental reconstruction. During training, we apply voice activity detection to remove non-audio segments, optimizing learning efficiency. For each batch, 1-second audio segments are randomly selected from each instance and are zero-padded if shorter than 1 second. No additional data augmentation is applied to maintain simplicity in the experiment.
+
+Model and training details. In our experiments, models were trained on two A800 GPUs for 300K iterations with a learning rate of 2e-4 and a batch size of 20 per GPU. All quantizers utilize a 9-bit code lookup from the EMA codebook. Additionally, we developed a low token-rate compression variant of MUFFIN to leverage the model's efficiency in capturing non-redundant information across different frequency bands. The highly compressed MUFFINs increase the downsampling rate by  $960 \times$  and  $1920 \times$  in the encoding layers, producing latent representations at  $25\mathrm{Hz}$  and  $12.5\mathrm{Hz}$ , respectively. This configuration achieves audio quantization at 150 and 100 tokens per second, utilizing additional residual codebooks with a total of six and eight vector quantization layers. The same MBS-RVQ configuration is used, maintaining a 4;2;1 frequency ratio relative to the sampled frequency of the latent embeddings (partitioned on a logarithmic scale) for the default band splits. MUFFIN, operating at  $12.5\mathrm{Hz}$ , is trained on 2-second audio segments and requires four A800 GPUs to support a batch size of 10 per GPU. Using a lower learning rate is crucial to prevent gradient explosion. The high-compression MUFFIN model comprises approximately 50.6M parameters, with 36.5M in the encoder and 14.1M in the decoder (both models share the same architectural depth, differing only in downsampling rates). It achieves significantly lower MACs at 17.9G per second of audio, enabling faster inference.
+
+Since our training setup closely adheres to Hifi-Codec, one of the leading codec model in the field, we establish it as our baseline by retraining the model with the same configuration, allowing for accurate performance comparison. Additionally, we benchmarked our method against other prominent
+
+codecs, including OPUS, Encodec and DAC, all reconfigured to a transmission rate of  $3.0\mathrm{kB / s}$  to assess performance at similar transmission rate as our codec reconstruction model. Furthermore, we evaluate high-compression MUFFIN variant against Mimi, the recent SOTA codec model for  $12.5\mathrm{Hz}$ , to highlight the strengths of our work.
+
+Evaluation objectives. We utilize the objective evaluation metrics outlined in codec-SUPERB (Wu et al., 2024) to assess the perceptual quality of different audio domains. Specifically, we incorporate metrics such as the Perceptual Evaluation of Speech Quality (PESQ) (Rix et al., 2001), Short-Time Objective Intelligibility (STOI) (Taal et al., 2010), STFT distance (Alsteris & Paliwal, 2007), Mel distance (Kubichek, 1993), and F0CORR (F0 Pearson Correlation Coefficient) (Jadoul et al., 2018). The selection of these metrics for the corresponding audio domain is justified by the inclusion-exclusion criteria discussed in codec-SUPERB. Additionally, we employ two automated Mean Opinion Score (MOS) evaluation metrics, UTMOS (Saeki et al., 2022) and ViSQOL (Hines et al., 2015), to assess the perceptual quality of the CODECs. These metrics are designed to closely approximate subjective listening tests, providing a more accurate and robust evaluation of codec performance. For speech evaluation, we use the test-clean and test-other set from LibriTTS and evaluate emotional speech reconstruction with IEMOCAP (Busso et al., 2008). For music, we employ the GTZAN dataset (Sturm, 2013), while environmental sounds are evaluated using audio from the BBC sound effects (LAION-AI, 2022).
+
+# 4.1. Experimental Results
+
+Table 1 and 2 compare the speech reconstruction quality of our MUFFINs against existing top performing NACs. We indicate that NACs listed in the table operate at a bandwidth of  $3.0\mathrm{kB / s}$ , except for Mimi at  $1.0\mathrm{kB / s}$ , MUFFIN (default) at  $2.7\mathrm{kB / s}$ , MUFFIN  $(\nabla)$  at  $1.35\mathrm{kB / s}$ , and MUFFIN  $(\triangle)$  at  $0.9\mathrm{kB / s}$ . We use LibriTTS evaluation dataset, with 4,837 samples for test-clean and 5,120 for test-other. Notably, MUFFIN achieves superior reconstruction fidelity, achieving the lowest distance error and outperforming competing NACs across various objective metrics. This includes HiFi-Codec, which serves as a robust baseline by being retrained under the same conditions, thereby confirming the effectiveness of the proposed framework. Furthermore, MUFFIN  $\nabla$  and  $\triangle$  achieve better UTMOS scores despite reductions in bandwidth and token rates. It appears that increasing the number of codebooks in our NAC system enhances the preservation of information from the original speech, resulting in improved perceptual quality as reflected by higher UTMOS scores. This suggests that with more codebooks, the system better captures nuanced details that contribute to the naturalness of the audio. However, while this strategy enhances perceptual naturalness, it may not uni
+
+formly improve all objective metrics. In fact, as the number of codebooks increases, especially at higher compression rates, some objective measures experience a decline. This decline may be attributed to the introduction of noise or artifacts by additional codebooks, which, while capturing more detail, also amplify aspects that negatively impact certain evaluation metrics. Thus, the relationship between increased codebooks and system performance exemplifies a trade-off between improved naturalness and the potential degradation of other audio quality metrics. Nevertheless, it is notable that MUFFIN  $\triangleleft$  remains competitive with leading NAC models and even outperforms Mimi in terms of naturalness and reconstruction fidelity, achieving significant gains in audio quality while maintaining efficient compression rates. Table 2 showcases MUFFIN's zero-shot reconstruc
+
+Table 1. Objective evaluation of reconstructed speech from the LibriTTS dataset using various neural audio codec models.  $GT$  refers to the abbreviation for ground truth. Except for highly-compressed MUFFIN and Mimi, which uses a compression rate of  $\nabla : \times 960$  (25.0 Hz) and  $\triangle : \times 1920$  (12.5 Hz), the others have the compression rate of  $\times 320$  (75 Hz). Note that HiFi-Codec was retrained using the official configuration settings, but on the same dataset as MUFFIN.  
+
+<table><tr><td colspan="7">Test-Clean (LibriTTS)</td></tr><tr><td>Model</td><td>STFT</td><td>MEL</td><td>PESQ</td><td>STOI</td><td>UTMOS</td><td>ViSQOL</td></tr><tr><td>GT</td><td>-</td><td>-</td><td>-</td><td>-</td><td>4.041</td><td>-</td></tr><tr><td>OPUS</td><td>5.728</td><td>2.796</td><td>1.132</td><td>0.715</td><td>1.264</td><td>2.878</td></tr><tr><td>Encodec</td><td>1.956</td><td>1.051</td><td>2.042</td><td>0.903</td><td>2.269</td><td>4.078</td></tr><tr><td>DAC</td><td>1.759</td><td>0.849</td><td>2.370</td><td>0.915</td><td>2.951</td><td>4.143</td></tr><tr><td>HiFi-Codec</td><td>1.618</td><td>0.765</td><td>2.712</td><td>0.943</td><td>3.831</td><td>4.410</td></tr><tr><td>Mimi▲</td><td>2.488</td><td>1.706</td><td>1.715</td><td>0.620</td><td>2.966</td><td>3.791</td></tr><tr><td>MUFFIN</td><td>1.555</td><td>0.692</td><td>2.996</td><td>0.954</td><td>4.017</td><td>4.516</td></tr><tr><td>MUFFIN ▽</td><td>1.626</td><td>0.755</td><td>2.525</td><td>0.937</td><td>4.035</td><td>4.345</td></tr><tr><td>MUFFIN ▲</td><td>1.663</td><td>0.807</td><td>2.360</td><td>0.932</td><td>4.074</td><td>4.225</td></tr><tr><td colspan="7">Test-Other (Libri-TTS)</td></tr><tr><td>GT</td><td>-</td><td>-</td><td>-</td><td>-</td><td>3.453</td><td>-</td></tr><tr><td>OPUS</td><td>5.390</td><td>2.703</td><td>1.143</td><td>0.695</td><td>1.271</td><td>2.815</td></tr><tr><td>Encodec</td><td>1.998</td><td>1.119</td><td>1.960</td><td>0.888</td><td>2.026</td><td>4.017</td></tr><tr><td>DAC</td><td>1.813</td><td>0.913</td><td>2.220</td><td>0.897</td><td>2.497</td><td>4.053</td></tr><tr><td>HiFi-Codec</td><td>1.681</td><td>0.840</td><td>2.419</td><td>0.919</td><td>3.216</td><td>4.296</td></tr><tr><td>Mimi▲</td><td>2.515</td><td>1.688</td><td>1.611</td><td>0.612</td><td>2.498</td><td>3.679</td></tr><tr><td>MUFFIN</td><td>1.615</td><td>0.758</td><td>2.658</td><td>0.934</td><td>3.444</td><td>4.454</td></tr><tr><td>MUFFIN ▽</td><td>1.681</td><td>0.817</td><td>2.232</td><td>0.914</td><td>3.516</td><td>4.276</td></tr><tr><td>MUFFIN ▲</td><td>1.725</td><td>0.875</td><td>2.086</td><td>0.904</td><td>3.560</td><td>4.129</td></tr></table>
+
+tion capability on the full 12-hour IEMOCAP dataset, which contains expressive emotional speech that was not used during training. While reconstruction fidelity declines across the NACs, as shown in the objective metrics, both the default and high-compression variants of MUFFIN demonstrate superior robustness, achieving higher naturalness in human-perceived audio quality based on UTMOS scores compared to the ground truth references, despite the high compression. Nevertheless, the prominent drop in reconstruction fidelity for emotional content highlights a challenge in preserving emotional nuances, which could potentially impair emotional recognition in downstream tasks.
+
+Table 2. Objective evaluation of the reconstructed speech from the IEMOCAP dataset was conducted using various neural audio codec models, following the same setup as before.  
+
+<table><tr><td>Model</td><td>STFT</td><td>MEL</td><td>PESQ</td><td>STOI</td><td>UTMOS</td><td>ViSQOL</td></tr><tr><td>GT</td><td>-</td><td>-</td><td>-</td><td>-</td><td>1.859</td><td>-</td></tr><tr><td>OPUS</td><td>2.361</td><td>1.586</td><td>1.207</td><td>0.478</td><td>1.242</td><td>2.642</td></tr><tr><td>Encodec</td><td>2.150</td><td>1.290</td><td>1.649</td><td>0.746</td><td>1.321</td><td>3.501</td></tr><tr><td>DAC</td><td>1.553</td><td>0.781</td><td>1.867</td><td>0.763</td><td>1.316</td><td>3.774</td></tr><tr><td>HiFi-Codec</td><td>1.447</td><td>0.755</td><td>1.998</td><td>0.763</td><td>1.564</td><td>3.651</td></tr><tr><td>Mimi▲</td><td>2.112</td><td>0.755</td><td>1.433</td><td>0.494</td><td>1.427</td><td>2.801</td></tr><tr><td>MUFFIN</td><td>1.399</td><td>0.675</td><td>2.178</td><td>0.806</td><td>1.903</td><td>4.000</td></tr><tr><td>MUFFIN ▽</td><td>1.392</td><td>0.703</td><td>1.844</td><td>0.748</td><td>2.026</td><td>3.612</td></tr><tr><td>MUFFIN▲</td><td>1.429</td><td>0.754</td><td>1.726</td><td>0.723</td><td>2.019</td><td>3.376</td></tr></table>
+
+Table 3. Objective evaluation of the reconstructed music from the GTZAN dataset using different NACs.  
+
+<table><tr><td>Model</td><td>STFT</td><td>MEL</td><td>PESQ</td><td>STOI</td><td>F0CORR</td><td>ViSQOL</td></tr><tr><td>OPUS</td><td>7.786</td><td>3.462</td><td>1.081</td><td>0.424</td><td>0.727</td><td>2.414</td></tr><tr><td>Encodec</td><td>2.712</td><td>1.016</td><td>1.684</td><td>0.756</td><td>0.882</td><td>4.247</td></tr><tr><td>DAC</td><td>2.493</td><td>0.928</td><td>1.709</td><td>0.741</td><td>0.867</td><td>4.220</td></tr><tr><td>HiFi-Codec</td><td>2.517</td><td>0.954</td><td>1.674</td><td>0.727</td><td>0.899</td><td>4.170</td></tr><tr><td>MUFFIN</td><td>2.360</td><td>0.866</td><td>1.815</td><td>0.760</td><td>0.896</td><td>4.298</td></tr><tr><td>MUFFIN ∇</td><td>2.492</td><td>0.928</td><td>1.474</td><td>0.674</td><td>0.879</td><td>4.273</td></tr><tr><td>MUFFIN ▲</td><td>2.550</td><td>0.987</td><td>1.409</td><td>0.642</td><td>0.872</td><td>4.223</td></tr></table>
+
+Tables 3 and 4 present zero-shot reconstruction results on full data samples for music and audio, specifically from the GTZAN and BBC datasets. We observe a general decrease in fidelity for music reconstruction, as the task becomes more challenging due to the need to reconstruct multiple instrumental audio channels. From the table, we observe that HiFi-Codec achieves a higher F0CORR, indicating superior pitch accuracy and suggesting that its model structure better preserves vocal quality compared to other NACs. However, the difference in F0CORR between MUFFIN and HiFi-Codec is minimal, down to the finer decimal places, and MUFFIN consistently outperforms other NACs across the remaining metrics. Moreover, while MUFFIN  $\nabla \triangle$  achieves a higher compression rate and learns to encode more efficiently with decent reconstruction fidelity for music, as confirmed by its competitive ViSQOL scores, we observed that PESQ and STOI were significantly lower, as reflected in Table 2. This suggests that highly compressed models face challenges in preserving fine information and are more vulnerable to reduced generalizability in zero-shot inference.
+
+Table 4. Objective evaluation of the reconstructed speech from the BBC dataset was conducted using various neural audio codec models, following the same setup as before.  
+
+<table><tr><td>Model</td><td>STFT</td><td>MEL</td><td>ViSQOL</td></tr><tr><td>OPUS</td><td>6.093</td><td>2.984</td><td>1.000</td></tr><tr><td>Encodec</td><td>1.998</td><td>1.011</td><td>3.852</td></tr><tr><td>DAC</td><td>1.846</td><td>0.784</td><td>3.995</td></tr><tr><td>HiFi-Codec</td><td>1.773</td><td>0.795</td><td>4.009</td></tr><tr><td>MUFFIN</td><td>1.658</td><td>0.720</td><td>4.065</td></tr><tr><td>MUFFIN ∇</td><td>1.700</td><td>0.748</td><td>4.010</td></tr><tr><td>MUFFIN ▲</td><td>1.706</td><td>0.777</td><td>3.997</td></tr></table>
+
+Table 4 showcases the strong generalizability of zero-shot reconstruction on general audio from the BBC dataset, un
+
+derscoring the robustness and efficiency of MUFFINs when compared to other NACs. The results consistently demonstrate the improved quality of our neural codec for general audio reconstruction.
+
+# 4.2. Ablation Studies of Deconstructing MBS Codes
+
+Auditory feature in codebook representations. In this section, we examine the information encoded in MUFFIN's learned codebooks, which correspond to different auditory frequency bands. MUFFIN's codebooks focus on isolating the perceptual characteristics of speech attributes, guided by psychoacoustic research, and do so without the need for label-targeted supervision. To enhance understanding, we will also provide demos (Section F) of audio reconstructed from each codebook, accessible via the link.
+
+Codebook 1 (Low-frequency bands,  $0 - 18.75\mathrm{Hz}$ ) contains fundamental frequencies and strong harmonic content necessary for conveying core speech information. However, they primarily capture broad aspects of speech and lack detail in articulating speech content. This approach contrasts with previous NAC models where full-band RVQ often consolidates most speech information into the first codebook. To assess semantic content preservation in these low frequencies, we measure the STOI and word error rate (WER) using the pre-trained Whisper-large V3 model (Radford et al., 2023), which analyzes audio reconstructed by the NAC model. For compatibility with the ASR model, trained on  $16\mathrm{kHz}$  audio, we use the LibriSpeech test-clean dataset (Panayotov et al., 2015), consisting of 2,620 samples that are resampled during reconstruction. Each sample's speech decoded from individual codebooks is processed by the ASR model. We also evaluate other NAC models, including MUFFIN with vanilla RVQ, to explore how different codebooks affect the semantic information in speech.
+
+Table 5. The table presents the WER of ASR performance on reconstructed speech from each NAC's codebook using the Whisper-large V3 pre-trained model.  
+
+<table><tr><td></td><td colspan="2">MUFFIN</td><td colspan="2">RVQ</td><td colspan="2">Hifi-Codec</td><td colspan="2">DAC</td><td colspan="2">Encodec</td></tr><tr><td>GT</td><td colspan="10">WER: 2.41; STOI: -</td></tr><tr><td></td><td>WER</td><td>STOI</td><td>WER</td><td>STOI</td><td>WER</td><td>STOI</td><td>WER</td><td>STOI</td><td>WER</td><td>STOI</td></tr><tr><td>All</td><td>2.67</td><td>0.940</td><td>2.72</td><td>0.930</td><td>3.00</td><td>0.919</td><td>3.53</td><td>0.901</td><td>3.15</td><td>0.900</td></tr><tr><td>Code 1</td><td>70.3</td><td>0.644</td><td>75.6</td><td>0.702</td><td>154</td><td>0.572</td><td>36.1</td><td>0.731</td><td>33.7</td><td>0.764</td></tr><tr><td>Code 2</td><td>114</td><td>0.379</td><td>141</td><td>0.426</td><td>139</td><td>0.454</td><td>132</td><td>0.148</td><td>159</td><td>0.199</td></tr><tr><td>Code 3</td><td>191</td><td>0.436</td><td>100</td><td>0.157</td><td>100</td><td>0.090</td><td>100</td><td>0.079</td><td>153</td><td>0.121</td></tr><tr><td>Code 4</td><td>107</td><td>0.082</td><td>101</td><td>0.086</td><td>112</td><td>0.129</td><td>100</td><td>0.049</td><td>147</td><td>0.094</td></tr></table>
+
+Table 5 highlights the crucial role of Codebook 1 in the MUFFIN model, as it becomes significantly challenging to recognize content using any codebook other than Codebook 1, with recognition errors exceeding 100. Unlike vanilla RVQ, which typically consolidates most speech content into the first codebook, MUFFIN strategically distributes information between Codebooks 1 and 2. The main speech intelligibility is allocated to Codebook 1, while details related
+
+to articulation are captured in the mid-frequency range of Codebook 2, which provides it with more contextual information than Codebook 3. Notably, MUFFIN's Codebooks 1 and 2 together achieve an STOI of 0.729 and WER of 19.2. In contrast, NACs that employ traditional RVQ often show a progressive decline in semantic content across subsequent codebooks, accompanied by a noticeable decrease in STOI.
+
+Codebook 3 (High-frequency bands,  $37.5 - 75\mathrm{Hz}$ ) captures essential auditory cues, such as speaker identity, pitch, and timbre, which are crucial for distinguishing speakers and enriching the depth of reconstructed audio. To assess the effectiveness of each codebook in preserving speaker information, we randomly select 600 speech files from the VoxCeleb dataset (Nagrani et al., 2017), each representing one of six distinct speakers. Latent features are extracted from each codebook and then average-pooled to form a vector representation for each speech sample. Using t-SNE (Van der Maaten & Hinton, 2008), we visualize these vector representations in a two-dimensional space to identify potential clusters corresponding to different speakers, as presented in Figure 5 (Appendix E, Figure 5).
+
+From Figure 5, Codebook 1 exhibits a broad, dispersed distribution with some cluster overlap, suggesting it encodes foundational speech content with substantial variance across samples. Codebook 2 forms a more concentrated central cluster with significant overlap between speakers, implying shared features likely related to vowel and consonant articulation. Codebook 3, in contrast, shows well-separated clusters with the most distinct boundaries among speakers and minimal overlap, indicating its focus on capturing speaker-specific features. This distinct clustering should lead to a low distance error in speaker group classification, suggesting that Codebook 3 is highly effective at distinguishing between different speakers based on their unique vocal attributes. Additionally, we strongly suggest comparing the auditory results between Codebook 3 and others using the provided demos to gain a more intuitive understanding. Lastly, Codebook 4 presents a compact distribution with considerable class overlap, suggesting it primarily encodes random or residual features that contribute minimally to core speech information. Additionally, the variations and the distributive patterns in the t-SNE plots suggest that each codebook captures slightly different information.
+
+# 4.3. Zero-shot TTS with MUFFIN tokens
+
+We further evaluate the performance of different codec systems in a zero-shot text-to-speech (TTS) setting. For this evaluation, we use VALL-E (Wang et al., 2023) as the TTS framework, implementing it based on a popular release<sup>3</sup>, and trained it on LibriTTS dataset. VALL-E employ a decoder-only language model, predicting the first layer of acoustic
+
+Table 6. The zero-shot TTS results include word error rate (WER), mean opinion score (MOS) for speech quality and naturalness, and speaker mean opinion score (S-MOS) for speaker similarity, evaluated for VALL-E with three different codecs. MOS and S-MOS are reported with  $95\%$  confidence interval.
+
+<table><tr><td>Systems</td><td>WER (%)</td><td>MOS</td><td>S-MOS</td><td>SECS</td></tr><tr><td>VALL-E w/ Encodec</td><td>21.05</td><td>3.91 ± 0.287</td><td>3.70 ± 0.368</td><td>0.5914</td></tr><tr><td>VALL-E w/ HiFi-Codec</td><td>32.35</td><td>4.00 ± 0.532</td><td>4.04 ± 0.333</td><td>0.5874</td></tr><tr><td>VALL-E w/ MUFFIN</td><td>12.20</td><td>4.18 ± 0.278</td><td>4.19 ± 0.288</td><td>0.6099</td></tr></table>
+
+codes autoregressively, while the remaining layers are predicted non-autoregressively. It has gained popularity as a zero-shot TTS framework and is frequently used as a benchmark in various studies (Du et al., 2024a; Shen et al., 2023).
+
+We follow the experimental settings in (Zhou et al., 2024b) and test 3 different VALL-E systems on the test-clean set from LibriTTS. For each speaker in the test set, one speech sample from the same speaker is randomly selected as the prompt, ensuring it is distinct from the speech to be synthesized. All the synthesized speech are ranging from 3 to 10 seconds. We first calculate the word error rate (WER) of the synthesized speech, as shown in Table 6. Compared to VALL-E using Encodec or Hifi-Codec, VALL-E with MUFFIN achieves the lowest WER results, demonstrating superior robustness in the synthesized speech. We then report the subjective evaluation results in Table 6, where 10 native speakers evaluated a total of 80 synthesized speech samples. Each speech sample was rated on a 5-point scale for speech quality and naturalness (MOS), as well as speaker similarity to the prompt speech (S-MOS). Consistent with the objective results, the VALL-E system using MUFFIN achieves the highest scores for both speech quality and speaker similarity. These findings suggest that MUFFIN effectively disentangles between semantic and acoustic information, thereby reducing error propagation during the TTS training and resulting in improved speech quality during inference.
+
+# 5. Conclusion
+
+In conclusion, we introduced MUFFIN, a neural psychoaudio codec that offers a novel perspective on quantizing units with the proposed MBS-RVQ within the latent space. By strategically aligning the codec architecture with psychoacoustic principles, MUFFIN achieves an optimal balance between compression efficiency and perceptual fidelity, tackling longstanding challenges in the domain. Extensive evaluations demonstrate MUFFIN's SOTA performance across diverse audio types reconstruction and downstream zero-shot TTS task. Furthermore, the development of a 1920 times highly compressed MUFFIN variant underscores its ability to maintain perceptual quality even under extreme compression settings. This study lays the groundwork for future advancements in real-time low latency neural audio coding and its integration with LLMs, providing a robust and scalable solution for speech-LLMs model applications.
+
+# Acknowledgements
+
+This research was conducted at the Alibaba-NTU Singapore Joint Research Institute.
+
+# Impact Statement
+
+The development of MUFFIN, a high-fidelity Neural Psychoacoustic Coding (NPC) framework, represents a novel and effective alternative to conventional speech encoders (Hsu et al., 2021; Chen et al., 2022; Ng et al., 2023). This framework offers broad applicability across domains including media streaming, telecommunications, and assistive technologies. By incorporating psychoacoustic principles to guide perceptual compression, MUFFIN enables the transmission of high-quality audio at significantly lower bitrates. This advancement facilitates wider accessibility in bandwidth-constrained environments, thereby promoting digital inclusion and improving access to communication and entertainment services in regions with limited internet infrastructure. In addition to its compression efficiency, MUFFIN introduces a mechanism for disentangling speaker identity from speech content. This capability offers new opportunities for personalization and content manipulation but also raises critical ethical considerations related to privacy, data security, and the potential for misuse in synthetic or manipulated speech, such as deepfake audio. These concerns highlight the urgent need for responsible deployment practices and appropriate regulatory oversight. Moreover, the integration of MUFFIN with large-scale language models has the potential to significantly enhance human-computer interaction in areas such as creativity support, education, and accessibility-driven applications. This synergy positions MUFFIN as a foundational component in the next generation of AI-powered multimedia systems. Future research should focus on developing robust bias mitigation strategies and establishing clear ethical frameworks to guide the deployment of neural audio coding technologies in a fair and secure manner across diverse global populations.
+
+# References
+
+Ai, Y., Jiang, X.-H., Lu, Y.-X., Du, H.-P., and Ling, Z.-H. Apocodec: A neural audio codec with parallel amplitude and phase spectrum encoding and decoding. arXiv preprint arXiv:2402.10533, 2024.  
+Alsteris, L. D. and Paliwal, K. K. Short-time phase spectrum in speech processing: A review and some experimental results. Digital signal processing, 17(3):578-616, 2007.  
+Arjovsky, M., Chintala, S., and Bottou, L. Wasserstein generative adversarial networks. In International conference on machine learning, pp. 214-223. PMLR, 2017.  
+Bartlett, P. L. and Mendelson, S. Rademacher and gaussian
+
+complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3(Nov):463-482, 2002.  
+Belkin, M. and Niyogi, P. Laplacian eigenmaps for dimensionality reduction and data representation. Neural computation, 15(6):1373-1396, 2003.  
+Bengio, Y., Léonard, N., and Courville, A. Estimating or propagating gradients through stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013.  
+Bosi, M., Brandenburg, K., Quackenbush, S., Fielder, L., Akagiri, K., Fuchs, H., and Dietz, M. Iso/iec MPEG-2 advanced audio coding. Journal of the Audio engineering society, 45(10):789-814, 1997.  
+Brandenburg, K. and Bosi, M. Overview ofmpeg audio: Current and future standards for low bit-rate audio coding. Journal of the Audio Engineering Society, 45(1/2):4-21, 1997.  
+Busso, C., Bulut, M., Lee, C.-C., Kazemzadeh, A., Mower, E., Kim, S., Chang, J. N., Lee, S., and Narayanan, S. S. Iemocap: Interactive emotional dyadic motion capture database. Language resources and evaluation, 42:335-359, 2008.  
+Byun, J., Shin, S., Sung, J., Beack, S., and Park, Y. Optimization of deep neural network (dnn) speech coder using a multi time scale perceptual loss function. In Interspeech, pp. 4411-4415, 2022.  
+Chen, H.-J., Meng, Y., and Lee, H.-y. Once-for-all sequence compression for self-supervised speech models. In ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 1-5. IEEE, 2023.  
+Chen, J., Dai, Z., Ye, Z., Tan, X., Liu, Q., Guo, Y., and Xue, W. Pyramidcodec: Hierarchical codec for longform music generation in audio domain. In *Findings of the Association for Computational Linguistics: EMNLP* 2024, pp. 4253-4263, 2024.  
+Chen, S., Wang, C., Chen, Z., Wu, Y., Liu, S., Chen, Z., Li, J., Kanda, N., Yoshioka, T., Xiao, X., et al. Wavlm: Large-scale self-supervised pre-training for full stack speech processing. IEEE Journal of Selected Topics in Signal Processing, 16(6):1505-1518, 2022.  
+Cover, T. M. Elements of information theory. John Wiley & Sons, 1999.  
+Dao, T. and Gu, A. Transformers are ssms: Generalized models and efficient algorithms through structured state space duality. arXiv preprint arXiv:2405.21060, 2024.
+
+Défossez, A., Copet, J., Synnaeve, G., and Adi, Y. High fidelity neural audio compression. arXiv preprint arXiv:2210.13438, 2022.  
+Défossez, A., Mazaré, L., Orsini, M., Royer, A., Pérez, P., Jégou, H., Grave, E., and Zeghidour, N. Moshi: a speech-text foundation model for real-time dialogue. arXiv preprint arXiv:2410.00037, 2024.  
+Du, Z., Chen, Q., Zhang, S., Hu, K., Lu, H., Yang, Y., Hu, H., Zheng, S., Gu, Y., Ma, Z., Gao, Z., and Yan, Z. Cosyvoice: A scalable multilingual zero-shot text-to-speech synthesizer based on supervised semantic tokens, 2024a.  
+Du, Z., Zhang, S., Hu, K., and Zheng, S. Funcodec: A fundamental, reproducible and integrable open-source toolkit for neural speech codec. In ICASSP 2024-2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 591-595. IEEE, 2024b.  
+Dunn, H. and White, S. Statistical measurements on conversational speech. The Journal of the Acoustical Society of America, 11(3):278-288, 1940.  
+Evans, Z., Parker, J. D., Carr, C., Zukowski, Z., Taylor, J., and Pons, J. Long-form music generation with latent diffusion. arXiv preprint arXiv:2404.10301, 2024.  
+French, N. R. and Steinberg, J. C. Factors governing the intelligibility of speech sounds. The journal of the Acoustical society of America, 19(1):90-119, 1947.  
+Gulrajani, I., Ahmed, F., Arjovsky, M., Dumoulin, V., and Courville, A. C. Improved training of wasserstein gans. Advances in neural information processing systems, 30, 2017.  
+Hager, W. W. Lipschitz continuity for constrained processes. SIAM Journal on Control and Optimization, 17(3):321-338, 1979.  
+Han, D., Wang, Z., Xia, Z., Han, Y., Pu, Y., Ge, C., Song, J., Song, S., Zheng, B., and Huang, G. Demystify mamba in vision: A linear attention perspective. arXiv preprint arXiv:2405.16605, 2024.  
+Herre, J. and Dick, S. Psychoacoustic models for perceptual audio coding—a tutorial review. Applied Sciences, 9(14): 2854, 2019.  
+Hines, A., Skoglund, J., Kokaram, A. C., and Harte, N. Visqol: an objective speech quality model. EURASIP Journal on Audio, Speech, and Music Processing, 2015: 1-18, 2015.
+
+Hsu, W.-N., Bolte, B., Tsai, Y.-H. H., Lakhotia, K., Salakhutdinov, R., and Mohamed, A. Hubert: Self-supervised speech representation learning by masked prediction of hidden units. IEEE/ACM transactions on audio, speech, and language processing, 29:3451-3460, 2021.  
+Jadoul, Y., Thompson, B., and De Boer, B. Introducing parsleymouth: A python interface to praat. Journal of Phonetics, 71:1-15, 2018.  
+Johnston, J. D. Transform coding of audio signals using perceptual noise criteria. IEEE Journal on selected areas in communications, 6(2):314-323, 1988.  
+Ju, Z., Wang, Y., Shen, K., Tan, X., Xin, D., Yang, D., Liu, Y., Leng, Y., Song, K., Tang, S., et al. Naturalspeech 3: Zero-shot speech synthesis with factorized codec and diffusion models. arXiv preprint arXiv:2403.03100, 2024.  
+Kong, J., Kim, J., and Bae, J. Hifi-gan: Generative adversarial networks for efficient and high fidelity speech synthesis. Advances in neural information processing systems, 33:17022-17033, 2020.  
+Kubichek, R. Mel-cepstral distance measure for objective speech quality assessment. In Proceedings of IEEE pacific rim conference on communications computers and signal processing, volume 1, pp. 125-128. IEEE, 1993.  
+Kumar, K., Kumar, R., De Boissiere, T., Gestin, L., Teoh, W. Z., Sotelo, J., De Brebisson, A., Bengio, Y., and Courville, A. C. Melgan: Generative adversarial networks for conditional waveform synthesis. Advances in neural information processing systems, 32, 2019.  
+Kumar, R., Seetharaman, P., Luebs, A., Kumar, I., and Kumar, K. High-fidelity audio compression with improved rvqgan. Advances in Neural Information Processing Systems, 36, 2024.  
+LAION-AI. BBC Sound Effects Library data card. https: //github.com/LAION-AI.audio-dataset/ blob/main/data_card/BBC.md, 2022. Accessed: [October, 7, 2024].  
+Langman, R., Jukić, A., Dhawan, K., Koluguri, N. R., and Ginsburg, B. Spectral codes: Spectrogram-based audio codes for high quality speech synthesis. arXiv preprint arXiv:2406.05298, 2024.  
+Lee, S.-g., Ping, W., Ginsburg, B., Catanzaro, B., and Yoon, S. Bigvgan: A universal neural vocoder with large-scale training. In The Eleventh International Conference on Learning Representations, 2022.  
+Lim, J. H. and Ye, J. C. Geometric gan. arXiv preprint arXiv:1705.02894, 2017.
+
+Liu, Q., Wang, W., Jackson, P. J., and Tang, Y. A perceptually-weighted deep neural network for monaural speech enhancement in various background noise conditions. In 2017 25th European Signal Processing Conference (EUSIPCO), pp. 1270-1274. IEEE, 2017.  
+Liu, Z., Mao, H., Wu, C.-Y., Feichtenhofer, C., Darrell, T., and Xie, S. A convnet for the 2020s. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 11976-11986, 2022.  
+Luo, Y. and Yu, J. Music source separation with band-split rnn. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 31:1893-1901, 2023.  
+Luo, Y., Yu, J., Chen, H., Gu, R., and Weng, C. Gull: A generative multifunctional audio codec. arXiv preprint arXiv:2404.04947, 2024.  
+Moore, B. C. and Glasberg, B. R. Suggested formulae for calculating auditory-filter bandwidths and excitation patterns. The journal of the acoustical society of America, 74(3):750-753, 1983.  
+Nagrani, A., Chung, J. S., and Zisserman, A. Voxceleb: a large-scale speaker identification dataset. arXiv preprint arXiv:1706.08612, 2017.  
+Ng, D., Zhang, R., Yip, J. Q., Yang, Z., Ni, J., Zhang, C., Ma, Y., Ni, C., Chng, E. S., and Ma, B. De'hubert: Disentangling noise in a self-supervised model for robust speech recognition. In ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 1-5. IEEE, 2023.  
+Nishimura, Y., Hirose, T., Ohi, M., Nakayama, H., and Inoue, N. Hall-e: hierarchical neural codec language model for minute-long zero-shot text-to-speech synthesis. arXiv preprint arXiv:2410.04380, 2024.  
+Panayotov, V., Chen, G., Povey, D., and Khudanpur, S. Librispeech: an asr corpus based on public domain audio books. In 2015 IEEE international conference on acoustics, speech and signal processing (ICASSP), pp. 5206-5210. IEEE, 2015.  
+Petermann, D., Jang, I., and Kim, M. Native multi-band audio coding within hyper-autocoded reconstruction propagation networks. In ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 1-5. IEEE, 2023.  
+Piczak, K. J. Esc: Dataset for environmental sound classification. In Proceedings of the 23rd ACM international conference on Multimedia, pp. 1015-1018, 2015.  
+Pulkki, V. and Karjalainen, M. Communication acoustics: an introduction to speech, audio and psychoacoustics. John Wiley & Sons, 2015.
+
+Qian, K., Zhang, Y., Gao, H., Ni, J., Lai, C.-I., Cox, D., Hasegawa-Johnson, M., and Chang, S. Contentvec: An improved self-supervised speech representation by disentangling speakers. In International Conference on Machine Learning, pp. 18003-18017. PMLR, 2022.  
+Radford, A., Kim, J. W., Xu, T., Brockman, G., McLeavey, C., and Sutskever, I. Robust speech recognition via largescale weak supervision. In International conference on machine learning, pp. 28492-28518. PMLR, 2023.  
+Richter, J., Wu, Y.-C., Krenn, S., Welker, S., Lay, B., Watanabe, S., Richard, A., and Gerkmann, T. Ears: An anechoic fullband speech dataset benchmarked for speech enhancement and dereverberation. arXiv preprint arXiv:2406.06185, 2024.  
+Rix, A. W., Beerends, J. G., Hollier, M. P., and Hekstra, A. P. Perceptual evaluation of speech quality (pesq)-a new method for speech quality assessment of telephone networks and CODECs. In 2001 IEEE international conference on acoustics, speech, and signal processing. Proceedings (Cat. No. 01CH37221), volume 2, pp. 749-752. IEEE, 2001.  
+Saeki, T., Xin, D., Nakata, W., Koriyama, T., Takamichi, S., and Saruwatari, H. Utmos: Utokyo-sarulab system for voicemos challenge 2022. arXiv preprint arXiv:2204.02152, 2022.  
+San Roman, R., Adi, Y., Deleforge, A., Serizel, R., Synnaeve, G., and Defossez, A. From discrete tokens to high-fidelity audio using multi-band diffusion. Advances in Neural Information Processing Systems, 36, 2024.  
+Santana, I. A. P., Pinhelli, F., Donini, J., Catharin, L., Mangolin, R. B., Feltrim, V. D., Domingues, M. A., et al. Music4all: A new music database and its applications. In 2020 International Conference on Systems, Signals and Image Processing (IWSSIP), pp. 399-404. IEEE, 2020.  
+Shen, K., Ju, Z., Tan, X., Liu, Y., Leng, Y., He, L., Qin, T., Zhao, S., and Bian, J. Naturalspeech 2: Latent diffusion models are natural and zero-shot speech and singing synthesizers, 2023. URL https://arxiv.org/abs/2304.09116.  
+Sturm, B. L. The gtzan dataset: Its contents, its faults, their effects on evaluation, and its future use. arXiv preprint arXiv:1306.1461, 2013.  
+Taal, C. H., Hendriks, R. C., Heusdens, R., and Jensen, J. A short-time objective intelligibility measure for time-frequency weighted noisy speech. In 2010 IEEE international conference on acoustics, speech and signal processing, pp. 4214-4217. IEEE, 2010.
+
+Van Den Oord, A., Vinyals, O., et al. Neural discrete representation learning. Advances in neural information processing systems, 30, 2017.  
+Van der Maaten, L. and Hinton, G. Visualizing data using t-sne. Journal of machine learning research, 9(11), 2008.  
+Variš, D. and Bojar, O. Sequence length is a domain: Length-based overfitting in transformer models. arXiv preprint arXiv:2109.07276, 2021.  
+Wang, C., Chen, S., Wu, Y., Zhang, Z., Zhou, L., Liu, S., Chen, Z., Liu, Y., Wang, H., Li, J., et al. Neural codec language models are zero-shot text to speech synthesizers. arXiv preprint arXiv:2301.02111, 2023.  
+Wu, H., Chung, H.-L., Lin, Y.-C., Wu, Y.-K., Chen, X., Pai, Y.-C., Wang, H.-H., Chang, K.-W., Liu, A. H., and Lee, H.-y. Codec-superb: An in-depth analysis of sound codec models. arXiv preprint arXiv:2402.13071, 2024.  
+Wu, Y.-C., Gebru, I. D., Marković, D., and Richard, A. Audiodec: An open-source streaming high-fidelity neural audio codec. In ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 1-5. IEEE, 2023.  
+Xiao, W., Liu, W., Wang, M., Yang, S., Shi, Y., Kang, Y., Su, D., Shang, S., and Yu, D. Multi-mode neural speech coding based on deep generative networks. In Proc. Interspeech, pp. 819-823, 2023.  
+Yang, D., Liu, S., Huang, R., Tian, J., Weng, C., and Zou, Y. Hifi-codec: Group-residual vector quantization for high fidelity audio codec. arXiv preprint arXiv:2305.02765, 2023.  
+Yip, J. Q., Zhao, S., Ng, D., Chng, E. S., and Ma, B. Towards audio codec-based speech separation. In Proc. Interspeech 2024, pp. 2190-2194, 2024.  
+Yu, W. and Wang, X. Mambaout: Do we really need mamba for vision? arXiv preprint arXiv:2405.07992, 2024.  
+Zeghidour, N., Luebs, A., Omran, A., Skoglund, J., and Tagliasacchi, M. Soundstream: An end-to-end neural audio codec. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 30:495-507, 2022.  
+Zen, H., Dang, V., Clark, R., Zhang, Y., Weiss, R. J., Jia, Y., Chen, Z., and Wu, Y. Libritts: A corpus derived from librispeech for text-to-speech. arXiv preprint arXiv:1904.02882, 2019.  
+Zhang, D., Li, S., Zhang, X., Zhan, J., Wang, P., Zhou, Y., and Qiu, X. Speechgpt: Empowering large language models with intrinsic cross-modal conversational abilities. arXiv preprint arXiv:2305.11000, 2023a.
+
+Zhang, X., Zhang, D., Li, S., Zhou, Y., and Qiu, X. Speechtokenizer: Unified speech tokenizer for speech large language models. arXiv preprint arXiv:2308.16692, 2023b.  
+Zhang, X., Zhang, D., Li, S., Zhou, Y., and Qiu, X. Speechtokenizer: Unified speech tokenizer for speech language models. In The Twelfth International Conference on Learning Representations, 2024.  
+Zhen, K., Lee, M. S., Sung, J., Beack, S., and Kim, M. Psychoacoustic calibration of loss functions for efficient end-to-end neural audio coding. IEEE Signal Processing Letters, 27:2159-2163, 2020.  
+Zhou, K., Zhang, Y., Zhao, S., Wang, H., Pan, Z., Ng, D., Zhang, C., Ni, C., Ma, Y., Nguyen, T. H., et al. Emotional dimension control in language model-based text-to-speech: Spanning a broad spectrum of human emotions. arXiv preprint arXiv:2409.16681, 2024a.  
+Zhou, K., Zhao, S., Ma, Y., Zhang, C., Wang, H., Ng, D., Ni, C., Hieu, N. T., Yip, J. Q., and Ma, B. Phonetic enhanced language modeling for text-to-speech synthesis. In Proc. INTERSPEECH, 2024b.  
+Ziyin, L., Hartwig, T., and Ueda, M. Neural networks fail to learn periodic functions and how to fix it. Advances in Neural Information Processing Systems, 33:1583-1594, 2020.  
+Zwicker, E. and Fastl, H. Psychoacoustics: Facts and models, volume 22. Springer Science & Business Media, 2013.  
+Zwicker, E. and Terhardt, E. Analytical expressions for critical-band rate and critical bandwidth as a function of frequency. The Journal of the Acoustical Society of America, 68(5):1523-1525, 1980.
+
+# A. Psychoacoustic evidence of perceptual speech characteristics
+
+![](images/73f6ffd72c384b397466f3d084eed8febed02d27108329a6778235339cf8f937.jpg)  
+Fig. 1
+
+![](images/373889a322e7ea9db2519aa235489a13f01502080b1e91dc171464ad9d89e301.jpg)  
+Fig. 2
+
+![](images/b9ff4684062446ebe17004baa8f560558180de112e84f16f6625f7ac8ed70cca.jpg)  
+Fig. 3
+
+![](images/5fc16f25f575350945b5c3e9973d74e692cc9a5229b6069731755edebff29ed7.jpg)  
+Fig. 4
+
+![](images/3522139e7a36afb1dc8c4726451719b92747c988996c4fc9193e2325fc1768da.jpg)  
+Fig. 5
+
+![](images/0e2557bc654592792e620944258eff1374d5c6b7a431b7e0796c1ba83825d82a.jpg)  
+Fig. 6  
+Figure 3. The figures have been sourced from French & Steinberg (1947), which discusses how speech sounds are recognized by the ear. The data were collected from microphones based on human speech and then analyzed with computational tools to derive the intensity and sound pressure levels. (1) Comparison of Speech Spectra. (2) Idealized Long Average Speech Spectrum at one meter from lips. (3) R.m.s. pressure of speech at  $30\mathrm{cm}$  from lips. (4) Articulation test with low pass filters. (5) Articulation test with high pass filters. (6) Syllable articulation versus cut-off frequency.
+
+From the presented illustration, it becomes clear that the core components of speech intelligibility are predominantly emphasized within the lower frequency bands where the majority of vocal energy is concentrated. Fig. 1 demonstrates the variation in intensity levels across the frequency spectrum, measured in decibels relative to a standard sound pressure level. This figure contrasts intensity levels at a near-field distance, merely two inches from the speaker's lips, with adjustments made for gender differences in voice power and frequency content. The data reveal a pronounced convex curve with higher intensities noted in the lower frequencies, which gradually decrease as the frequency increases. This pattern supports our initial assertions and is corroborated by Fig. 2 and 3. These figures extend the observations to longer ranges (1 meter) and include measurements of RMS pressure, following the experimental setup described by Dunn & White (1940) for six male subjects.
+
+Then, Fig. 4 and 5 present the results from articulation tests employing low-pass and high-pass filters across varying cutoff frequencies, demonstrating the impact of frequency range restrictions on speech intelligibility. Notably, a reduction in the cutoff frequency through low-pass filters correlates with a decrease in intelligibility, highlighting the critical role of higher frequencies in the recognition of consonants and the differentiation of similar-sounding syllables. In contrast, high frequencies are pivotal for capturing the intricate details and nuances that enhance speech clarity and comprehensibility.
+
+Interestingly, although high-pass filtering up to a specific cutoff frequency can yield improvements in speech intelligibility, performance degrades when lower frequencies are excessively attenuated. This trend underscores the critical role of low-frequency components—primarily conveyed through vowel sounds—in preserving the spectral power and tonal richness of speech. These vowel sounds are fundamental to intelligibility, as they provide essential acoustic energy and rhythmic structure.
+
+Fig. 6 integrates the findings from Fig. 4 and 5, combining the effects of high-pass and low-pass filtering to offer a more
+
+holistic view of how frequency components influence speech intelligibility across various system settings. The intersection of the filter curves in Fig. 6 identifies critical bands, pinpointing a crucial frequency range central to maintaining speech intelligibility. This analysis elucidates that speech articulation predominantly occurs within the mid-frequency bandwidth, providing key insights into the frequency-dependent nature of speech processing.
+
+# B. Validity of MBS-RVQ at latent representation space.
+
+We note the theoretical distinction in performing band splitting at the input space versus applying it to the latent representations of an autoencoder. This raises key considerations regarding the preservation of psychoacoustic properties within the latent representation space. However, we argue that latent representations retain the psychoacoustic properties of speech is supported by the Lipschitz continuity of the encoder.
+
+Formally, if  $f: \mathbb{R}^n \to \mathbb{R}^m$  is Lipschitz continuous with constant  $L$ , then for any two speech signals  $\mathbf{x}_1, \mathbf{x}_2$  (Hager, 1979):
+
+$$
+\left\| f \left(\mathbf {x} _ {1}\right) - f \left(\mathbf {x} _ {2}\right) \right\| \leq L \left\| \mathbf {x} _ {1} - \mathbf {x} _ {2} \right\| \tag {3}
+$$
+
+Psychoacoustic cues such as formant positions, harmonic relationships, and energy distributions in critical frequency bands are primarily reflected in subtle variations of the speech signal's waveform. Lipschitz continuity ensures that these modest yet perceptually crucial differences are neither excessively magnified nor erased when the signal is transformed into the latent domain. In other words, two psychoacoustically similar speech signals cannot become drastically separated in latent space (Arjovsky et al., 2017; Bartlett & Mendelson, 2002). Furthermore, empirical evidence in representation learning supports this notion, demonstrating that neural networks constrained by Lipschitz continuity typically learn more structured representations, within which subtle perceptual attributes remain discernible (Belkin & Niyogi, 2003; Gulrajani et al., 2017).
+
+Consequently, if an autoencoder's encoder maintains Lipschitz continuity, the latent embeddings it generates for speech signals can be expected to closely reflect the psychoacoustic characteristics present in the original waveform. Minor spectral changes perceived by listeners, such as slight shifts in vowels or sibilants, correspond to small changes in latent space, thus helping to preserve the overall psychoacoustic signature of the speech. This argument provides the basis for the notion that, although a strict one-to-one psychoacoustic fidelity is not mathematically guaranteed, in practice, Lipschitz continuity significantly mitigates the risk of losing important auditory details in the encoder's output.
+
+In this work, we introduce the MUFFIN encoder, a novel architecture that is provably Lipschitz continuous. The core design leverages primarily linear components—namely convolutional layers and fully connected (linear) layers—alongside a modified Snake activation function. Convolutional and linear layers are intrinsically linear transformations, ensuring that the overall network adheres to Lipschitz continuity.
+
+For modified snake activation function,  $f(x) = x + \frac{\beta}{\alpha}\sin^2(\alpha x) + \gamma$ , the derivative is presented as:
+
+$$
+f ^ {\prime} (x) = \frac {d}{d x} \left[ x + \frac {\beta}{\alpha} \sin^ {2} (\alpha x) + \gamma \right] = 1 + 2 \beta \sin (\alpha x) \cos (\alpha x) = 1 + \beta \sin (2 \alpha x).
+$$
+
+Since  $\sin (2\alpha x)$  is bounded between  $-1$  and  $1$ , we have
+
+$$
+\left| f ^ {\prime} (x) \right| = | \beta \sin (2 \alpha x) | \leq | \beta |.
+$$
+
+A function whose derivative is bounded by  $L$  is  $L$ -Lipschitz. Therefore,
+
+$$
+\left| f ^ {\prime} (x) \right| \leq | \beta | \quad \Longrightarrow \quad f \text {i s} | \beta | - L i p s c h i t z.
+$$
+
+Hence, our modified snake activation function is Lipschitz continuous with a Lipschitz constant  $|\beta|$ . This property ensures stability in our model, as the layer computations with the activation functions preserve the distances between input features.
+
+# C. On the effect of MBS-RVQ compared to vanilla RVQ.
+
+In this section, we conduct an ablation study by comparing the performance with disabling multi-band spectral residual vector quantization, reverting to vanilla residual vector quantization, to assess the contribution of MBS-RVQ in enhancing
+
+generative quality under the constraint imposed by perceptual entropy. In addition to the results presented in Table 5, which analyze the individual contributions of each codebook and the full MBS-RVQ quantizer configuration with respect to word error rate and short-time objective intelligibility, we further report detailed reconstruction performance of the neural psychoacoustic codec. The following tables present reconstruction quality evaluated using the same metrics employed in the main results. Complementary to the table-based evaluation, we also provide audio samples on our demo page to audibly highlight the differences between MBS-RVQ and vanilla RVQ.
+
+Table 7. The table presents the objective evaluation of the reconstructed speech from listed evaluation set using full codebook quantizers of MBS-RVQ versus vanilla RVQ.  
+
+<table><tr><td colspan="7">Test-Clean (LibriTTS)</td></tr><tr><td>Model</td><td>STFT</td><td>MEL</td><td>PESQ</td><td>STOI</td><td>UTMOS</td><td>ViSQOL</td></tr><tr><td>MUFFIN</td><td>1.555</td><td>0.692</td><td>2.996</td><td>0.954</td><td>4.017</td><td>4.516</td></tr><tr><td>Vanilla-RVQ</td><td>1.627</td><td>0.768</td><td>2.856</td><td>0.940</td><td>3.875</td><td>4.328</td></tr><tr><td>MUFFIN (12.5 Hz)</td><td>1.663</td><td>0.807</td><td>2.360</td><td>0.932</td><td>4.074</td><td>4.225</td></tr><tr><td>Vanilla-RVQ (12.5 Hz)</td><td>1.755</td><td>0.879</td><td>2.260</td><td>0.924</td><td>3.785</td><td>4.017</td></tr><tr><td colspan="7">Test-Other (LibriTTS)</td></tr><tr><td>MUFFIN</td><td>1.615</td><td>0.758</td><td>2.658</td><td>0.934</td><td>3.444</td><td>4.454</td></tr><tr><td>RVQ</td><td>1.683</td><td>0.810</td><td>2.544</td><td>0.917</td><td>3.318</td><td>4.268</td></tr><tr><td>MUFFIN (12.5 Hz)</td><td>1.725</td><td>0.875</td><td>2.086</td><td>0.904</td><td>3.560</td><td>4.129</td></tr><tr><td>RVQ (12.5 Hz)</td><td>1.863</td><td>0.963</td><td>1.940</td><td>0.815</td><td>3.399</td><td>3.993</td></tr><tr><td colspan="7">IEMOCAP</td></tr><tr><td>MUFFIN</td><td>1.399</td><td>0.675</td><td>2.178</td><td>0.806</td><td>1.903</td><td>4.000</td></tr><tr><td>RVQ</td><td>1.510</td><td>0.793</td><td>2.039</td><td>0.715</td><td>1.805</td><td>3.883</td></tr><tr><td>MUFFIN (12.5 Hz)</td><td>1.429</td><td>0.754</td><td>1.726</td><td>0.723</td><td>2.026</td><td>3.612</td></tr><tr><td>RVQ (12.5 Hz)</td><td>1.584</td><td>0.835</td><td>1.644</td><td>0.645</td><td>1.917</td><td>3.455</td></tr></table>
+
+# D. Periodic Activation Function - Modified Snake Activation Function
+
+In the course of modeling, Figure 4 depicts four distinct data scenarios, each presenting challenging frequency details where periodic patterns appear odd and highly abrupt, complicating the modelling. It is observed that the standard activation function inadequately models regions exhibiting high-frequency patterns, likely due to trade-offs involving amplitude preservation. Although the introduction of the amplitude,  $\beta$  parameter, partially mitigates this issue, it simultaneously introduces regions of elevated variance, which compromise overall stability. In contrast, we observe that incorporating a bias term  $\gamma$  provides a more robust solution by effectively stabilizing the model and reducing the variance associated with overestimating or underestimating outcomes.
+
+In addition to the figure above, we present an ablation study examining the impact of each additive term on reconstruction quality in Table 8. These results are evaluated using the same metrics as those employed in the main table and the table elucidates the distinct contributions of each additive term to the reconstruction quality of the codec (i.e., the component with amplitude and bias).
+
+Table 8. The table presents the objective evaluation of the reconstructed speech from Test-Clean (LibriTTS) with additive term of the modifications to the vanilla snake activation.  
+
+<table><tr><td>Model (MUFFIN)</td><td>STFT</td><td>MEL</td><td>PESQ</td><td>STOI</td><td>UTMOS</td><td>ViSQOL</td></tr><tr><td>Added amplitude &amp; bias (Ours)</td><td>1.555</td><td>0.692</td><td>2.996</td><td>0.954</td><td>4.017</td><td>4.516</td></tr><tr><td>Added amplitude</td><td>1.603</td><td>0.744</td><td>2.928</td><td>0.945</td><td>3.943</td><td>4.448</td></tr><tr><td>Vanilla</td><td>1.635</td><td>0.760</td><td>2.876</td><td>0.940</td><td>3.905</td><td>4.409</td></tr></table>
+
+![](images/7205685178e6a6d9371d8e651f1eeefd9f2cc6c4532bd135964e56a06f77d531.jpg)
+
+![](images/9a8798a6837bd84643435bb47f0cc6153fdcf1617ba93d5a0494ad18146c9441.jpg)
+
+![](images/f4687afcc72441319b4376abf9e7bfa2cbf9627eafc5d8c3acdca10b2c7fd3f3.jpg)
+
+![](images/9c98bba2058c84f8c6bb6d01dc68d8d7726eea73a00242fe75d1e68f64853c5e.jpg)  
+Data Modelling (Application)
+
+![](images/df3aa7183e9e9ebf525c8987e854e404fc5e90bef0a08b6530c8d701bdbeb37b.jpg)
+
+# Modified Snake(x) Activation
+
+$$
+\begin{array}{l} x + \frac {1}{\alpha} \sin^ {2} (\alpha x) \\ x + \frac {\beta}{\alpha} \sin^ {2} (\alpha x) \\ x + \frac {\beta}{\alpha} \sin^ {2} (\alpha x) + \gamma \\ \bullet \quad \text {G r o u n d T r u t h (d a t a)} \\ \end{array}
+$$
+
+![](images/afa25260917b4617458e91b3607a43e235333e4e2756edd1a1185272fa788a81.jpg)  
+Figure 4. Illustration of our proposed modifications to the vanilla snake activation and its behavior in actual modeling for different sequential data.
+
+![](images/40c4e3e5e791bef022239f2a10d29dea73949ffade8670664e8d8af5bfdd1c30.jpg)
+
+# E. Illustrations of the auditory feature across various codebook representations.
+
+In this section, we conduct a comprehensive examination of the perceptual characteristics inherent to each distinct quantized frequency band, as revealed by the learned codebooks without target supervision. We note that the cutoff for the frequency band split follows the relative logarithmic scale of the latent sampling rate, according to the psychoacoustic studies. Our analysis utilizes multiple visualization tools to elucidate these characteristics, which include t-SNE plots, randomly sampled spectrograms, and elbow plot.
+
+These visualizations are derived from the test-clean set of LibriSpeech and a subset of a popular speaker recognition VoxCeleb dataset, allowing us to explore the specific semantic and speaker traits captured within the learned representations. The combined use of these diverse methodologies not only underscores the discriminative power of the representations but also enhances our understanding of their underlying structure and variability. However, it is important to note that the latent average-pooled representations of each sampled utterance were not specifically trained for the speaker recognition (classification) task. Consequently, the optimization objectives did not aim to achieve highly deterministic speaker vectors but rather to encode sequential acoustic and semantic content. This may inherently limits the zero-shot performance of the system on deriving the vector representations for each utterance.
+
+In Figure 5, we justify that among all frequency bands, those in the high-frequency range of  $37.5 - 75\mathrm{Hz}$  distinctly demarcate speaker boundaries with minimal overlap, indicating that codebook 3 naturally disentangled speech information to quantize speaker information without target supervision. This observation is further substantiated by the distances between clusters in the t-SNE plot, where these high-frequency representations are the furthest apart based on the coordinate axis compared to those from the low frequency range of  $0 - 18.75\mathrm{Hz}$  (codebook 1), mid frequency range of  $18.75 - 37.5\mathrm{Hz}$  (codebook 2), and the residuals (codebook 4).
+
+![](images/97634f16be15d5fffb444a2966311c9bf9e5d8a7aadaf111439c1e3de7392d41.jpg)
+
+![](images/2858ca6a40d21f3ad836226d1a713a7828d61a7e55fbb0b8854a193141004616.jpg)
+
+![](images/7dedac51b0d121c1d539c74d5b16fa8a070e14c2c49f5475ee1d4a9e328f1632.jpg)  
+Codebook 3
+
+![](images/e405466e4bb929debd5c85ed1772079872fb5225b1c19bedf40356618465d3b2.jpg)  
+Codebook 4  
+Figure 5. A t-SNE plot showcasing each codebook, with speech randomly sampled from VoxCeleb, effectively represents six distinct speakers of the color code.
+
+Furthermore, we stress that the reconstructed audio from the demos presented in Section (F) offers compelling evidence that Codebook 3 effectively quantizes speaker attributes from the phonetic content. This is demonstrated by the permutation of the reconstructed audio from different codebooks excluding Codebook 3, is bound to a uniformly flat pitch. Such audio remains invariant to both gender and distinct speaker characteristics, underscoring the unique role of Codebook 3 in capturing these nuances. This resulting outcome can also be used potentially to obtain data samples for normalizing speaker speech, serving the pre-training of applications such as ContentVec (Qian et al., 2022).
+
+Likewise, in Figure 6, we present the spectrogram of a randomly sampled speech utterance from the LibriSpeech dataset, alongside its decomposition into different codebooks. This reconstruction incrementally utilizes quantized codebooks. The spectrogram clearly illustrates the previously mentioned flat pitch contours, showing no significant variation from the original waveform when Codebook 3 is omitted from the reconstruction. This effect is evident when only Codebooks 1 and 2 are utilized.
+
+From the spectrogram, we also observe that the frequency of formants is emphasized when combining Codebooks 1 and 2, which reveals clearer phonetic articulation patterns distinguishing vowels and consonants. This observation allows us to appreciate that Codebook 2 is responsible for quantizing the articulation and respiratory paralanguage (formant) of the speech.
+
+Next, we plot an elbow curve of the Word Error Rate (WER) as codebooks are added incrementally in Figure 7, illustrating the contribution of each codebook to preserving speech content. A noticeable decrease in the error rate indicates enhanced clarity and intelligibility of speech, attributed to improved articulation. We posit that Codebook 3 does not focus on
+
+"BUT NOW HERE IS A SUBJECT OF WHICH YOU WILL WONDER AT FIRST WHY TURNER DREW IT AT ALL"
+
+![](images/afc990f22c5e38d2956a1a16bf21221a40d2ce706ffe7bb0fb3ba642c7d97f56.jpg)  
+Original waveform
+
+![](images/0b6b87ae41ef1ec4f54c33eee792d9787486961ce9ce141e1dee977e52560b2a.jpg)  
+Codebook 1
+
+![](images/b11111c722e644c5050bedce2c1f62ea1095d805588f6900eb10a85f1b98a3b2.jpg)  
+Codebook 1 + 2
+
+![](images/3a6dc10571f43f0857a4314783ec2b2d28647554cb44142358327fdf7388f9ac.jpg)  
+Codebook 1 + 2 + 3
+
+![](images/a879a74491f4baaadb49fdb2425a9cb4ef0008547e3acea3ab03aaf4f9071b12.jpg)  
+Figure 6. An illustration depicts a randomly sampled speech utterance alongside its reconstruction using incremental codebooks.  
+Figure 7. The elbow plot of the word error rate from whisper-large model, utilizing the same setup of incremental codebooks.
+
+contextual speech content; using Codebook 3 alone results in high recognition errors, suggesting its limited contribution to
+
+core speech intelligibility, while the quantized information in codebook 4 is simply error residual. This analysis, along with auditory demos in Section F, further supports the assertion that Codebooks 1 and 2 are primarily responsible for enhancing speech intelligibility and articulation details.
+
+Consequently, our neural psychoacoustic coding with MUFFIN offers a novel perspective that facilitates the natural disentanglement of speech attributes, guided by label-free psychoacoustic studies. This uniquely positions our approach as an innovative alternative to FACodec (Ju et al., 2024), achieving similar goals of obtaining factorized information through a considerably simpler optimization process that could spur further investigation to the advancement of low-resource factorized speech representation learning.
+
+While psychoacoustic studies have primarily focused on speech, applying similar analysis to non-stationary or transient sounds, such as those in music, is both important and intriguing. To explore this, we extended our decomposition approach to a variety of musical genres, including singing, classical, jazz, and symphonic music. Consistent with the psychoacoustic framework used in speech analysis, we observed that:
+
+- Codebook 1: primarily captures vocal content and coarse rhythmic beats.  
+- Codebook 2: emphasizes vocal clarity and mid-frequency information.  
+Codebook 3: encodes pitch details reflective of the singer's unique characteristics.
+
+The above characteristics are demonstrated with samples of music audio presented in our demo page. Interestingly, instrumental content does not clearly separate across Codebooks 2 and 3, suggesting that our psychoacoustic-guided representation is particularly effective in disentangling vocal attributes (speech and singing), but less so for purely instrumental channels. This finding reinforces the theoretical value of psychoacoustic principles for modeling vocal properties, an area that remains underexplored in neural codes. While applying this framework to instrumental music remains challenging, we believe this opens new research directions. Further investigations, beyond the scope of the current study, may be investigated in our future work.
+
+# F. Discussion on zero-shot text-to-speech synthesis.
+
+# F.1. Training details
+
+We evaluate the performance of MUFFIN within the VALL-E framework (Wang et al., 2023). In contrast to the original study, we train the model with fewer than 600 hours of data. Additionally, during inference, we select a random speech segment from the same speaker to use as the prompt, rather than using the first 1-3 seconds of the speech to be synthesized, as the original study did. This latter approach typically provides more consistent speaker information and constitutes an easier task. These modifications likely account for our lower performance compared to the results reported in the original paper.
+
+# F.2. Discussion
+
+During our experiments, we observed that the open-sourced VAL-E configuration did not integrate effectively with MUFFIN operating at a  $12.5\mathrm{Hz}$  sampling rate. This discrepancy highlighted the sensitivity of prompt sequence length in learning sequential decoding information. Longer prompts tended to simplify the task, leading the model to undergeneralize, while shorter prompts provided insufficient information, causing the model to collapse prematurely. Balancing the length of prompts is crucial, particularly as each frame now encapsulates more complex information due to high compression. Consequently, noise in the output sequence reduced speech clarity. Despite these issues, the naturalness of the synthesized speech was reasonably good, demonstrating the benefits of using tokenized audio units. These units separate speech intelligibility from speaker information into distinct, independent codebooks, which are unaffected by adverse conditional computations. This observation emphasizes the importance of the speech codec's sampling rate in text-to-speech (TTS) systems, directly influencing the quality and intelligibility of synthesized speech. Future work will further investigate this trade-off through systematic experimentation to find optimal configurations that balance resource efficiency with high-quality speech synthesis.
+
+Furthermore, it would also be highly beneficial to consider weakly supervised training for each psychoacoustic codebook by providing targeted labels in small volumes. Now that we have a clearer understanding of the purpose of each codebook,
+
+it becomes more intuitive to apply appropriate supervision with minimal effort, optimizing the quality of the embedding networks while reducing the cost of collecting extensive labeled data for the disentanglement of speech information on factorizing attribute codebook. We anticipate that this will further enhance performance on TTS (Zhou et al., 2024a) or codec-based speech separation (Yip et al., 2024) downstream tasks.
+
+# G. Details of the hyperparameters and specifications of NACs at 24 kHz Sampling Audio
+
+Note that the MACs (associated with real-time latency) is computed based on a 1-second audio waveform sampled at 24 kHz, using the tool available at https://github.com/sovrasov/flops-counter.pytorch/tree/master. The table below demonstrates that MUFFIN achieves significantly lower MACs, particularly as compression increases with higher downsampling rates, compared to existing CODECs. This indicates that MUFFIN offers a lower latency rate than other CODECs, including the baseline HiFi-Codec, thereby supporting improved real-time applications.
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">Num. of Params (M)</td><td colspan="2">MACs (G)</td><td rowspan="2">Encoding Rate</td><td rowspan="2">Downsampling Rate</td><td rowspan="2">Frame Rate (Hz)</td><td rowspan="2">Bandwidth (kB/s)</td><td rowspan="2">Token/s</td></tr><tr><td>Encoder</td><td>Decoder</td><td>Total</td><td>Encoder</td><td>Decoder</td></tr><tr><td>MUFFIN</td><td>34.2</td><td>11.9</td><td>46.1</td><td>14.7</td><td>16.9</td><td>(2, 4, 5, 8)</td><td>320</td><td>75</td><td>2.7</td><td>300</td></tr><tr><td>MUFFIN ▽</td><td>34.3</td><td>11.9</td><td>46.2</td><td>5.85</td><td>8.9</td><td>(4, 5, 6, 8)</td><td>960</td><td>25</td><td>1.35</td><td>150</td></tr><tr><td>MUFFIN ▲</td><td>36.5</td><td>14.1</td><td>50.6</td><td>6.88</td><td>11.1</td><td>(3, 5, 8, 16)</td><td>1920</td><td>12.5</td><td>0.9</td><td>100</td></tr><tr><td>Encodec</td><td>7.43</td><td>7.43</td><td>14.9</td><td>1.51</td><td>4.10</td><td>(2, 4, 5, 8)</td><td>320</td><td>75</td><td>3.0</td><td>300</td></tr><tr><td>DAC</td><td>21.5</td><td>52.3</td><td>73.8</td><td>18.4</td><td>64.9</td><td>(2, 4, 5, 8)</td><td>320</td><td>75</td><td>3.0</td><td>300</td></tr><tr><td>Hifi-Codec</td><td>47.2</td><td>14.3</td><td>61.5</td><td>20.6</td><td>23.8</td><td>(2, 4, 5, 8)</td><td>320</td><td>75</td><td>3.0</td><td>300</td></tr></table>

2025/Multi-band Frequency Reconstruction for Neural Psychoacoustic Coding/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:05a5e54f1b8f0ebcffbcced967f892b45cd2905d8d06c3b4822606719eb2b056
+size 966317

2025/Multi-objective Linear Reinforcement Learning with Lexicographic Rewards/66599d63-c75d-4e63-a2bb-e5bb4f0dd0cd_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a88fec003e8a5fa7d705866e7015ec252ab03fba85da5b566c0261294b724e2d
+size 472588

2025/Multi-objective Linear Reinforcement Learning with Lexicographic Rewards/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:dbde9c263f8144ec7f39f3443ad0e3aafeeeb6955bbbd0a832dd751076230cca
+size 1109683

2025/MultiPDENet_ PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation/fc6a1b12-5b25-4f65-b6e6-903ab0a4ef17_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:19c9783ed9d5087ee89ab02d9d9f3fb8160285093622c0e48f035915d57b5410
+size 11643511

2025/MultiPDENet_ PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation/full.md

@@ -0,0 +1,590 @@
+# MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation
+
+Qi Wang $^{1}$  Yuan Mi $^{1}$  Haoyun Wang $^{1}$  Yi Zhang $^{2}$  Ruizhi Chengze $^{2}$  Hongsheng Liu $^{2}$  Ji-Rong Wen $^{1}$  Hao Sun $^{1}$
+
+# Abstract
+
+Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but struggle with weak generalizability, interpretability, and data dependency, as well as suffer in long-term prediction. To this end, we propose a PDE-embedded network with multiscale time stepping (MultiPDENet), which fuses the scheme of numerical methods and machine learning, for accelerated simulation of flows. In particular, we design a convolutional filter based on the structure of finite difference stencils with a small number of parameters to optimize, which estimates the equivalent form of spatial derivative on a coarse grid to minimize the equation's residual. A Physics Block with a 4th-order Runge-Kutta integrator at the fine time scale is established that embeds the structure of PDEs to guide the prediction. To alleviate the curse of temporal error accumulation in long-term prediction, we introduce a multiscale time integration approach, where a neural network is used to correct the prediction error at a coarse time scale. Experiments across various PDE systems, including the Navier-Stokes equations, demonstrate that MultiPDENet can accurately predict long-term spatiotemporal dynamics, even given small and incomplete training data, e.g., spatiotemporally down-sampled datasets. MultiPDENet achieves the state-of-the-art performance compared with other neural baseline models, also with clear speedup compared to classical numerical methods.
+
+# 1. Introduction
+
+Complex spatiotemporal dynamical systems, e.g., climate system (Schneider et al., 2017) and fluid dynamics (Ferziger et al., 2019), are fundamentally governed by partial differential equations (PDEs). To capture the intricate behaviors of these systems, various numerical methods have been developed. Direct Numerical Simulation (DNS) is a widely used method for solving PDEs. It requires specifying initial conditions (ICs), boundary conditions (BCs), and PDE parameters, followed by discretizing the equations on a grid using techniques like finite difference (FD), finite element (FE), finite volume (FV), or spectral methods. Despite their accuracy, traditional numerical methods face key challenges of high computational costs (Goc et al., 2021), when addressing with high-dimensional problems or necessitating fine spatial and temporal resolutions.
+
+Recent advances in deep learning have introduced neural-based approaches (Lu et al., 2021; Li et al., 2021; Gupta & Brandstetter, 2023) for solving PDEs. These data-driven methods eliminate the need for explicit theoretical formulations, enabling networks to learn underlying patterns directly from data through end-to-end training. While promising, these approaches face notable challenges, including a heavy dependence on large training datasets and limited generalization. For instance, achieving accurate predictions becomes particularly challenging when models encounter unseen ICs or scenarios beyond the training distribution.
+
+A notable progress lies in the paradigm of Physics-informed neural networks (PINNs) (Raissi et al., 2019), which incorporates physical prior knowledge (such as PDE residuals and I/BCs) as constraints within the loss function. This approach allows the network to fit the data while simultaneously maintaining a certain degree of physical consistency. Variants of PINNs (Raissi et al., 2020; Wang et al., 2020; Eshkofti & Hosseini, 2023) have shown notable success across various domains, reducing the dependency on extensive datasets to some degree. However, such methods still face scalability and generalizability challenges when applied to complex nonlinear dynamical systems. Additionally, optimizing complex loss functions (Rathore et al., 2024) and ensuring model interpretability remain challenges.
+
+A series of approaches have been proposed to integrate physics into neural networks (NNs) to overcome the above challenges. For instance, PeRCNN (Rao et al., 2022; 2023), which uses feature map multiplication to construct polynomial combinations for approximating the underlying PDEs, can capture the latent spatiotemporal dynamics even with low-resolution, noisy, and coarse data, demonstrating strong generalizability. Nevertheless, this method suffers from error accumulation, degrading its performance in long-term predictions. Another approach (Kochkov et al., 2021; Sun et al., 2023), combining NNs with numerical methods, aims to accelerate the simulation process on coarse grids. These hybrid methods leverage traditional solvers for stability and NNs for accuracy. However, they often rely heavily on NN capabilities and often requires large amounts of data.
+
+To overcome these limitations, we propose MultiPDENet, a PDE-embedded network that incorporates multiscale timestepping (as shown in Figure 1), to efficiently simulate spatiotemporal dynamics, e.g., turbulent fluid flows, on coarse spatial and temporal grids with limited data. Notably, it integrates a trainable neural solver for precise predictions at micro time scales, while employing a NN to correct errors at macro time steps. Additionally, by embedding PDEs, MultiPDENet offers enhanced generalizability. The primary contributions of this work are summarized as follows:
+
+- We developed MultiPDENet, a PDE-embedded network with multiscale time-stepping, for accelerated flow simulations on spatiotemporal coarse grids. By integrating neural solver with PDEs, MultiPDENet achieves great generalizability and efficiency.  
+- Leveraging the structure of FD stencils, we introduced a symmetric convolutional filter that approximates the equivalent form of derivatives on coarse grids, aiming to reduce the residual error of the governing PDEs.  
+- Experimental results across various datasets, covering 1D and 2D equations (e.g., reaction-diffusion processes and turbulent flows), demonstrate the effectiveness of MultiPDENet in accelerating long-term simulations.
+
+# 2. Related Work
+
+Related works on numerical, machine learning, physics-inspired learning, and hybrid learning methods for simulation of PDE systems are given in Appendix A.
+
+# 3. Methodology
+
+# 3.1. Problem Description
+
+Let's consider a general spatiotemporal dynamical system governed by the following PDE:
+
+$$
+\mathbf {u} _ {t} = \mathcal {F} (\mathbf {u}, \mathbf {u} ^ {2}, \dots , \nabla \mathbf {u}, \Delta \mathbf {u}, \dots ; \boldsymbol {\lambda}) + \mathbf {f}, \tag {1}
+$$
+
+where  $\mathbf{u}(\mathbf{x},t)\in \mathbb{R}^n$  denotes the physical state in the spatiotemporal domain  $\Omega \times [0,T]$ ;  $\mathbf{u}_t$  the first-order time derivative term;  $\mathcal{F}(\cdot)$  a linear/nonlinear functional parameterized by PDE parameters  $\lambda$  (e.g., the Reynolds number  $Re$ );  $\pmb{\nabla}$  the Nabla operator is defined as  $[\partial x,\partial y,\dots ]^{\mathrm{T}}$ ; and  $\mathbf{f}$  the source term. Additionally, we define  $\mathcal{I}(\mathbf{u},\mathbf{u}_t;\mathbf{x}\in \Omega ,t = 0) = \mathbf{0}$  and  $\mathtt{B}(\mathbf{u},\nabla \mathbf{u},\dots ;\mathbf{x}\in \partial \Omega) = \mathbf{0}$  specified ICs and BCs, where  $\partial \Omega$  represents the domain boundary.
+
+We aim to accelerate the simulation of fluid flows by using a PDE-embedded network with multiscale time stepping based on a limited training data (coarse in both spatial and temporal scales). The model is capable of rapid simulation, achieving high solution accuracy while demonstrating strong generalizability across varying ICs, source terms, complex domains, and PDE parameters.
+
+# 3.2. Model Architecture
+
+In this section, we introduce MultiPDENet and show how our model efficiently captures the underlying spatiotemporal dynamics. As illustrated in Figure 1(a), predicting  $\mathbf{u}_{k + 1}$  from the input  $\mathbf{u}_k$  involves two main components: the Physics Block and the  $\mathrm{M_aNN}$  Block.
+
+# 3.2.1. MULTI-SCALE FORWARD TIME STEPPING SCHEME
+
+While the learnable neural solver can be used independently, its accuracy for long-term prediction is limited due to error accumulation. To address this issue, we introduce a multiscale time stepping scheme, incorporating micro-scale and macro-scale steps, to improve predictive accuracy and enables fast prediction of PDE solutions on coarse spatiotemporal grids. Specifically, we define two types of time stepping: micro-scale step and macro-scale step, to enhance the performance of spatiotemporal dynamics prediction. At the macro scale, given the coarse solution  $\mathbf{u}_k$  at time  $t_k$ , MultiPDENet is expected to predict the next-step solution  $\mathbf{u}_{k + 1}$  at  $t_{k + 1}$ , which can be expressed as:
+
+$$
+\mathbf {u} _ {k + 1} = \mathbf {u} _ {k} + \sum_ {m = 1} ^ {M} \delta \bar {\mathbf {u}} _ {m} ^ {k} + \mathrm {M} _ {\mathrm {a}} \mathrm {N N} (\mathbf {u} _ {k}, \Delta t, d x), \tag {2}
+$$
+
+where  $\Delta t$  is the macro-scale time interval, and  $dx$  the spatial resolution of mesh grid. Here,  $\delta \bar{\mathbf{u}}_m^k$  is the incremental update by the Physics Block (see Section 3.2.2) at each micro step, as shown in Eq. (3), where  $M$  denotes the number of micro-scale time steps in one macro-scale step (e.g.,  $M = 4$  herein). The  $\mathrm{M_aNN}$  Block (see Section 3.2.4) refines these incremental updates generated by the Physics Block on coarse grids, yielding the final update for the macro step.
+
+# 3.2.2. PHYSICS BLOCK: LEARNABLE NEURAL SOLVER
+
+To accurately predict at the micro-scale step, we developed a neural solver, referred to as the Physics Block, as illustrated
+
+![](images/ee32be079634b55498c3da656f9835a417bb12e87193d77772bad409c2fa4da7.jpg)
+
+![](images/e2e65ec6874df9ad0e1d32c7dad296a25f850d8f33c8d40fca7c932a70f9f948.jpg)
+
+![](images/f3e5398a816d6bb0d6c158cfbdecec5617f4d49f41f113a4ffdf74ba4abc5cd3.jpg)  
+Figure 1. Schematic of MultiPDENet for learning turbulent flows. (a), Model architecture. (b), Physics Block. (c), Learnable PDE block.
+
+in Figure 1(b). This solver is designed to ensure the stability (Hoffman & Frankel, 2018), accuracy, and efficiency of its predictions by adhering to the Courant-Friedrichs-Lewy (CFL) conditions (LeVeque, 2007). The Physics Block comprises three main components: the Poisson Block, the PDE Block, and the  $\mathbf{M}_{\mathrm{i}}\mathbf{N}\mathbf{N}$  Block. The solution update for each micro-scale time step can be describe as  $\bar{\mathbf{u}}_{m + 1}^{k} = \bar{\mathbf{u}}_{m}^{k} + \delta \bar{\mathbf{u}}_{m}^{k}$ , where
+
+$$
+\begin{array}{l} \delta \bar {\mathbf {u}} _ {m} ^ {k} = \int_ {t _ {k} + (m - 1) \delta t} ^ {t _ {k} + m \delta t} \left[ \mathcal {B} (\tilde {\mathbf {u}} (\tau), \boldsymbol {\nabla} \tilde {\mathbf {u}} (\tau), \dots ; \boldsymbol {\lambda}) + \mathbf {f} (\tau) \right] d \tau \\ + \mathbf {M} _ {\mathrm {i}} \mathbf {N N} \left(\bar {\mathbf {u}} _ {m} ^ {k}, \Xi_ {m} ^ {k} \left(p, \hat {\nabla} \hat {\mathbf {u}}, \hat {\nabla} ^ {2} \hat {\mathbf {u}}, \hat {\nabla} p, \mathbf {f}, R e\right)\right). (3) \\ \end{array}
+$$
+
+Here,  $\bar{\mathbf{u}}_m^k$  represents the intermediate state at  $m$ -th microscale step initialized at time  $t_k$  (note that  $\bar{\mathbf{u}}_1^k = \mathbf{u}_k$ ). We denote  $\tilde{\mathbf{u}}(\tau) \triangleq \mathbf{u}(\tilde{\mathbf{x}}, \tau)$ , where  $\tilde{\mathbf{x}}$  depicts the coordinates of coarse grid. Moreover,  $\lambda$  can be set as trainable if unknown.  $\mathcal{B}$  represents the PDE block, used for approximating  $\mathcal{F}$  in Eq. (1). To keep the accuracy and ensure the stability, the PDE block is designed based on the RK4 integrator (see Appendix B.3) and consists of the Correction Block and a trainable filter bank. Since the considered micro-scale time interval is relatively large, the  $\mathrm{M_iNN}$  block is used as a corrector to refine the solution. More details can be found in Appendix B. In fact, the Physics Block can be used for prediction independently (e.g., the quantitative results for the NSE dataset predictions using purely the Physics Block are presented in Table 3, labeled Model C).
+
+PDE Block. The PDE block computes the residual of the governing PDEs. It incorporates a learnable filter bank with
+
+symmetry constraints, which calculates derivative terms based on the corrected solution produced by a Correction Block. These terms are then combined into the governing PDEs, a learnable form of  $\mathcal{F}$  in Eq. (1). This process is incorporated into the RK4 integrator (see Appendix B.3) for solution update which can be expressed as
+
+$$
+\begin{array}{l} \mathcal {F} \left(\bar {\mathbf {u}} _ {m} ^ {k}, \dots , \hat {\nabla} \hat {\bar {\mathbf {u}}} _ {m} ^ {k}, \hat {\nabla} ^ {2} \hat {\bar {\mathbf {u}}} _ {m} ^ {k}, \dots ; \boldsymbol {\lambda}\right) \tag {4} \\ \leftarrow^ {a p p r o x.} \mathcal {B} \left(\bar {\mathbf {u}} _ {m} ^ {k}, \dots , \nabla \bar {\mathbf {u}} _ {m} ^ {k}, \nabla^ {2} \bar {\mathbf {u}} _ {m} ^ {k}, \dots ; \boldsymbol {\lambda}\right), \\ \end{array}
+$$
+
+where  $\mathcal{B}$  denotes the PDE block, and  $\bar{\mathbf{u}}_m^k$  the coarse solution (aka, solution on coarse grids) at micro-scale time  $t_k + m\delta t$ . Here,  $\hat{\bar{\mathbf{u}}}_m^k$  refers to the neural-corrected state of the coarse solution, which is obtained through the Correction Block (see Appendix B.1 for details). This corrected state  $\hat{\bar{\mathbf{u}}}_m^k$  is used to estimate spatial derivatives, namely,  $\hat{\bar{\mathbf{u}}}_m^k = \mathrm{NN}(\bar{\mathbf{u}}_m^k)$ . Note that  $\hat{\nabla}$  and  $\hat{\nabla}^2$  represent trainable Nabla and Laplace operators, respectively, each consisting of a symmetrically constrained convolution filter, e.g., an enhanced FD kernel to approximate spatial equivalent derivatives. By utilizing the RK4 integrator, we can project the coarse solution to the subsequent micro-scale time step. Despite the reduced resolution causing some information loss, this learnable PDE block enables a closer approximation of the equivalent form of the derivatives on coarse grids. This addition serves as a fully interpretable "white box" element within the overall network structure.
+
+Poisson Block. In solving incompressible NSE, the pressure term,  $p$ , is obtained by solving an associated Poisson equation. To compute the pressure field, we imple
+
+Table 1. Overview of datasets and training configurations. Note that “ $\rightarrow$ ” denotes the downsampling process from the high resolution (simulation) to the low resolution (training and testing).  
+
+<table><tr><td>Dataset</td><td>Numerical Method</td><td>Spatial Grid</td><td>Time Steps (Temporal Grid)</td><td># of Training Trajectories</td><td># of Testing Trajectories</td><td>Macro-step Rollout</td><td>Micro-step Rollout</td></tr><tr><td>KdV</td><td>Spectral</td><td>256 → 64</td><td>10000 → 2000</td><td>3</td><td>10</td><td>10</td><td>4</td></tr><tr><td>Burgers</td><td>FD</td><td>1002 → 252</td><td>2000 → 200</td><td>5</td><td>10</td><td>10</td><td>4</td></tr><tr><td>GS</td><td>FD</td><td>1282 → 322</td><td>4000 → 200</td><td>3</td><td>10</td><td>1</td><td>4</td></tr><tr><td>NSE</td><td>FV</td><td>20482 → 642</td><td>153600 → 1200</td><td>5</td><td>10</td><td>1</td><td>4</td></tr></table>
+
+mented a pressure-solving module shown in Figure S1(a), which solves the Poisson equation,  $\Delta p = \psi (\mathbf{u})$ , where  $\psi (\mathbf{u}) = 2(u_xv_y - u_yv_x)$  for 2D problems (the subscripts indicate the spatial derivatives along  $x$  or  $y$  directions). To compute the pressure, we employ a spectral method (Poisson solver) based on  $\psi (\bar{\mathbf{u}}_m^k)$  to calculate  $\bar{p}_m^k$ . As shown in Figure S1(b), this approach dynamically estimates the pressure field from the velocity inputs, removing the need for labeled pressure data.
+
+# 3.2.3. ADAPTIVE FILTER WITH CONSTRAINT
+
+Traditional FD methods often yield inaccurate derivatives on coarse grids. To address this, we propose a learnable filter with constraints that approximates equivalent derivatives on coarse grids, minimizing the PDE residuals during training and thereby improving the model's predictive accuracy. By leveraging the symmetry of central difference stencils, our filter maintains structural integrity while enhancing network flexibility. As shown in Figure 2, we construct two  $5 \times 5$  symmetric matrices, each requiring only six learnable parameters due to symmetry constraints. These matrices are designed to compute the first-order  $(g^{\prime})$  and second-order  $(g^{\prime \prime})$  derivatives, respectively. In the matrix of  $g^{\prime \prime}$ ,  $s = 4 \times (a_{3} + a_{4} + a_{5} + a_{6}) + 2 \times (a_{1} + a_{2})$ . This design leverages the structural properties of central difference methods. By satisfying the Order of Sum Rules (Long et al., 2018), this filter can achieve up to 4th-order accuracy in approximating the derivatives via optimization of trainable parameters.
+
+# 3.2.4. NN BLOCK
+
+To alleviate the error accumulation during long-term predictions on coarse grids, we introduce the  $\mathrm{M_iNN}$  and  $\mathrm{M_aNN}$  Blocks, operating at micro- and macro-scales, respectively. The MiNN Block employs a lightweight model (e.g., FNO, DenseCNN (Liu et al., 2024a)) for efficient micro-step predictions, whereas the  $\mathrm{M_aNN}$  Block delivers more accurate predictions at larger steps (Gupta & Brandstetter, 2023). In this study, we utilized FNO as the  $\mathrm{M_iNN}$  Block and UNet as the  $\mathrm{M_aNN}$  Block. The significance of these blocks is evident from the ablation studies presented in Table 3.
+
+$\mathbf{M}_{\mathrm{i}}\mathbf{N}\mathbf{N}$  Block. The  $\mathbf{M}_{\mathrm{i}}\mathbf{N}\mathbf{N}$  block is designed to rectify error accumulation during micro-scale time step predictions. As
+
+![](images/491f806e87ae36a39ea16f79799b2bfcd0433c0e62714be563ecf1325da1b693.jpg)  
+Figure 2. Symmetric filter
+
+![](images/156988b45635fc94889d82ee309103437fab19af176081e676c86a5f26a000b2.jpg)
+
+shown in Figure 1(b) in the upper path,  $\bar{\mathbf{u}}_m^k$  is first corrected by the Correction Block, and  $\vec{p}_m^k$  is computed by the Poisson Block. Inputs, including solution states  $\{\bar{\mathbf{u}}_m^k,\vec{p}_m^k\}$  and their derivative terms, forcing term, and Reynolds number, are fed into the M_iNN Block (see Figure S1(d)). The M_iNN Block continuously refines the PDE block's outputs on the fly. For detailed information of the M_iNN Block settings, please refer to Appendix Table S5.
+
+$\mathbf{M}_{\mathrm{a}}\mathbf{N}\mathbf{N}$  Block. Although the Physics Block offers real-time corrections for the  $\mathbf{M}_{\mathrm{i}}\mathbf{N}\mathbf{N}$  outputs, errors still accumulate in long-term predictions. To mitigate error accumulation in long-term predictions given training data sampled at large time steps (e.g.,  $128\Delta t$  for the NSE dataset), we introduce the  $\mathbf{M}_{\mathrm{a}}\mathbf{N}\mathbf{N}$  Block. As depicted in Figure 1(a), the  $\mathbf{M}_{\mathrm{a}}\mathbf{N}\mathbf{N}$  Block takes the current velocity field  $\mathbf{u}_k$  as input, and updates the solution  $\mathbf{u}_{k + 1}$  which is obtained by integrating the outputs from both the upper and lower paths. During the backpropagation, the  $\mathbf{M}_{\mathrm{a}}\mathbf{N}\mathbf{N}$  Block learns to correct the coarse solution output of the Physics Block in real time, ensuring that their combined results more closely align with the ground truth. The configuration details for this block are found in Appendix Table S6.
+
+# 4. Experiment
+
+We validate the performance of our method against baseline models on various PDE datasets. We then perform generalization tests across different external forces (f), Reynolds numbers  $(Re)$ , and domain sizes on the Kolmogorov flow (KF) dataset. Finally, we present ablation studies to demonstrate the contributions of each component in our model.
+
+Table 2. Results of MultiPDENet and baselines. For KdV, Burgers, and GS, we inferred upper time limits of  $50~\mathrm{s}$ ,  $1.4~\mathrm{s}$ , and  $1200~\mathrm{s}$ , for the test set as the system dynamics stabilized within these trajectories. These time limits were used to calculate HCT.  
+
+<table><tr><td>Case</td><td>Model</td><td>RMSE (↓)</td><td>MAE (↓)</td><td>MNAD (↓)</td><td>HCT (s)</td></tr><tr><td rowspan="6">KdV</td><td>FNO</td><td>0.9541</td><td>0.4607</td><td>0.3469</td><td>10.0833</td></tr><tr><td>PINO</td><td>0.4120</td><td>0.3022</td><td>0.2139</td><td>13.90</td></tr><tr><td>UNet</td><td>1.9887</td><td>1.5722</td><td>1.6158</td><td>3.1250</td></tr><tr><td>DeepONet</td><td>NaN</td><td>NaN</td><td>NaN</td><td>0.1500</td></tr><tr><td>MultiPDENet</td><td>0.1536</td><td>0.1110</td><td>0.0833</td><td>39.8</td></tr><tr><td>Improvement (↑)</td><td>62.7%</td><td>63.3%</td><td>61.1%</td><td>186.3%</td></tr><tr><td rowspan="7">Burgers</td><td>FNO</td><td>0.0980</td><td>0.0762</td><td>0.0620</td><td>0.3000</td></tr><tr><td>PINO</td><td>0.0832</td><td>0.0749</td><td>0.0599</td><td>0.5546</td></tr><tr><td>UNet</td><td>0.3316</td><td>0.2942</td><td>0.2556</td><td>0.0990</td></tr><tr><td>DeepONet</td><td>0.2522</td><td>0.2106</td><td>0.1692</td><td>0.0020</td></tr><tr><td>PeRCNN</td><td>0.0967</td><td>0.1828</td><td>0.1875</td><td>0.4492</td></tr><tr><td>MultiPDENet</td><td>0.0057</td><td>0.0037</td><td>0.0031</td><td>1.4000</td></tr><tr><td>Improvement (↑)</td><td>93.1%</td><td>95.1%</td><td>94.8%</td><td>152.4%</td></tr><tr><td rowspan="7">GS</td><td>FNO</td><td>8774</td><td>1303</td><td>1303</td><td>270</td></tr><tr><td>PINO</td><td>0.5721</td><td>0.3579</td><td>0.3520</td><td>510</td></tr><tr><td>UNet</td><td>NaN</td><td>NaN</td><td>NaN</td><td>20</td></tr><tr><td>DeepONet</td><td>0.4113</td><td>0.2961</td><td>0.2898</td><td>568</td></tr><tr><td>PeRCNN</td><td>0.1763</td><td>0.1198</td><td>0.1198</td><td>640</td></tr><tr><td>MultiPDENet</td><td>0.0573</td><td>0.0294</td><td>0.0298</td><td>1400.0</td></tr><tr><td>Improvement (↑)</td><td>67.5%</td><td>75.5%</td><td>75.1%</td><td>118.8%</td></tr><tr><td rowspan="7">NSE</td><td>FNO</td><td>1.0100</td><td>0.7319</td><td>0.0887</td><td>2.5749</td></tr><tr><td>UNet</td><td>0.8224</td><td>0.5209</td><td>0.0627</td><td>3.9627</td></tr><tr><td>LI</td><td>NaN</td><td>NaN</td><td>NaN</td><td>3.5000</td></tr><tr><td>TSM</td><td>NaN</td><td>NaN</td><td>NaN</td><td>3.7531</td></tr><tr><td>DeepONet</td><td>2.1849</td><td>1.0227</td><td>0.1074</td><td>0.1126</td></tr><tr><td>MultiPDENet</td><td>0.1379</td><td>0.0648</td><td>0.0077</td><td>8.3566</td></tr><tr><td>Improvement (↑)</td><td>83.2%</td><td>87.6%</td><td>87.7%</td><td>110.9%</td></tr></table>
+
+# 4.1. Setup
+
+Dataset. We generate the data using high-order FD/FV methods with high resolution under periodic boundary conditions and then downsample it spatially and temporally to a coarse grid. The low-resolution dataset is used for both training and testing. We consider four distinct dynamical systems: Korteweg-de Vries (KdV), Burgers, Gray-Scott (GS), and Navier-Stokes equations (NSE). Each dataset is divided into  $90\%$  for training and  $10\%$  for validation. We segment trajectories into data series, where each sample includes multi snapshots (e.g., for the KdV dataset, the sample length is set to 10, as detailed in Table 1) separated by a time step  $\Delta t$ , the 2nd to the last snapshot serves as the training labels. During training, we use only 3-5 trajectories for each system, and evaluate them on 10 distinct trajectories. For further details, please refer to Appendix C.
+
+Model training. Our objective is to accelerate flow simulations with all computations anchored to coarse grids. During training, the model solely predicts the solutions for subsequent time steps, employing Mean Squared Error (MSE) as the loss metric. Unlike PINNs, our MultiPDENet directly embeds PDEs into its architecture, resulting in a loss function that exclusively comprises data loss, given by:  $\mathcal{J}(\boldsymbol{\lambda}) = \frac{1}{BN}\sum_{i=1}^{B}\sum_{j=1}^{N}MSE\left(\tilde{\mathbf{H}}_{ij},\mathbf{H}_{ij}\right)$ , where  $\tilde{\mathbf{H}}_{ij}$  denotes the coarse solution predicted by model rollout for the  $j$ -th sample in the  $i$ -th batch, and  $\mathbf{H}_{ij}$  is the correspond
+
+ing ground truth. Here,  $N$  denotes the number of batches,  $B$  the batch size, and  $\lambda$  the trainable PDE parameters.
+
+Model generalization. The generalization of MultiPDENet is evaluated across ICs, PDE parameters (e.g.,  $Re$ ), force terms, and computational domain sizes (e.g., different mesh grids). The model integrates ICs through its time-marching mechanism, ensuring robust generalization when trained effectively. The Reynolds number ( $Re$ ) is represented via a two-dimensional embedding,  $Re_{\mathrm{emb}} = \frac{1}{Re} \cdot (\mathbf{a} \otimes \mathbf{b})$ , using trainable vectors  $\mathbf{a}$  and  $\mathbf{b}$ . This embedding, applied in both the PDE and MiNN blocks, reduces error propagation from the diffusion term on coarse grids and enhances generalization across  $Re$  values. The force term is incorporated into the learnable PDE block and the MiNN block, where it serves as both a PDE feature and an input feature map as shown in Figure S1, enabling joint learning of force variations for better generalization.
+
+Evaluation metrics. We evaluate the performance of our model using four metrics: Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Mean Normalized Absolute Difference (MNAD), and High Correlation Time (HCT). For detailed definitions, please refer to Appendix E.
+
+Baseline models. To ensure a comprehensive comparison, we selected several baseline models, including FNO (Li et al., 2021), PINO (Li et al., 2024c), UNet (Gupta & Brandstetter, 2023), TSM (Sun et al., 2023), LI (Kochkov et al., 2021), DeepONet (Lu et al., 2021), and PeRCNN (Rao et al., 2023). Details are found in Appendix F.
+
+# 4.2. Solving PDE Systems
+
+KdV. The primary challenge of this dataset lies in accurately capturing the complex interplay between nonlinearity and dispersion, leading to phenomena like soliton formation (Gardner et al., 1967). As shown in Figure 3(a), each baseline model struggles to produce accurate predictions, with DeepONet exhibiting significant divergence. In contrast, MultiPDENet demonstrates superior accurate predictions for ICs outside the training range. The correlation curve in Figure 3(b) highlights the significantly higher correlation of MultiPDENet compared to the baselines. The error distribution in Figure 3(c) confirms its lower error levels. Table 2 shows our model's generalizability, with performance improvements ranging from  $61.1\%$  to  $186.3\%$ .
+
+Burgers. As shown in Figure 3(d), the solution snapshots predicted by MultiPDENet are significantly more accurate than those of the baseline models. The baseline models, limited by the coarse training data, produce incorrect predictions. The correlation curve in Figure 3(e) shows that MultiPDENet maintains a high correlation with the ground truth throughout the prediction, while other baseline models diverge. This is further evidenced by the error distribution in Figure 3(f), demonstrating that MultiPDENet's error is
+
+![](images/eaa58e9d993d56eee187b4fd632b9fbbf7a57023b78f72b987d348d7457fba3f.jpg)
+
+![](images/b86674c858e74e90c3f3726729f4005cd40029a3c107af3f65653f6f193831e8.jpg)
+
+![](images/c75824f07ac14ceabc8b12c68fade1cb8a9eaf1e90b713d6649302b55858c199.jpg)
+
+![](images/221ac856e3485b5520d8565c6bb4aa1552044ba09446e2bf990d680385f3c8d0.jpg)
+
+![](images/7b758322b8c555359a28eeb906de162635b77cf53ad1c77a2f971a93a5d644e9.jpg)
+
+![](images/f31f0776294cb17a293212cf8427ecb2c64613feb5f4988f5dd8a390aaca2e90.jpg)
+
+![](images/84f8e9a67ab11f8ad3e4edea02490d6b1c98827133cd91e311cf1644a17144a5.jpg)  
+Figure 3. An overview of the comparison between our MultiPDENet and baselines, including predicted solutions (left), correlation curve (middle), and error distributions (right). (a)-(c) show the qualitative results on KdV. (d)-(f) show the qualitative results on Burgers. (g)-(i) show the qualitative results on GS. These PDE systems are trained with grid sizes of 64,  $25 \times 25$ ,  $32 \times 32$ , respectively.
+
+![](images/83cab5e2c5fe2e46409c53589926d699f45dee1e38f7849f6fb4a18ecf2f42a2.jpg)
+
+![](images/00e71b8ddf4e3487b0e50075acff35ad30726ee70224f905e954f51ccdf488c6.jpg)
+
+over an order of magnitude lower than that of the baselines. Table 2 confirms these findings with MultiPDENet's improvements exceeding  $94.1\%$  across all evaluation metrics.
+
+GS. This reaction-diffusion system is nonlinear, making it challenging to capture its complex patterns (see Figure 3(g)). Only MultiPDENet accurately predicts the trajectory evolution. The baseline models struggle to learn the spatiotemporal dynamics, and even PeRCNN, despite its embedded physics, produces inaccurate predictions due to the limited and coarse training data. Figure 3(h) demonstrates the superior correlation of MultiPDENet's predictions with the ground truth. The error analysis in Figure 3(i) reveals significantly lower error levels for MultiPDENet, often by 1 to 2 orders of magnitude smaller compared to the baselines. Table 2 further validates this observation, with MultiPDENet's improvements of  $67.5\%$  to  $118.8\%$  over the best baseline.
+
+NSE. We evaluate a KF with  $Re = 1000$  across different ICs, governed by the NSE. Figure 4(a) shows the trajectory snapshots predicted by MultiPDENet and the baseline models over  $10 \, \text{s}$ . Our model outperforms DNS 512, accurately capturing both global and local correction patterns. The neural methods, particularly FNO, exhibit poor generalization, producing granular and erroneous solutions. Among the Physics + ML baselines, TSM performs the best, but
+
+starts to produce incorrect patterns at  $t = 5$  s due to error accumulation. The correlation curve in Figure 4(b) supports these findings. Our model also achieves a spectrum energy distribution closely matching the ground truth (see Figure 4(c)). Table 2 highlights a performance improvement of over  $83.2\%$ . Even with  $20\%$  less training data, our model maintains strong generalizability (see Appendix Table S3).
+
+# 4.3. Model Generalization
+
+We conducted generalization tests on the KF flow dataset to assess our model's ability to capture the underlying dynamics. The model was initially trained on 5 sets of trajectories, where the forcing term is defined as  $\mathbf{f} = \sin (4y)\pmb{\eta}_x - 0.1\mathbf{u}$  with  $Re = 1000$  and  $\pmb{\eta}_{x} = [1,0]^{T}$ . After training, we tested MultiPDENet on 10 different sets of trajectories, each with varying Reynolds numbers  $(Re)$  and forcing terms (f), to evaluate its performance across a range of different ICs.
+
+Test on various Reynolds numbers. Firstly, we evaluate the generalizability of MultiPDENet across four different Reynolds numbers:  $Re = 500, 800, 1600, 2000$ . The varying  $Re$  values result in trajectories with differing levels of complexity. Figure 5(a) displays the accurate predictions made by our model at time step 300 for different  $Re$ . Figure 5(b) shows the correlation curves between the predicted
+
+![](images/4c05d4466a8c7a9ebe1e07ea8240589624243a9ce2b734d70b7717f82d4ac8fc.jpg)  
+Figure 4. Comparison of MultiPDENet and baseline models on Kolmogorov flow with  $Re = 1000$ . (a) shows the evolution of predicted vorticity fields for reference, MultiPDENet and baselines, starting from the same initial velocities. (b) shows the correlation curve across 500 time steps. (c) shows the scaled energy spectrum scaled by  $k^5$  averaged between time steps 100 and 500.
+
+![](images/991c5fae0b0f3fce81a0dcaf07bf13dbaa9808ac2dd0a19923d4f27467b0eed7.jpg)
+
+![](images/2d3dbff6655e79819f697e4467491544ca61301ee6355efb0fa472f8fdf95dbe.jpg)
+
+and ground truth trajectories, while Figure 5(c) highlights the error distributions, which remain consistently below 0.1, indicating a low error level.
+
+Test on external forces. Next, we performed the generalization test using 4 distinct  $\mathbf{f}$ . As shown in Figure 5(d), MultiPDENet accurately predicts the trajectories for all forces. Notably, the downward trend in the correlation curve (in Figure 5 (e)) for  $\mathbf{f}_2$  parallels the behavior observed in Figure 4(b), likely because  $\mathbf{f}$  alters only the periodic function without changing the wave number. The error analysis in Figure 5(f) confirms that the error levels remain below 0.1.
+
+Test on flow with  $Re = 4000$ . Turbulence at high  $Re$ 's presents significant challenges for prediction due to its nonlinearity and complex vortex structures. To further demonstrate the superior capability of our model, we conducted an additional experiment with a high  $Re = 4000$  (see details in Table S2) with the experimental setup in Section 4.1. After
+
+training, the model was tested on 10 trajectories with new ICs. Appendix Figure 6(a) illustrates the snapshots predicted by MultiPDENet over 600 timesteps, demonstrating sustained accuracy even at time step 450. The correlation curve in Appendix Figure 6(b) highlights the superiority of our model compared to DNS 1024. The error analysis in Appendix Figure 6(c) confirms this performance, with errors consistently below 0.01. These results demonstrate the effectiveness of MultiPDENet for higher Reynolds number, e.g.,  $Re = 4000$ , within domain  $(0, 2\pi)^2$ .
+
+Test on flow within larger domains. We extended the spatial domain from  $(0,2\pi)^{2}$  to  $(0,4\pi)^{2}$  to further evaluate our model's generalizability over larger mesh grids in a more complex scenario. Larger domains introduce diverse physical phenomena, challenging the model to capture global and local dynamics on coarse grids. Using the same  $64 \times 64$  grid, we tested our trained model on 10 unseen trajectories
+
+![](images/748fb40694b8212936c13aee5dba5bb11dbc96bdf0d8c8a66c214960901b0e0e.jpg)
+
+![](images/7aaa41aa391d810af693c531111093502a6d6e6658bf5e8e5c9dfaaf5248ab11.jpg)
+
+![](images/c50bf1467bf5434ef0d2149218e97ed34dff22e0669bfce6387b2bd05bbd819a.jpg)
+
+![](images/2679071c29f92daa5fe17fb8a7ce9528259492eb121a0972119529cccfc7e1e4.jpg)  
+Figure 5. MultiPDENet can generalize to simulate different Reynolds numbers and external forcings without retraining. Vorticity snapshots predicted by MultiPDENet and ground truth at timestep = 300 (left), correlation curve over 600 timesteps (middle), error distribution (right). (a-c) display results for different Reynolds numbers, (d-f) show results for varying external forcings.
+
+![](images/2a4f1608aedbc570ab30aa7f3b8029671854a6ad932f94cea18a7dfdccd6b6fb.jpg)
+
+![](images/cbbce576979c880e0e612d8cd145463c0ed7ed3e305f62962358e54d6de556a0.jpg)
+
+![](images/2c6dcf264a5453adb1c84f5ccb8de7c6ef7d3261eb9a54c7a2642e0945f1fafb.jpg)  
+Figure 6. MultiPDENet is applicable to high Reynolds number turbulence. (a) Trajectories predicted by MultiPDENet at  $Re = 4000$ . (b-c) Correlation and error distribution comparison between MultiPDENet and numerical method.
+
+![](images/49505044de7d8d5fff4c61197bd6ef6f223f0a8bc8603c0efc76c9360c6944eb.jpg)
+
+![](images/978dd535815195b92e16d0dedfef2458a6e27902899045d50ea33fa79a87fe72.jpg)
+
+(details in Appendix C). As shown in Appendix Figure 7(a), the snapshots over 300 time steps remain accurate. The correlation curve in Appendix Figure 7(b) depicts our model's performance closely matches or exceeds DNS 1024. The error distribution in Appendix Figure 7(c) shows error levels comparable to DNS 1024. Notably, our model achieves a speedup of  $6 \times$  compared to the FV DNS method.
+
+In summary, MultiPDENet demonstrates remarkable generalizability, showcasing its ability to capture the underlying dynamics across multiple temporal scales. The embedded learnable PDE module within our model is crucial for enabling robust and accurate predictions.
+
+# 4.4. Ablation Study
+
+To quantify the contribution of each module, we conducted ablation experiments on the KF dataset. Specifically, we compared the following model variations: (1) Model A (no
+
+Poisson Block); (2) Model B (no filter structure constraint); (3) Model C (only Physics Block for prediction); (4) Model D (FD convolution instead of symmetric filter); (5) Model E (no Correction Block); (6) Model F (no  $\mathbf{M}_{\mathrm{i}}\mathbf{N}\mathbf{N}$  Block); (7) Model G (no  $\mathbf{M}_{\mathrm{a}}\mathbf{N}\mathbf{N}$  Block); (8) Model H (no Physics Block); (9) Model I (forward Euler); and (10) MultiPDENet (the full model). The results are shown in Table 3.
+
+Removing the Poisson Block impairs our model's performance, confirming  $p$ -u decoupling necessity in the NSE. Relaxing the filter structure constraint degrades the result, validating our proposed kernel's efficacy. The Physics Block alone yields worse prediction compared to the full model. Replacing the Conv kernel with FD stencils results in poorer performance, indicating that fixed-value FD kernels are unsuitable for coarse grids. Omitting the Correction Block also degrades the model prediction, highlighting the necessity of field correction. While the model can still accurately predict up to 5.9 s without the M_iNN Block, the error increases by
+
+![](images/e736428bf000076291827a70166b2ba2490b4be4dcac5a1b23144bcb4a8b8c2a.jpg)  
+Figure 7. MultiPDENet applied to a larger domain. (a) Predicted trajectories within  $4\pi \times 4\pi$ . (b-c) Correlation and error distribution comparison with the numerical method.
+
+![](images/8664a8fb017e165ab3dc7494f189c2052815573ecf1a6e670f336525da56141f.jpg)
+
+![](images/4f1a0c681980efd016cf0492812f354c48e1977245ee952fe3211a4a9fff8527.jpg)
+
+Table 3. Results of the ablation study.  
+
+<table><tr><td>Ablated Model</td><td>RMSE (↓)</td><td>MAE (↓)</td><td>MNAD (↓)</td><td>HCT (↑)</td></tr><tr><td>Model A</td><td>0.1601</td><td>0.0711</td><td>0.0085</td><td>7.904</td></tr><tr><td>Model B</td><td>0.2432</td><td>0.1156</td><td>0.0137</td><td>7.8633</td></tr><tr><td>Model C</td><td>0.2632</td><td>0.1402</td><td>0.0186</td><td>7.3146</td></tr><tr><td>Model D</td><td>0.2503</td><td>0.1410</td><td>0.0145</td><td>7.0783</td></tr><tr><td>Model E</td><td>0.2958</td><td>0.1453</td><td>0.0180</td><td>6.6856</td></tr><tr><td>Model F</td><td>0.4338</td><td>0.2401</td><td>0.0285</td><td>5.9768</td></tr><tr><td>Model G</td><td>NaN</td><td>NaN</td><td>NaN</td><td>1.4193</td></tr><tr><td>Model H</td><td>1.2023</td><td>0.9256</td><td>0.1122</td><td>0.6227</td></tr><tr><td>Model I</td><td>0.4357</td><td>0.2321</td><td>0.0278</td><td>6.2481</td></tr><tr><td>MultiPDENet</td><td>0.1379</td><td>0.0648</td><td>0.0077</td><td>8.3566</td></tr></table>
+
+$4 \times$ , suggesting the need for the  $\mathrm{M_iNN}$  Block.
+
+Removing the  $\mathrm{M_aNN}$  Block restricts the micro-scale accuracy and causes macro-scale instability due to error accumulation. This underscores the importance of the  $\mathrm{M_aNN}$  Block for long-term stability. The omission of the Physics Block (e.g., solely U-Net) cripples the prediction with limited coarse training data, proving its critical role. Moreover, the use of RK4 outperforms forward Euler in terms of stability and accuracy. Hence, all the components are essential and contribute meaningfully to the MultiPDENet model.
+
+# 5. Conclusion
+
+We introduce an end-to-end physics-encoded network (aka, MultiPDENet) with multi-scale time stepping for accelerated simulation of spatiotemporal dynamics such as turbulent flows. MultiPDENet consists of a multi-scale temporal learning architecture, a learnable Physics Block for solution prediction at the fine time scale, where trainable symmetric filters are designed for improved derivative approximation on coarse spatial grids. Such a method is capable of long-term prediction on coarse grids given very limited training data (see the data size scaling test in Appendix D.4). MultiPDENet outperforms other baselines through extensive tests on fluid dynamics and reaction-diffusion equations. In particular, such a model excels in generalizability over ICs, Reynolds numbers, and external forces in the turbulent flow experiments. MultiPDENet also exhibits strong stabil
+
+ity in long-term prediction of turbulent flows, effectively capturing both global and local patterns in larger domains.
+
+We also tested the computational efficiency of trained MultiPDENet for accelerated flow prediction (more details shown in Appendix G.2). For a certain given accuracy (e.g., correlation  $\geq 0.8$ ), MultiPDENet achieves  $\geq 5 \times$  speedup compared with GPU-accelerated DNS (Appendix Table S8), e.g., JAX-CFD, where all the tests were performed on a single Nvidia A100 80G GPU. However, MultiPDENet still faces some unresolved challenges. Firstly, the model currently only handles regular grids, due to the limitation of convolution operation used in the model. In the future, we aim to address this issue by incorporating graph neural networks to manage irregular grids. Secondly, the model has only been currently tested on 1D and 2D problems. We will extend it to 3D systems in our future work.
+
+# Acknowledgment
+
+The work is supported by the National Science and Technology Major Project (No. 2022ZD0117804), the National Natural Science Foundation of China (No. 62276269 and No. 92270118), the Beijing Natural Science Foundation (No. 1232009), and the Fundamental Research Funds for the Central Universities (No. 202230265). Our source codes are available in the following GitHub repository: https://github.com/intell-sci-comput/MultiPDENet.
+
+# Impact statement
+
+This work aims to develop a PDE-embedded network with multiscale time stepping, integrating numerical schemes and machine learning to accelerate flow simulations. This approach has the potential to benefit various fields, including weather forecasting and turbulent flow prediction. This research is solely intended for scientific purposes and does not present any foreseeable ethical risks.
+
+# References
+
+Arcomano, T., Szunyogh, I., Wikner, A., Pathak, J., Hunt, B. R., and Ott, E. A hybrid approach to atmospheric modeling that combines machine learning with a physics-based numerical model. Journal of Advances in Modeling Earth Systems, 14(3):e2021MS002712, 2022.  
+Azizzadenesheli, K., Kovachki, N., Li, Z., Liu-Schiaffini, M., Kossaifi, J., and Anandkumar, A. Neural operators for accelerating scientific simulations and design. Nature Reviews Physics, pp. 1-9, 2024.  
+Bhatnagar, S., Afshar, Y., Pan, S., Duraisamy, K., and Kaushik, S. Prediction of aerodynamic flow fields using convolutional neural networks. Computational Mechanics, 64:525-545, 2019.  
+Bonev, B., Kurth, T., Hundt, C., Pathak, J., Baust, M., Kashinath, K., and Anandkumar, A. Spherical fourier neural operators: Learning stable dynamics on the sphere. In International Conference on Machine Learning, pp. 2806-2823. PMLR, 2023.  
+Brandstetter, J., Welling, M., and Worrall, D. E. Lie point symmetry data augmentation for neural pde solvers. pp. 2241-2256, 2022a.  
+Brandstetter, J., Worrall, D. E., and Welling, M. Message passing neural pde solvers. In International Conference on Learning Representations, 2022b.  
+Cao, S. Choose a transformer: Fourier or galerkin. Advances in Neural Information Processing Systems, 34:24924-24940, 2021.  
+Chen, H., Liu, Y., and Sun, H. PINP: Physics-informed neural predictor with latent estimation of fluid flows. In International Conference on Learning Representations, 2025.  
+Dresdner, G., Kochkov, D., Norgaard, P. C., Zepeda-Nunez, L., Smith, J., Brenner, M., and Hoyer, S. Learning to correct spectral methods for simulating turbulent flows. Transactions on Machine Learning Research, 2023.  
+Eshkofti, K. and Hosseini, S. M. A gradient-enhanced physics-informed neural network (gpinn) scheme for the coupled non-fickian/non-fourierian diffusion-thermoelasticity analysis: A novel gpinn structure. Engineering Applications of Artificial Intelligence, 126:106908, 2023.  
+Ferziger, J. H., Perić, M., and Street, R. L. Computational methods for fluid dynamics. Springer, 2019.  
+Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M. Method for solving the korteweg-devries equation. Physical Review Letters, 19(19):1095, 1967.
+
+Geneva, N. and Zabaras, N. Transformers for modeling physical systems. Neural Networks, 146:272-289, 2022.  
+Goc, K. A., Lehmkuhl, O., Park, G. I., Bose, S. T., and Moin, P. Large eddy simulation of aircraft at affordable cost: a milestone in computational fluid dynamics. *Flow*, 1:E14, 2021.  
+Gupta, J. K. and Brandstetter, J. Towards multispatiotemporal-scale generalized PDE modeling. Transactions on Machine Learning Research, 2023.  
+He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 770-778, 2016.  
+Hernández, Q., Badías, A., Chinesta, F., and Cueto, E. Thermodynamics-informed neural networks for physically realistic mixed reality. Computer Methods in Applied Mechanics and Engineering, 407:115912, 2023. ISSN 0045-7825.  
+Hoffman, J. D. and Frankel, S. Numerical methods for engineers and scientists. CRC press, 2018.  
+Kochkov, D., Smith, J. A., Alieva, A., Wang, Q., Brenner, M. P., and Hoyer, S. Machine learning-accelerated computational fluid dynamics. Proceedings of the National Academy of Sciences, 118(21):e2101784118, 2021.  
+Kossaczka, T., Ehrhardt, M., and Gunther, M. Enhanced fifth order weno shock-capturing schemes with deep learning. Results in Applied Mathematics, 12:100201, 2021.  
+Kumar, N. K. A review on burgers' equations and it's applications. Journal of Institute of Science and Technology, 28(2):49-52, 2023.  
+LeVeque, R. J. Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. SIAM, 2007.  
+Li, Z., Kovachki, N. B., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A. Fourier neural operator for parametric partial differential equations. In International Conference on Learning Representations, 2021.  
+Li, Z., Han, W., Zhang, Y., Fu, Q., Li, J., Qin, L., Dong, R., Sun, H., Deng, Y., and Yang, L. Learning spatiotemporal dynamics with a pretrained generative model. Nature Machine Intelligence, 6(12):1566-1579, 2024a.  
+Li, Z., Shu, D., and Barati Farimani, A. Scalable transformer for pde surrogate modeling. Advances in Neural Information Processing Systems, 36, 2024b.
+
+Li, Z., Zheng, H., Kovachki, N., Jin, D., Chen, H., Liu, B., Azizzadenesheli, K., and Anandkumar, A. Physics-informed neural operator for learning partial differential equations. ACM/JMS Journal of Data Science, 1(3):1-27, 2024c.  
+Liu, X.-Y., Zhu, M., Lu, L., Sun, H., and Wang, J.-X. Multiresolution partial differential equations preserved learning framework for spatiotemporal dynamics. Communications Physics, 7(1):31, 2024a.  
+Liu, Y., Hu, T., Zhang, H., Wu, H., Wang, S., Ma, L., and Long, M. itransformer: Inverted transformers are effective for time series forecasting. International conference on machine learning, 2024b.  
+Liu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., Zhang, Z., Lin, S., and Guo, B. Swin transformer: Hierarchical vision transformer using shifted windows. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 10012-10022, 2021.  
+Long, Z., Lu, Y., Ma, X., and Dong, B. Pde-net: Learning pdes from data. In International Conference on Machine Learning, pp. 3208-3216, 2018.  
+Long, Z., Lu, Y., and Dong, B. Pde-net 2.0: Learning pdes from data with a numeric-symbolic hybrid deep network. Journal of Computational Physics, 399:108925, 2019.  
+Lu, L., Jin, P., Pang, G., Zhang, Z., and Karniadakis, G. E. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3(3):218-229, 2021.  
+Lu, Y., Zhong, A., Li, Q., and Dong, B. Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In International Conference on Machine Learning, pp. 3276-3285, 2018.  
+Mi, Y., Ren, P., Xu, H., Liu, H., Wang, Z., Guo, Y., Wen, J.-R., Sun, H., and Liu, Y. Conservation-informed graph learning for spatiotemporal dynamics prediction. In Proceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining V.1, pp. 1056-1067, 2025.  
+Moukalled, F., Mangani, L., Darwish, M., Moukalled, F., Mangani, L., and Darwish, M. The finite volume method. Springer, 2016.  
+Rahman, M. A., Ross, Z. E., and Azizzadenesheli, K. U-NO: U-shaped neural operators. Transactions on Machine Learning Research, 2023. ISSN 2835-8856.  
+Raissi, M., Perdikaris, P., and Karniadakis, G. E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear
+
+partial differential equations. Journal of Computational Physics, 378:686-707, 2019.  
+Raissi, M., Yazdani, A., and Karniadakis, G. E. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science, 367(6481):1026-1030, 2020.  
+Rao, C., Ren, P., Liu, Y., and Sun, H. Discovering nonlinear pdes from scarce data with physics-encoded learning. In International Conference on Learning Representations, 2022.  
+Rao, C., Ren, P., Wang, Q., Buyukozturk, O., Sun, H., and Liu, Y. Encoding physics to learn reaction-diffusion processes. Nature Machine Intelligence, 5(7):765-779, 2023.  
+Rathore, P., Lei, W., Frangella, Z., Lu, L., and Udell, M. Challenges in training pinns: A loss landscape perspective. CoRR, 2024.  
+Ren, P., Rao, C., Liu, Y., Ma, Z., Wang, Q., Wang, J.-X., and Sun, H. Physr: Physics-informed deep superresolution for spatiotemporal data. Journal of Computational Physics, 492:112438, 2023.  
+Ren, P., Song, J., Rao, C., Wang, Q., Guo, Y., Sun, H., and Liu, Y. Learning spatiotemporal dynamics from sparse data via a high-order physics-encoded network. Computer Physics Communications, 312:109582, 2025.  
+Ronneberger, O., Fischer, P., and Brox, T. U-net: Convolutional networks for biomedical image segmentation. In Proceedings of the 18th International Conference Medical Image Computing and Computer-Assisted Intervention, part III 18, pp. 234-241, 2015.  
+Schneider, T., Teixeira, J., Bretherton, C. S., Brient, F., Pressel, K. G., Schär, C., and Siebesma, A. P. Climate goals and computing the future of clouds. Nature Climate Change, 7(1):3-5, 2017.  
+So, C. C., Li, T. O., Wu, C., and Yung, S. P. Differential spectral normalization (dsn) for PDE discovery. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 9675-9684, 2021.  
+Stachenfeld, K., Fielding, D. B., Kochkov, D., Cranmer, M., Pfaff, T., Godwin, J., Cui, C., Ho, S., Battaglia, P., and Sanchez-Gonzalez, A. Learned coarse models for efficient turbulence simulation. In International Conference on Learning Representations, 2022.  
+Sun, Z., Yang, Y., and Yoo, S. A neural PDE solver with temporal stencil modeling. In International Conference on Machine Learning, pp. 33135-33155, 2023.
+
+Takamoto, M., Praditia, T., Leiteritz, R., MacKinlay, D., Alesiani, F., Pflüger, D., and Niepert, M. Pdebench: An extensive benchmark for scientific machine learning. Advances in Neural Information Processing Systems, 35: 1596-1611, 2022.  
+Tang, K., Zhai, J., Wan, X., and Yang, C. Adversarial adaptive sampling: Unify pinn and optimal transport for the approximation of pdes. In International Conference on Learning Representations, 2024.  
+Thomas, J. W. Numerical partial differential equations: finite difference methods, volume 22. Springer Science & Business Media, 2013.  
+Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., and Polosukhin, I. Attention is all you need. Advances in Neural Information Processing Systems, 2017.  
+Wang, Q., Ren, P., Zhou, H., Liu, X.-Y., Deng, Z., Zhang, Y., Chengze, R., Liu, H., Wang, Z., Wang, J.-X., et al.  $\mathrm{P}^2\mathrm{C}^2$  Net: PDE-Preserved Coarse Correction Network for Efficient Prediction of Spatiotemporal Dynamics. Advances in Neural Information Processing Systems, 37: 68897-68925, 2024.  
+Wang, R., Kashinath, K., Mustafa, M., Albert, A., and Yu, R. Towards physics-informed deep learning for turbulent flow prediction. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 1457-1466, 2020.  
+Wang, R., Walters, R., and Yu, R. Incorporating symmetry into deep dynamics models for improved generalization. In International Conference on Learning Representations, 2021.  
+Wu, H., Luo, H., Ma, Y., Wang, J., and Long, M. Ropinn: Region optimized physics-informed neural networks. Advances in Neural Information Processing Systems, 2024.  
+Wu, T., Wang, Q., Zhang, Y., Ying, R., Cao, K., Sosic, R., Jalali, R., Hamam, H., Maucec, M., and Leskovec, J. Learning large-scale subsurface simulations with a hybrid graph network simulator. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 4184-4194, 2022.  
+Xing, L., Wu, H., Ma, Y., Wang, J., and Long, M. Helmholtz dynamics for interpretable fluid prediction. International Conference on Machine Learning, 2024.  
+Zeng, B., Wang, Q., Yan, M., Liu, Y., Chengze, R., Zhang, Y., Liu, H., Wang, Z., and Sun, H. PhyMPGN: Physics-encoded message passing graph network for spatiotemporal PDE systems. In International Conference on Learning Representations, 2025.
+
+Zeng, D., Liu, W., Chen, W., Zhou, L., Zhang, M., and Qu, H. Substructure aware graph neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pp. 11129-11137, 2023.  
+Zhuang, J., Kochkov, D., Bar-Sinai, Y., Brenner, M. P., and Hoyer, S. Learned discretizations for passive scalar advection in a two-dimensional turbulent flow. Physical Review Fluids, 6(6):064605, 2021.  
+Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z. The finite element method: its basis and fundamentals. Elsevier, 2005.
+
+# APPENDIX
+
+# A. Related work
+
+Numerical Methods. Numerical methods have been extensively applied to solve PDEs. Approaches such as FD (Thomas, 2013), FE (Zienkiewicz et al., 2005), and FV methods (Moukalled et al., 2016) discretize the continuous domain into mesh grids, transforming PDEs into algebraic equations that can be solved with high accuracy. However, these methods often require fine spatiotemporal grids and substantial computational resources to achieve accurate solutions, particularly in high-dimensional spaces. This leads to two main challenges: (1) the need for repeated computations when conditions change (e.g., ICs); (2) the demand for fast simulations in many industrial applications.
+
+Machine Learning Methods. Building on the success of machine learning in fields like natural language processing (Vaswani et al., 2017) and computer vision (He et al., 2016), these techniques have also been applied to solving PDEs. With abundant labeled data, it is possible to train end-to-end models to predict solutions. Representative works include ResNet (Lu et al., 2018), CNN-based models (Bhatnagar et al., 2019; Stachenfeld et al., 2022; Gupta & Brandstetter, 2023), Transformer-based models (Cao, 2021; Geneva & Zabaras, 2022; Liu et al., 2024b; Li et al., 2024b), Graph-based models (Brandstetter et al., 2022b; Wu et al., 2022; Zeng et al., 2023), and pretrained diffusion models (Li et al., 2024a). Many notable neural operators (Lu et al., 2021; Li et al., 2021; Rahman et al., 2023; Bonev et al., 2023; Azizzadenesheli et al., 2024), which learn a mapping between functional spaces, enable the approximation of complex relationships in PDEs. While these methods show promise in learning complex dynamics and approximating solutions, they often require substantial amounts of labeled data.
+
+Physics-inspired Learning Methods. Recently, physics-inspired learning methods have demonstrated impressive capabilities in solving PDEs, which can be classified into two categories according to the way of adding prior knowledge: physics-informed and physics-encoded. The physics-informed methods take PDEs and I/BCs as a part of the loss functions (e.g., the family of PINN (Raissi et al., 2019; 2020; Wang et al., 2020; Tang et al., 2024; Wu et al., 2024; Chen et al., 2025), PhySR (Ren et al., 2023)). On the other hand, the physics-encoded methods employ a different approach that preserves the structure of PDEs, ensuring that the model adheres to the given equations to capture the underlying dynamics, e.g., EquNN (Wang et al., 2021), TiGNN (Hernández et al., 2023), PeRCNN (Rao et al., 2022; 2023; Ren et al., 2025), PhyMPGN (Zeng et al., 2025), CiGNN (Mi et al., 2025), and HelmFluid (Xing et al., 2024). In addition, other related studies (Long et al., 2018; 2019; Kossaczká et al., 2021; So et al., 2021) have explored the use of CNN as alternative spatial derivative operators for approximating derivatives and capturing the dynamics of interest.
+
+Hybrid Learning Methods. Hybrid learning methods combine the strengths of numerical approaches and NNs to improve prediction accuracy. For efficient modeling of spatiotemporal dynamics, these methods can be trained on coarse grids. Representative methods include FV-based neural methods (Kochkov et al., 2021; Sun et al., 2023), FD-based neural methods (Zhuang et al., 2021; Liu et al., 2024a; Wang et al., 2024), and spectral-based neural methods (Dresdner et al., 2023; Arcomano et al., 2022). While these approaches show efficacy in modeling spatiotemporal dynamics, their representation capacities are often limited by the fixed structure of their numerical components. As a result, most of these models still require sufficiently large amounts of training data.
+
+# B. The Details of MultiPDENet
+
+# B.1. Correction Block
+
+The Correction Block leverages a neural network to refine the coarse solution, with the Fourier Neural Operator (FNO) (Li et al., 2021) as the correction mechanism within this block. FNO functions by decomposing the input field into frequency components, processing each frequency individually, and reconstructing the modified spectral information back into the physical domain via the Fourier transform. This layer-wise update process is expressed as:
+
+$$
+\mathbf {v} ^ {l + 1} (\tilde {\mathbf {x}}) = \sigma \left(\mathbf {W} ^ {l} \mathbf {v} ^ {l} (\tilde {\mathbf {x}}) + \left(\mathcal {K} (\phi) \mathbf {v} ^ {l}\right) (\tilde {\mathbf {x}})\right), \tag {S1}
+$$
+
+where  $\mathbf{v}^l (\tilde{\mathbf{x}})$  denotes the latent feature map at the  $l$ -th layer, defined on the coarse grid  $\tilde{\mathbf{x}}$ . The initial feature map is  $\mathbf{v}^0 (\tilde{\mathbf{x}}) = \mathcal{P}(\bar{\mathbf{u}}_m^k)$ , where  $\mathcal{P}$  is a local mapping function that projects  $\bar{\mathbf{u}}_m^k$  into a higher-dimensional space. The kernel integral transformation is defined as  $\mathcal{K}(\phi)(\mathbf{z}) = \mathrm{iFFT}(\mathbf{R}_{\phi}\cdot \mathrm{FFT}(\mathbf{z}))$ , which applies the Fourier transform, spectral
+
+filtering via  $\mathbf{R}_{\phi}$ , convolution in the frequency domain, and the inverse Fourier transform to the latent feature map  $\mathbf{z}$ . Here,  $\phi$  represents the trainable parameters,  $\sigma(\cdot)$  is the GELU activation function, and  $\mathbf{W}^l$  denotes the weights of the linear layer. After passing through an  $L$ -layer FNO, the refined coarse solution is computed as  $\hat{\tilde{\mathbf{u}}}_m^k = \mathcal{Q}(\mathbf{v}^L(\tilde{\mathbf{x}}))$ , where  $\mathcal{Q}$  projects the latent representation of the final layer back into the original solution space.
+
+In the Correction Block, we set  $L = 2$ . For the Burgers equation, we configure the model with modes  $= 12$ , width  $= 12$ , and a projection from 12 channels to 50 channels. For the GS case, we use the same configuration: modes  $= 12$ , width  $= 20$ , with a projection from 20 channels to 50 channels. For the KdV equation, the setup is defined as modes  $= 32$  and width  $= 64$ , with a projection from 64 channels to 128 channels. The NSE case, however, requires a different setup: modes  $= 25$ , width  $= 20$ , with a projection from channel  $= 20$  to channel  $= 128$ .
+
+# B.2. Physics Block
+
+To accurately predict at the micro-scale step, we developed a neural solver called the Physics Block, ensuring stability, accuracy, and efficiency through adherence to the Courant-Friedrichs-Lewy (CFL) conditions (LeVeque, 2007). The Physics Block comprises three key components: the Poisson Block (Figure S1(a)-b), the PDE Block (Figure 1(c)), and the  $\mathbf{M}_{\mathrm{i}}\mathbf{N}\mathbf{N}$  Block (Figure S1(c)).
+
+Poisson Solver. The pressure field is computed using the spectral method, which involves solving the Poisson equation:
+
+$$
+\Delta p = \psi (\mathbf {u}). \tag {S2}
+$$
+
+Here,  $\psi (\mathbf{u}) = 2\left(u_{x}v_{y} - u_{y}v_{x}\right)$  represents the source term for the pressure.
+
+Applying the Fast Fourier Transform (FFT) to Eq. (S2), we obtain:
+
+$$
+- \left(\varphi_ {x} ^ {2} + \varphi_ {y} ^ {2}\right) p ^ {*} = \psi^ {*} (\mathbf {u}), \tag {S3}
+$$
+
+where  $\varphi_{x}$  and  $\varphi_{y}$  are the wavenumbers in the  $x$  and  $y$  directions, respectively. Assuming  $\varphi_x^2 +\varphi_y^2\neq 0$ , we can solve for the pressure in the frequency domain:
+
+$$
+p ^ {*} = \frac {\psi^ {*} (\mathbf {u})}{- \left(\varphi_ {x} ^ {2} + \varphi_ {y} ^ {2}\right)}. \tag {S4}
+$$
+
+Finally, the pressure field is recovered in the spatial domain using the inverse FFT (iFFT):
+
+$$
+p = \mathrm {i F F T} [ p ^ {*} ]. \tag {S5}
+$$
+
+This spectral method offers an efficient approach to calculating the pressure field without the need for labeled data or training.
+
+BC encoding. To ensure that the solution obeys the given periodic boundary conditions and that the feature map shape remains unchanged after differentiation, we employ periodic BC padding (see Figure S2) in our architecture. This method of hard encoding padding not only guarantees that the boundary conditions are periodic, but also improves accuracy.
+
+![](images/1302c244713618e62dbd387f36fdd115dcbc01f75f4fe6282afa4b922c3c77e9.jpg)  
+Figure S1. Components of Physics Block. (a), Poisson block. (b), Poisson solver. (c),  $\mathbf{M}_{\mathrm{i}}\mathbf{N}\mathbf{N}$  block.
+
+![](images/4f998ea8e6e2c8cd1f19359f05fa29870b8b5e422b276320e3d3b7fadcf05ab1.jpg)
+
+![](images/fcae23255b14e44eb2ee4fa0f7c299ac44eedacbef7dcdfd9723fe7101e2d673.jpg)
+
+![](images/f12eff396927b126b9a47dc68171c238804f3f2597331d080e1ac12d037e3938.jpg)  
+Figure S2. Periodic BC padding.
+
+# B.3. RK4 Integration Scheme
+
+RK4 is a widely used numerical integration method for solving ordinary differential equations (ODEs) and PDEs, commonly employed as a time integration solver. It provides a balance between computational efficiency and accuracy by calculating intermediate slopes at various points within each time step. The general numerical integration method for time marching from  $\mathbf{u}_{t_j}$  to  $\mathbf{u}_{t_{j + 1}}$  can be written as:
+
+$$
+\mathbf {u} _ {j + 1} = \mathbf {u} _ {j} + \int_ {t _ {j}} ^ {t _ {j + 1}} \mathcal {B} \left(\mathbf {u} _ {j} \left(\tilde {\mathbf {x}}, \tau\right)\right) \mathrm {d} \tau . \tag {S6}
+$$
+
+Among them,  $\mathbf{u}_{j + 1}$  and  $\mathbf{u}_j$  are solutions at time  $j + 1$  and  $j$ . RK4 is a high-order integration scheme, which divides the time interval into multiple equally spaced small time steps to approximate the integral. The final update of the above state change can be written as:
+
+$$
+\begin{array}{l} \mathbf {r} _ {1} = \mathcal {B} \left(\mathbf {u} _ {j}, t _ {j}\right), \\ \mathbf {r} _ {2} = \mathcal {B} \left(\mathbf {u} _ {j} + \frac {\delta t}{2} \times \mathbf {r} _ {1}, t _ {j} + \frac {\delta t}{2}\right), \\ \mathbf {r} _ {3} = \mathcal {B} \left(\mathbf {u} _ {j} + \frac {\delta t}{2} \times \mathbf {r} _ {2}, t _ {j} + \frac {\delta t}{2}\right), \tag {S7} \\ \mathbf {r} _ {4} = \mathcal {B} \left(\mathbf {u} _ {j} + \delta t \times \mathbf {r} _ {3}, t _ {j} + \delta t\right), \\ \mathbf {u} _ {j + 1} = \mathbf {u} _ {j} + \frac {1}{6} \delta t (\mathbf {r} _ {1} + 2 \mathbf {r} _ {2} + 2 \mathbf {r} _ {3} + \mathbf {r} _ {4}), \\ \end{array}
+$$
+
+where  $\delta t$  denotes the step size and  $\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3, \mathbf{r}_4$  represent four intermediate variables (slopes). The global error is proportional to the step size to the fourth power, i.e.,  $\mathcal{O}(\delta t^4)$ .
+
+# C. Data Details
+
+KdV. The Korteweg-de Vries (KdV) equation is a well-known nonlinear PDE used to describe the movement of shallow water waves with small amplitude in a channel. It models the dynamics of these waves and is particularly noted for its ability to represent solitary waves, or solutons, given by:
+
+$$
+\frac {\partial u}{\partial t} + u \frac {\partial u}{\partial x} + \frac {\partial^ {3} u}{\partial x ^ {3}} = 0, \tag {S8}
+$$
+
+Table S1. Settings for generating datasets.  
+
+<table><tr><td>Parameters / Case</td><td>KdV</td><td>Burgers</td><td>GS</td><td>NSE</td></tr><tr><td>DNS Method</td><td>Spectral</td><td>FD</td><td>FD</td><td>FV</td></tr><tr><td>Spatial Domain</td><td>[0, 64]</td><td>[0, 1]2</td><td>[0, 1]2</td><td>[0, 2π]2</td></tr><tr><td>Calculate Grid</td><td>256</td><td>1002</td><td>1282</td><td>20482</td></tr><tr><td>Training Grid</td><td>64</td><td>252</td><td>322</td><td>642</td></tr><tr><td>Simulation dt (s)</td><td>1.00 × 10-2</td><td>1.00 × 10-3</td><td>2.00 × 10-3</td><td>2.19 × 10-4</td></tr><tr><td>Warmup (s)</td><td>0</td><td>0.1</td><td>0</td><td>40</td></tr><tr><td>Training data group</td><td>3</td><td>5</td><td>3</td><td>5</td></tr><tr><td>Testing data group</td><td>10</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Spatial downsample</td><td>4×</td><td>16×</td><td>16×</td><td>1024×</td></tr><tr><td>Temporal downsample</td><td>5×</td><td>10×</td><td>20×</td><td>128×</td></tr></table>
+
+where  $u = u(x,t)$  represents the wave amplitude as a function of position  $x$  and time  $t$ .  $u\frac{\partial u}{\partial x}$  accounts for the nonlinear effects, while  $\frac{\partial^3u}{\partial x^3}$  captures the dispersive effects in the system. The equation illustrates how a balance between these two effects leads to the formation of solitary waves, or solitons, that maintain their shape over long distances.
+
+To generate the dataset, we employ the method of lines (MOL) using pseudospectral methods to compute the spatial derivatives (Brandstetter et al., 2022a). The dataset is initially generated on a grid of 256 points and then downsampled to a grid of 64 points for numerical experiments. The simulation timestep is set to  $dt = 1 \times 10^{-2}$  seconds, with the total simulation duration set to 100 seconds. For training, we use 3 sets of data, each comprising 1000 timesteps over  $\Delta t = 5dt$ , along with ten additional testing sets with different ICs.
+
+Burgers. This equation models the behavior of a viscous fluid (Kumar, 2023), incorporating both nonlinear dynamics and diffusion effects. It finds extensive applications across various scientific disciplines, including fluid mechanics, materials science, applied mathematics and engineering. The equation is expressed as follows:
+
+$$
+\frac {\partial \mathbf {u}}{\partial t} = \nu \nabla^ {2} \mathbf {u} - \mathbf {u} \cdot \nabla \mathbf {u}, \quad t \in [ 0, T ], \tag {S9}
+$$
+
+where  $\mathbf{u} = \{u,v\} \in \mathbb{R}^2$  represents the fluid velocities,  $\nu$  is the viscosity coefficient set to 0.002, and  $\Delta$  is the Laplacian operator.
+
+As shown in Table S1, we generate the dataset using the finite difference method with a 4th-order Runge-Kutta time integration (Rao et al., 2023) and periodic boundary conditions over the spatial domain  $\mathbf{x} \in [0,1]$ . The data is initially generated on a  $100^{2}$  grid and subsequently downsampled to a  $25^{2}$  grid for use in numerical experiments. The simulation timestep is set to  $dt = 1 \times 10^{-3}$  seconds, with a total duration of  $T = 1.4$  seconds. During the training stage, we employ five trajectories with  $\Delta t = 10\delta t$ , each consisting of 140 snapshots. In the testing stage, we use ten different trajectories, each containing 140 snapshots.
+
+GS. The Gray-Scott (GS) reaction-diffusion model is a system of PDEs that describes the interaction and diffusion of two reacting chemicals. It is known for its ability to produce intricate and evolving patterns, making it a popular model for studying pattern formation. It is widely used in fields such as chemistry, biology, and physics to simulate processes like chemical reactions and biological morphogenesis. The equation is expressed by:
+
+$$
+\frac {\partial u}{\partial t} = D _ {u} \Delta u - u v ^ {2} + \alpha (1 - u), \tag {S10}
+$$
+
+$$
+\frac {\partial v}{\partial t} = D _ {v} \Delta v + u v ^ {2} - (\alpha + \kappa) v,
+$$
+
+where  $u$  and  $v$  denote the concentrations of two distinct chemical species, with  $D_{u}$  and  $D_{v}$  indicating their respective diffusion coefficients. The first equation models the change in the concentration of  $u$  over time. The term  $D_{u}\Delta u$  represents the diffusion of  $u$ ,  $-uv^2$  describes the reaction between  $u$  and  $v$ , and  $\alpha (1 - u)$  represents the replenishment of  $u$  based on the feed rate  $\alpha$ . The second equation models the evolution of  $v$ , where  $D_{v}\Delta v$  accounts for diffusion,  $uv^2$  represents the creation of  $v$  from the reaction with  $u$ , and  $-(\alpha +\kappa)v$  describes the decay of  $v$ , with  $\kappa$  as the decay rates.
+
+Table S2. Settings for generating the NSE datasets.  
+
+<table><tr><td>Dataset</td><td>Grid</td><td>Spatial Domain</td><td>Re</td><td>Warmup time</td><td>dt</td><td>Innerstep</td></tr><tr><td>f1 ~ f4</td><td>20482 → 642</td><td>(0, 2π)2</td><td>1000</td><td>40</td><td>2.1914×10-4</td><td>32</td></tr><tr><td>Re = 500</td><td>20482 → 642</td><td>(0, 2π)2</td><td>500</td><td>40</td><td>2.1914×10-4</td><td>32</td></tr><tr><td>Re = 800</td><td>20482 → 642</td><td>(0, 2π)2</td><td>800</td><td>40</td><td>2.1914×10-4</td><td>32</td></tr><tr><td>Re = 1000</td><td>20482 → 642</td><td>(0, 2π)2</td><td>1000</td><td>40</td><td>2.1914×10-4</td><td>32</td></tr><tr><td>Re = 1600</td><td>20482 → 642</td><td>(0, 2π)2</td><td>1600</td><td>40</td><td>2.1914×10-4</td><td>32</td></tr><tr><td>Re = 2000</td><td>20482 → 642</td><td>(0, 2π)2</td><td>2000</td><td>40</td><td>2.1914×10-4</td><td>32</td></tr><tr><td>Re = 4000</td><td>40962 → 642</td><td>(0, 2π)2</td><td>4000</td><td>40</td><td>1.0957×10-4</td><td>32</td></tr><tr><td>Re = 1000</td><td>40962 → 642</td><td>(0, 4π)2</td><td>1000</td><td>40</td><td>1.0957×10-4</td><td>32</td></tr></table>
+
+We also utilize the RK4 time integration method for dataset generation. In this case, we assign the values  $D_{u} = 2.0 \times 10^{-5}$ ,  $D_{v} = 5.0 \times 10^{-6}$ ,  $\alpha = 0.04$ , and  $\kappa = 0.06$ . The dataset is generated using the finite difference method on a  $128^{2}$  grid with periodic boundary conditions, spanning the spatial domain  $\mathbf{x} \in [0,1]^{2}$ . To generate different ICs, we first define a grid based on the spatiotemporal resolution and initialize the concentrations of two chemicals. By setting different random seeds and adding varied random noise, we create unique ICs. The simulation uses a timestep of  $dt = 0.5$  s over a total duration of  $T = 1400$  seconds. The data is then downsampled to a  $32^{2}$  grid, and the timestep is increased to 10 seconds ( $\Delta t = 20dt$ ) for ground truth creation. We utilize three training trajectories, each with 180 snapshots, and ten additional testing sets with varying ICs.
+
+NSE. The Navier-Stokes equations (NSE) are fundamental to the study of fluid dynamics, governing the behavior of fluid motion. In this paper, we focus on a two-dimensional, incompressible Kolmogorov flow with periodic boundary conditions, expressed in velocity-pressure form as:
+
+$$
+\begin{array}{l} \frac {\partial \mathbf {u}}{\partial t} + (\mathbf {u} \cdot \nabla) \mathbf {u} = \frac {1}{R e} \nabla^ {2} \mathbf {u} - \nabla p + \mathbf {f}, \quad t \in [ 0, T ], \tag {S11} \\ \nabla \cdot \mathbf {u} = 0, \\ \end{array}
+$$
+
+where  $\mathbf{u} = \{u,v\} \in \mathbb{R}^2$  denotes the fluid velocity vector,  $p \in \mathbb{R}$  represents the pressure, and  $Re$  is the Reynolds number that characterizes the flow regime. The Reynolds number serves as a scaling factor in the NSE, balancing the inertial forces, represented by the advection term  $(\mathbf{u} \cdot \nabla)\mathbf{u}$ , with the viscous forces, captured by the Laplacian term  $\Delta \mathbf{u}$ . When  $Re$  is low, the flow remains predominantly laminar and smooth due to the dominance of the viscous forces. Conversely, at high Reynolds numbers, the inertial forces take precedence, leading to a more chaotic and turbulent flow behavior.
+
+To create the dataset, we follow the approach outlined in JAX-CFD (Kochkov et al., 2021). We simulate data using the Finite Volume Method (FVM) on a fine grid with a time step of  $dt$  (e.g.,  $Re = 1000$ ,  $2048 \times 2048$ ). This data is then downsampled to a coarse grid with  $\Delta t = 128dt$  (e.g.,  $Re = 1000$ ,  $64 \times 64$ ) to serve as the ground truth. Different ICs are generated by introducing random noise into each component of the velocity field and subsequently filtering it to obtain a divergence-free field with the desired properties. For training, we utilize only five groups of labeled data with 4800 snapshots, while testing involves ten sets of trajectories. The model performance tests include trajectories with different Reynolds numbers  $Re = 500$ , 800, 1600, 2000, 4000, different external forces  $\mathbf{f}_1 = \cos(2y)\pmb{\eta}_x - 0.1\mathbf{u}$ ,  $\mathbf{f}_2 = 0$ $\mathbf{f}_3 = \cos(4y)\pmb{\eta}_x - 0.1\mathbf{u}$ ,  $\mathbf{f}_4 = \sin(4y)\pmb{\eta}_x - 0.4\mathbf{u}$ , and a larger computational domain  $\mathbf{x} \in (0,4\pi)^2$ . The detailed dataset parameters are shown in Table S2.
+
+# D. Additional Experimental Results
+
+# D.1. Less Data and Added Noise
+
+To evaluate our model's robustness against missing data and noise, we conducted experiments on the NSE using five sets of trajectories  $(5\times 1200\times 2\times 64\times 64)$ . We tested two conditions: (1) randomly removing  $20\%$  of snapshots and (2) adding  $0.1\%$  Gaussian noise during training. As shown in Table S3, the model's performance was only slightly impacted in Experiment 1, with low error rates. In Experiment 2, HCT remained above 8 s. These results highlight the model's strong generalization ability even under challenging conditions.
+
+Table S3. Performance metrics under different noise levels during training.  
+
+<table><tr><td>Training</td><td>RMSE</td><td>MAE</td><td>MNAD</td><td>HCT (s)</td></tr><tr><td>- 20% data</td><td>0.1935</td><td>0.0958</td><td>0.0113</td><td>8.1392</td></tr><tr><td>+ 0.1% noise</td><td>0.2083</td><td>0.1014</td><td>0.0123</td><td>8.0431</td></tr><tr><td>normal</td><td>0.1379</td><td>0.0648</td><td>0.0077</td><td>8.3566</td></tr></table>
+
+Table S4. Performance metrics for different NN blocks.  
+
+<table><tr><td>Model</td><td>RMSE</td><td>MAE</td><td>MNAD</td><td>HCT</td></tr><tr><td>Model-a</td><td>NaN</td><td>NaN</td><td>NaN</td><td>0.8846</td></tr><tr><td>Model-b</td><td>NaN</td><td>NaN</td><td>NaN</td><td>5.2930</td></tr><tr><td>Model-c</td><td>0.2575</td><td>0.1507</td><td>0.0191</td><td>7.2930</td></tr><tr><td>Model-d</td><td>0.1564</td><td>0.0703</td><td>0.0083</td><td>8.0525</td></tr><tr><td>Model-e</td><td>0.2479</td><td>0.1242</td><td>0.0197</td><td>7.6346</td></tr><tr><td>MultiPDENet</td><td>0.1379</td><td>0.0648</td><td>0.0077</td><td>8.3566</td></tr></table>
+
+# D.2. Parametric Experiments on  $\mathbf{M}_{\mathrm{i}}\mathbf{N}\mathbf{N}$  Block and  $\mathbf{M}_{\mathrm{a}}\mathbf{N}\mathbf{N}$  Block
+
+To investigate the role of the NN blocks in our model, we conducted additional comparative experiments with the following configurations:
+
+- Model-a: the  $\mathrm{M_iNN}$  Block was set to UNet and the  $\mathrm{M_aNN}$  Block to FNO;  
+- Model-b: both the  $\mathrm{M_iNN}$  Block and the  $\mathrm{M_aNN}$  Block were set to FNO;  
+- Model-c: both blocks were set to FNO with roll-out training applied at the macro step (with an unrolled step size of 8);  
+- Model-d: the  $\mathrm{M_iNN}$  Block was set to DenseCNN and the  $\mathrm{M_aNN}$  Block to UNet;  
+- Model-e: the  $\mathrm{M_iNN}$  Block was set to FNO while the  $\mathrm{M_aNN}$  Block was set to Swin Transformer (Liu et al., 2021).
+
+All other experimental settings were kept consistent, and the results are presented in Table S4. Model-a and Model-b encountered NaN values, which can be attributed to the  $\mathrm{M_aNN}$  Block requiring a model capable of robust predictions at the macro step. Without such a model, multi-step roll-out training (as in Model-c) becomes necessary to enhance the model's stability in long-term predictions. When a strong predictive module is employed at the macro step (e.g., UNet), the  $\mathrm{M_iNN}$  Block can be replaced with a more parameter-efficient model, such as DenseCNN (Model-d). Setting the  $\mathrm{M_aNN}$  Block to Swin Transformer resulted in a slight decrease in accuracy, which can be attributed to the relatively small size of our dataset, as the Swin Transformer typically excels on larger datasets.
+
+# D.3. Generalization Test on flow with  $Re = 4000$ .
+
+To further demonstrate the superior capability of our model, we conducted an additional experiment with a high Reynolds number  $Re = 4000$  (see details in Table S2) maintaining the experimental setup of Section 4.1. The result is shown in Figure 6.
+
+# D.4. Scaling against Data Size
+
+As shown in Figure S3, our testing results demonstrate that the model exhibits a scaling law behavior, with the RMSE gradually decreasing as the amount of training data increases. Our model adheres to this scaling law as well. Moreover, even in scenarios with limited data regimes (e.g., only five trajectories), the model achieves a low level of error, highlighting its robustness and ability to learn from limited data.
+
+![](images/96f42f44694444fda16158972003853de3d1deac397e6540bb3b7b40dd9fb73c.jpg)  
+Figure S3. Comparison of PeRCNN and MultiPDENet across various training set sizes on the Burgers equation. The  $x$ -axis represents data volume, defined as the product of trajectory timesteps and the number of trajectories (corresponding to 5, 8, 16, 32, and 64 trajectories, respectively). Dotted lines denote the linear interpolation.
+
+# E. Evaluation Metrics
+
+We employ several metrics to assess the performance of the tested models, including Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Normalized Absolute Difference (MNAD), and High Correction Time (HCT) (Sun et al., 2023). RMSE quantifies the average magnitude of the errors between predicted and actual values, providing insight into the model's accuracy. MAE assesses the average absolute deviation between predicted and observed values, thereby indicating the scale of the errors. MNAD serves as an important metric for evaluating the consistency of model outputs over time, calculating the average discrepancy across temporal data points and offering a normalized measure of prediction error relative to the range of actual data. HCT gauges the model's capability for making reliable long-term predictions. These metrics are defined as follows:
+
+$$
+\mathrm {R M S E} = \sqrt {\frac {1}{n} \sum_ {i = 1} ^ {n} \left\| \mathbf {H} _ {i} - \check {\mathbf {H}} _ {i} \right\| ^ {2}}, \tag {S12}
+$$
+
+$$
+\mathrm {M A E} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left| \mathbf {H} _ {i} - \check {\mathbf {H}} _ {i} \right|, \tag {S13}
+$$
+
+$$
+\mathrm {M N A D} = \frac {1}{n} \sum_ {i = 1} ^ {n} \frac {\left\| \mathbf {H} _ {i} - \check {\mathbf {H}} _ {i} \right\|}{\left\| \mathbf {H} _ {i} \right\| _ {\max } - \left\| \mathbf {H} _ {i} \right\| _ {\min }}, \tag {S14}
+$$
+
+$$
+\mathrm {H C T} = \sum_ {i = 1} ^ {N} \Delta t \cdot \left[ P C C \left(\mathbf {H} _ {i}, \tilde {\mathbf {H}} _ {i}\right) > 0. 8 \right], \tag {S15}
+$$
+
+where
+
+$$
+P C C \left(\mathbf {H} _ {i}, \check {\mathbf {H}} _ {i}\right) = \frac {\operatorname {c o v} \left(\mathbf {H} _ {i} , \check {\mathbf {H}} _ {i}\right)}{\sigma_ {\mathbf {H} _ {i}} \sigma_ {\check {\mathbf {H}} _ {i}}}. \tag {S16}
+$$
+
+Here,  $n$  represents the number of trajectories;  $\mathbf{H}_i$  denotes the ground truth for each trajectory;  $\dot{\mathbf{H}}_i$  indicates the spatiotemporal sequence predicted by the model. The term "cov" refers to the covariance function, while "  $\sigma$  " represents the standard deviation of the respective sequence. The Iverson bracket returns a value of 1 when the condition  $(PCC(\mathbf{H}_i, \dot{\mathbf{H}}_i) > 0.8)$  is satisfied and 0 otherwise. The variable  $N$  signifies the total number of time steps.
+
+# F. Baseline Models
+
+Fourier Neural Operator (FNO). The FNO (Li et al., 2021) combines neural networks with Fourier transforms to effectively capture both global and local features of system dynamics. The architecture has two key components: First, it applies Fourier transforms to the system state, performing convolution operations in the frequency domain to capture global information. This is followed by an inverse Fourier transform which maps the data back to the spatial domain. Secondly, convolutional layers are employed to extract local features directly from the system state. The outputs from both global and local components are then integrated through the application of activation functions, which ultimately yield the final prediction.
+
+PINO. PINO (Li et al., 2024c) shares the same architecture as FNO but incorporates physics-based constraints by embedding governing equations into the loss function. The loss function can be expressed as:  $\mathcal{L}(\lambda) = \mathcal{L}_{Eq} + \mathcal{L}_{Data}$ . The first term of the loss function,  $\mathcal{L}_{Eq}$ , represent an MSE loss computed using the analytical expressions of the dynamics, as defined in Equation 1. The second term  $\mathcal{L}_{Data}$  is the data loss, which was previously discussed in Section 4.1.
+
+PeRCNN. PeRCNN (Rao et al., 2023) integrates physics-based principles directly into the learning framework by embedding governing equations into the neural network structure. The architecture features multiple parallel CNNs that model polynomial relationships via feature map multiplications. This incorporation of physical laws improves the model's generalization and extrapolation capabilities, enabling accurate predictions in dynamic systems governed by complex equations.
+
+UNet. The UNet architecture (Ronneberger et al., 2015) adopts a symmetric encoder-decoder structure originally designed for computer vision tasks. The encoder compresses the input by applying multiple downsampling layers to capture hierarchical features at various scales. Conversely, the decoder gradually restores the original spatial resolution using upsampling operations. Skip connections bridge the encoder and decoder, directly transferring feature maps to retain fine-grained details. This design allows UNet to merge high-resolution spatial information with deeper, abstract features, achieving accurate reconstructions of both local and global structures.
+
+DeepONet. DeepONet (Lu et al., 2021) is designed to approximate operators and map inputs directly to outputs by leveraging neural networks. The architecture consists of two main components: the trunk network, which processes domain-specific information, and the branch network, which handles the input functions. This dual-structure approach enables the efficient learning of complex functional relationships and enhances the model's capability to capture detailed operator mappings across various applications.
+
+Learned Interpolation (LI). The LI (Kochkov et al., 2021) employs a finite volume approach enhanced with neural networks as a replacement for conventional polynomial-based interpolation schemes in computing velocity tensor product. The network adapts to the local flow conditions by learning a dynamic interpolation mechanism that can adjust to the characteristics of the flow. This enables LI to provide accurate fluid dynamics predictions even on coarse grids, improving computational efficiency while maintaining prediction fidelity.
+
+Temporal Stencil Modeling (TSM). TSM (Sun et al., 2023) addresses time-dependent partial differential equations (PDEs) in conservation form by integrating time-series modeling with learnable stencil techniques. It effectively recovers information lost during downsampling, enabling enhanced predictive accuracy. TSM is particularly advantageous for machine learning models dealing with coarse-resolution datasets.
+
+# G. Computational Details
+
+# G.1. Training Details
+
+All experiments (both training and inference) in this study were conducted on a single Nvidia A100 GPU (with 80GB memory) running on a server with an Intel(R) Xeon(R) Platinum 8380 CPU (2.30GHz, 64 cores). All model training efforts were performed on coarse grids (see Table 1).
+
+MultiPDENet. The MultiPDENet architecture employs the Adam optimizer with a learning rate of  $5 \times 10^{-3}$ . The model is trained over 1000 epochs with a batch size of 90. Detailed settings for the rollout timestep can be found in Table 1. Additionally, we use the StepLR scheduler to adjust the learning rate by a factor of 0.96 every 200 steps. The model hyperparameters are listed in Tables S5 and S6.
+
+Table S5. Overview of hyperparameters used in the  ${\mathrm{M}}_{\mathrm{i}}\mathrm{{NN}}$  Block.  
+
+<table><tr><td>Case</td><td>Hyperparameters</td><td>Value</td></tr><tr><td rowspan="8">NSE</td><td>Network</td><td>FNO (Li et al., 2021)</td></tr><tr><td>Layers</td><td>6</td></tr><tr><td>Modes</td><td>30</td></tr><tr><td>Width</td><td>30</td></tr><tr><td>Blocks</td><td>1</td></tr><tr><td>Padding</td><td>periodical</td></tr><tr><td>σ</td><td>GELU</td></tr><tr><td>Inputs</td><td>{ūm,k,Ξm(k)(p,∇hatū,∇2hatū,∇p,f,Re)}</td></tr><tr><td rowspan="8">Burgers</td><td>Network</td><td>FNO (Li et al., 2021)</td></tr><tr><td>Layers</td><td>4</td></tr><tr><td>Modes</td><td>12</td></tr><tr><td>Width</td><td>12</td></tr><tr><td>Blocks</td><td>1</td></tr><tr><td>Padding</td><td>periodical</td></tr><tr><td>σ</td><td>GELU</td></tr><tr><td>Inputs</td><td>{ūk,m}</td></tr><tr><td rowspan="8">GS</td><td>Network</td><td>FNO (Li et al., 2021)</td></tr><tr><td>Layers</td><td>6</td></tr><tr><td>Modes</td><td>12</td></tr><tr><td>Width</td><td>22</td></tr><tr><td>Blocks</td><td>1</td></tr><tr><td>Padding</td><td>periodical</td></tr><tr><td>σ</td><td>GELU</td></tr><tr><td>Inputs</td><td>{ūk,m}</td></tr><tr><td rowspan="8">KdV</td><td>Network</td><td>FNO (Li et al., 2021)</td></tr><tr><td>Layers</td><td>4</td></tr><tr><td>Modes</td><td>32</td></tr><tr><td>Width</td><td>64</td></tr><tr><td>Blocks</td><td>1</td></tr><tr><td>Padding</td><td>periodical</td></tr><tr><td>σ</td><td>ReLU</td></tr><tr><td>Inputs</td><td>{ūk,m}</td></tr></table>
+
+FNO. The architecture of the FNO network closely follows that presented in the original study (Li et al., 2021), with the main adjustment being the adaptation of its training methodology to an autoregressive framework. The training utilizes the Adam optimizer with a learning rate of  $1 \times 10^{-3}$  and a batch size of 20. Training is carried out for 1000 epochs, and the rollout timestep matches MultiPDENet.
+
+UNet. We implement the modern UNet architecture (Gupta & Brandstetter, 2023) using its default settings, ensuring that the rollout timestep is consistent with that of the MultiPDENet. The StepLR scheduler is employed with a step size of 100 and a gamma of 0.96. The optimizer is Adam, with a learning rate of  $1 \times 10^{-3}$  and a batch size of 10. The model is trained for 1000 epochs.
+
+DeepONet. We utilize the default configuration of DeepONet (Lu et al., 2021) along with the Adam optimizer. The learning rate is established at  $5 \times 10^{-4}$ , with a decay factor of 0.9 applied every 5000 steps. The model is trained using a batch size of 16 over a total of 20000 epochs.
+
+PeRCNN. We maintain the standard architecture of PeRCNN (Rao et al., 2023). The optimization process is executed with the Adam optimizer and employs a StepLR scheduler that reduces the learning rate by a factor of 0.96 every 100 steps. The initial learning rate is set to 0.01, and the training is conducted over 1000 epochs with a batch size of 32.
+
+Table S6. Overview of hyperparameters used in the  ${\mathrm{M}}_{\mathrm{a}}\mathrm{{NN}}$  Block.  
+
+<table><tr><td>Case</td><td>Hyperparameters</td><td>Value</td></tr><tr><td rowspan="6">NSE</td><td>Network</td><td>U-Net (Gupta &amp; Brandstetter, 2023)</td></tr><tr><td>Hidden size</td><td>[128, 128, 256, 512]</td></tr><tr><td>Blocks</td><td>2</td></tr><tr><td>Padding</td><td>periodical</td></tr><tr><td>Inputs</td><td>{u, Δt, dx}</td></tr><tr><td>σ</td><td>GELU</td></tr><tr><td rowspan="6">Burgers</td><td>Network</td><td>U-Net (Gupta &amp; Brandstetter, 2023)</td></tr><tr><td>Hidden size</td><td>[64, 64, 128, 256]</td></tr><tr><td>Blocks</td><td>2</td></tr><tr><td>Padding</td><td>periodical</td></tr><tr><td>Inputs</td><td>{u, Δt, dx}</td></tr><tr><td>σ</td><td>GELU</td></tr><tr><td rowspan="6">GS</td><td>Network</td><td>U-Net (Gupta &amp; Brandstetter, 2023)</td></tr><tr><td>Hidden size</td><td>[64, 64, 128, 256]</td></tr><tr><td>Blocks</td><td>2</td></tr><tr><td>Padding</td><td>periodical</td></tr><tr><td>Inputs</td><td>{u, Δt, dx}</td></tr><tr><td>σ</td><td>GELU</td></tr><tr><td rowspan="6">KdV</td><td>Network</td><td>U-Net (Gupta &amp; Brandstetter, 2023)</td></tr><tr><td>Hidden size</td><td>[64, 64, 128, 256]</td></tr><tr><td>Blocks</td><td>2</td></tr><tr><td>Padding</td><td>periodical</td></tr><tr><td>Inputs</td><td>{u, Δt, dx}</td></tr><tr><td>σ</td><td>ReLU</td></tr></table>
+
+LI. We adopt the default network architecture and parameter settings for LI (Kochkov et al., 2021). The optimizer used is Adam with  $\beta_{1} = 0.9$  and  $\beta_{2} = 0.99$ . The batch size is configured to 8, along with a global gradient norm clipping threshold of 0.01. The learning rate is set to  $1 \times 10^{-3}$ , and weight decay is configured to  $1 \times 10^{-6}$ .
+
+TSM. We follow the default network architecture and parameter settings for TSM (Sun et al., 2023). The initial learning rate is set to  $1 \times 10^{-4}$ , with a weight decay of  $1 \times 10^{-4}$ . The gradient clipping norm is configured to be  $1 \times 10^{-2}$ . We use the Adam optimizer with  $\beta_{2} = 0.98$ , and the batch size is set to 8.
+
+# G.2. Computational Cost (Inference)
+
+Taking NSE as an example, we compared the inference time, RMSE, and HCT of MultiPDENet with the Direct Numerical Simulation (DNS) method across three cases. The comparison principle is based on the time required to simulate the same trajectory length ( $T = 8.4$  s) under identical experimental conditions (a single A100 GPU). The inference time is measured from the moment the initial conditions (IC) are fed into the model until the trajectory of the same length is predicted.
+
+The DNS settings follow JAX-CFD (Kochkov et al., 2021). According to the CFL condition, the simulated time step  $(dt)$  varies with the resolution of the DNS method, resulting in different numbers of timesteps required for calculation. DNS 2048, DNS 4096, and DNS 4096 are used as the ground truth for the three cases, respectively. Detailed comparison results are presented in Table S7. The computational time for a given accuracy (e.g., correlation  $\geq 0.8$ ) on the NSE dataset is depicted in Table S8.
+
+Nevertheless, we also would like to clarify that the DNS code used above was implemented in JAX, while our model was programmed in PyTorch. These two platforms have distinct efficiencies even for the same model. Typically, the codes under JAX environment runs much faster compared with PyTorch (up to  $6\times$ ) (Takamoto et al., 2022). We anticipate to achieve much higher speedup of our model if also implemented and optimized in JAX, which is, however, out of the scope of the present study.
+
+Table S7. Performance comparison of different methods on the NSE dataset across various cases.  
+
+<table><tr><td>Case</td><td>Method</td><td>Timestep</td><td>Infer Cost (s)</td><td>RMSE</td><td>HCT (s)</td></tr><tr><td>Re = 1000</td><td>DNS 2048</td><td>38400</td><td>260</td><td>0</td><td>8.4</td></tr><tr><td>Re = 1000</td><td>DNS 1024</td><td>19200</td><td>135</td><td>0.1267</td><td>8.4</td></tr><tr><td>Re = 1000</td><td>DNS 512</td><td>9600</td><td>52</td><td>0.2674</td><td>6.5</td></tr><tr><td>Re = 1000</td><td>DNS 64</td><td>1200</td><td>18</td><td>0.7818</td><td>2.7</td></tr><tr><td>Re = 1000</td><td>MultiPDENet</td><td>300</td><td>26</td><td>0.1379</td><td>8.4</td></tr><tr><td>Re = 4000</td><td>DNS 4096</td><td>76800</td><td>1400</td><td>0</td><td>8.4</td></tr><tr><td>Re = 4000</td><td>DNS 1024</td><td>19200</td><td>136</td><td>0.1463</td><td>6.8</td></tr><tr><td>Re = 4000</td><td>DNS 512</td><td>9600</td><td>52</td><td>0.2860</td><td>5.8</td></tr><tr><td>Re = 4000</td><td>DNS 128</td><td>2400</td><td>31</td><td>0.8658</td><td>3.6</td></tr><tr><td>Re = 4000</td><td>MultiPDENet</td><td>300</td><td>26</td><td>0.1685</td><td>6.4</td></tr><tr><td>x ∈ (0,4π)2</td><td>DNS 4096</td><td>75750</td><td>1280</td><td>0</td><td>8.4</td></tr><tr><td>x ∈ (0,4π)2</td><td>DNS 1024</td><td>19200</td><td>129</td><td>0.4638</td><td>6.6</td></tr><tr><td>x ∈ (0,4π)2</td><td>DNS 512</td><td>9600</td><td>50</td><td>0.6166</td><td>5.2</td></tr><tr><td>x ∈ (0,4π)2</td><td>DNS 128</td><td>2400</td><td>30</td><td>0.8835</td><td>2.3</td></tr><tr><td>x ∈ (0,4π)2</td><td>MultiPDENet</td><td>300</td><td>26</td><td>0.4577</td><td>6.7</td></tr></table>
+
+Table S8. Computational time for a given accuracy (e.g., correlation  $\geq 0.8$ ) on the NSE dataset.  
+
+<table><tr><td>Item</td><td>Re = 1000</td><td>Re = 4000</td><td>x ∈[0,4π]2</td></tr><tr><td>DNS 1024</td><td>135 s</td><td>130 s</td><td>133 s</td></tr><tr><td>MultiPDENet</td><td>26 s</td><td>19 s</td><td>21 s</td></tr><tr><td>Speed up</td><td>5×</td><td>7×</td><td>6×</td></tr></table>

2025/MultiPDENet_ PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:db5104e63f202f6c3d960be74f3fe3e7ebeb805bef8434f351c7a0901484daea
+size 1605668

2025/Multiaccuracy and Multicalibration via Proxy Groups/5d6e9d73-a3b3-40a7-bc11-bbcb3bfb308c_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2f2743e78fbd30e39a27d7cfccc03b2117725ffa562b892ff58ab1439e982bfb
+size 758610

2025/Multiaccuracy and Multicalibration via Proxy Groups/full.md

@@ -0,0 +1,776 @@
+# Multiaccuracy and Multicalibration via Proxy Groups
+
+Beepul Bharti  $^{1,2}$  Mary Versa Clemens-Sewall  $^{3}$  Paul Yi  $^{4}$  Jeremias Sulam  $^{1,25}$
+
+# Abstract
+
+As the use of predictive machine learning algorithms increases in high-stakes decision-making, it is imperative that these algorithms are fair across sensitive groups. However, measuring and enforcing fairness in real-world applications can be challenging due to missing or incomplete sensitive group information. Proxy-sensitive attributes have been proposed as a practical and effective solution in these settings, but only for parity-based fairness notions. Knowing how to evaluate and control for fairness with missing sensitive group data for newer, different, and more flexible frameworks, such as multiaccuracy and multicalibration, remains unexplored. In this work, we address this gap by demonstrating that in the absence of sensitive group data, proxy-sensitive attributes can provably be used to derive actionable upper bounds on the true multiaccuracy and multicalibration violations, providing insights into a predictive model's potential worst-case fairness violations. Additionally, we show that adjusting models to satisfy multiaccuracy and multicalibration across proxy-sensitive attributes can significantly mitigate these violations for the true, but unknown, sensitive groups. Through several experiments on real-world datasets, we illustrate that approximate multiaccuracy and multicalibration can be achieved even when sensitive group data is incomplete or unavailable.
+
+$^{1}$ Department of Biomedical Engineering, Johns Hopkins University, Baltimore, USA  $^{2}$ Mathematical Institute of Data Science, Johns Hopkins University, Baltimore, USA  $^{3}$ Department of Applied Mathematics & Statistics, Johns Hopkins University, Baltimore, USA  $^{4}$ St. Jude Children's Research Hospital, Arlington, USA  $^{5}$ Department of Computer Science, Johns Hopkins University, Baltimore, USA. Correspondence to: Beepul Bharti <bbharti1@jhu.edu>, Jeremias Sulam <jsulam1@jhu.edu>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+# 1. Introduction
+
+Predictive machine learning algorithms are increasingly being used in high-stakes decision-making contexts such as healthcare (Shailaja et al., 2018), employment (Freire & de Castro, 2021), credit scoring (Thomas et al., 2017), and criminal justice (Rudin et al., 2020). Although these predictive models demonstrate impressive overall performance, growing evidence indicates that they can often exhibit biases and discriminate against certain sensitive groups (Obermeyer & Mullainathan, 2019; Dastin, 2022; Li et al., 2023). For instance, ProPublica's investigation (Angwin et al., 2022) revealed significant racial disparities in recidivism risk assessment algorithms, which disproportionately classified African Americans as high-risk for re-offending. As the deployment of these algorithms increases, regulatory bodies worldwide, including the US Office of Science and Technology Policy (OSTP) (of Science & Policy, 2022), European Union (Commission, 2021), and United Nations (United Nations Educational & , UNESCO), have emphasized the importance of ensuring that predictive algorithms avoid discrimination and uphold fairness.
+
+These concerns have led to the emergence of algorithmic fairness, a field dedicated to ensuring that predictive models do not inadvertently discriminate against sensitive groups defined by sensitive attributes such as race, age, or biological sex. Unfortunately, measuring and controlling a model's fairness can be challenging in many real-world settings, as sensitive group information is often incomplete or unavailable (Holstein et al., 2019; Garin et al.; Yi et al., 2025). In certain contexts, like healthcare, privacy and legal regulations such as the HIPAA Privacy Rule restrict access to sensitive data. In other cases, the information was not collected because it was considered unnecessary (Weissman & Hasnain-Wynia, 2011; Fremont et al., 2016; Zhang, 2018). Despite these obstacles, it remains crucial to evaluate a model's fairness before and during its deployment. This raises the question: how can we evaluate and promote fairness when sensitive group information is imperfect or missing altogether?
+
+One popular approach, widely applied in healthcare (Brown et al., 2016), finance (Zhang, 2018), and politics (Imai & Khanna, 2016) is to utilize proxy attributes in place of true attributes. Proxy methods have been immensely effective
+
+in evaluating and controlling for traditional parity-based notions of fairness (Diana et al., 2022; Bharti et al., 2024; Awasthi et al., 2021; Prost et al., 2021; Zhao et al., 2022; Awasthi et al., 2020; Gupta et al., 2018; Kallus et al., 2022), such as demographic parity (Calders et al., 2009), equalized odds (Hardt et al., 2016), and disparate mistreatment (Zafar et al., 2017), which all aim to equalize model statistics across protected groups.
+
+While enforcing parity is desirable in some settings, it can also lead to undesirable trade-offs. For instance, in breast cancer screening, incidence rates vary by age, with older women generally at higher risk than younger women (Kim et al., 2025). Thus, equalizing a model's false positive rates across age groups might reduce the sensitivity of cancer detection in older women, who are more likely to have the disease. Conversely, equalizing false negative rates might lead to unnecessary biopsies by increasing false positive rates in certain groups. Instead, a more appropriate fairness criterion would be to ask that the model's risk predictions approximately reflect true probabilities within each age group.
+
+These types of domain-specific challenges have led to the development of two newer fairness notions: multiaccuracy and multicalibration (Kim et al., 2019; Gopalan et al., 2022; Hébert-Johnson et al., 2018). Instead of enforcing parity, these methods ensure that model predictions are unbiased and well-calibrated across groups. They can be applied to complex, overlapping sensitive groups—such as those defined by race and gender—while maintaining high predictive accuracy and ensuring the model remains useful in practice, offering significant advantages over traditional parity-based metrics. As a result, enforcing multiaccuracy or multicalibration is powerful and often preferable in many contexts (Kim et al., 2019; Hébert-Johnson et al., 2018). However, a key challenge remains: when sensitive group data is missing, how can we build provably multiaccurate and multicalibrated models leveraging proxies?
+
+Tackling this issue is essential for developing models that are fair across multiple complex groups without sacrificing accuracy or utility. In this work, we address this gap. We study how to estimate multiaccuracy and multicalibration fairness violations without access to true sensitive attributes. We show that proxy-sensitive attributes can be used to derive computable upper bounds on these violations, capturing the model's worst-case fairness. Additionally, we demonstrate that post-processing a model to satisfy multiaccuracy or multicalibration across proxies effectively reduces the worst-case fairness violations, offering practical insights. In conclusion, we demonstrate that even when sensitive information is incomplete or inaccessible, proxies can greatly help in providing approximate multiaccuracy and multicalibration protections in a useful and meaningful way.
+
+# 1.1. Related Work
+
+Using proxy-sensitive attributes to measure and enforce model fairness has been extensively studied for various parity-based fairness notions.
+
+Measuring fairness. Measuring a model's true fairness through proxies has become an important area of research. Chen et al. (2019) were among the first to tackle this challenge by studying the error in measuring demographic parity using proxies derived from thresholding the Bayes optimal predictor for the sensitive attribute. Awasthi et al. (2021) focus on equalized odds, identifying key properties that proxies must satisfy to accurately estimate true equalized odds disparities. Kallus et al. (2022) further examine the ability to identify traditional parity-based fairness violations. They demonstrate that, under general assumptions about the distribution and classifiers, it is usually impossible to pinpoint fairness violations accurately using proxies. Additionally, by assuming access to the Bayes optimal predictor for the sensitive attribute, they provide tight upper and lower bounds on various fairness criteria, thereby characterizing the feasible regions for these violations. More generally, considering any proxy model instead of the Bayes optimal, Zhu et al. (2023) shows that estimating true parity-based fairness disparities using proxies results in errors proportional to the proxy error and the true fairness disparity. Most recently, Bharti et al. (2024) address a setting with more limited information compared to Zhu et al. (2023), providing computable and actionable upper bounds on true equalized odds disparities based on the proxy's misclassification error and proxy group-wise predictor statistics.
+
+Enforcing fairness. An equally important question is how to ensure fairness using proxies. Awasthi et al. (2020) examine the post-processing method for equalized odds (Hardt et al., 2016) when noisy proxies are used instead of true sensitive attributes. They show that, under conditional independence assumptions, using proxies in the post-processing method results in a predictor with reduced equalized odds disparity. Wang et al. (2020), working with a slightly different noise model, propose robust optimization approaches to train fair models using noisy sensitive features. Having proven that fairness violations are often unidentifiable, Kallus et al. (2022) take a different approach and focus on reducing the worst-case violations. Under additional smoothness assumptions they derive tighter feasible regions for fairness disparities, offering improved worst-case guarantees for fairness violations. More recently, Bharti et al. (2024) characterize the predictor that has optimal worst-case violations and provide a generalized version of Hardt et al. (2016)'s method that returns such a predictor. Taking a different perspective, Lahoti et al. (2020) avoid relying on proxies altogether. Instead, they propose solving a minimax optimization problem over a vast set of subgroups, reasoning
+
+that any good proxy for a sensitive feature would naturally be included in this set. Diana et al. (2022) address the problem of learning proxies that enable downstream model developers to train models that satisfy common parity-based fairness notions. They demonstrate that this entails constructing a multiaccurate proxy and introduce a general oracle-efficient algorithm to learn such proxies.
+
+# 1.2. Our Contributions
+
+There exists a rich line of work that studies how to evaluate and enforce parity-based notions of fairness when sensitive attribute data is missing via proxies. These, however, do not extend to settings where multiaccuracy and multicalibration are more appropriate, limiting their applicability in data-scare regimes. In this work, we address this issue. Our main contributions are the following:
+
+1. We study the problem of estimating multiaccuracy and multicalibration violations of a predictive model without access to sensitive group information.  
+2. We derive computable upper bounds for multiaccuracy and multicalibration violations using proxy-sensitive attributes.  
+3. We show that post-processing a model to satisfy multi-accuracy and multicalibration across proxies reduces the worst-case violations, allowing us to provide meaningful fairness guarantees without access to sensitive group data.
+
+Organization. The remainder of the paper is structured as follows. In Section 2, we introduce the necessary notation and formalize the setting. Section 3 provides background on multiaccuracy and multicalibration. Our main theoretical results are presented in Section 4 and Section 5, where we establish computable upper bounds on the multiaccuracy and multicalibration violations and demonstrate how to minimize them. Experimental results are detailed in Section 6. Finally, in Section 7 we discuss the implications of our work and provide closing remarks.
+
+# 2. Preliminaires
+
+Notation. We consider a binary classification setting  ${}^{1}$  with a data distribution  $\mathcal{D}$  supported on  $\mathcal{X} \times \mathcal{Z} \times \mathcal{Y}$ , where  $\mathcal{X} \subseteq \mathbb{R}^d$  and  $\mathcal{Z} \subseteq \mathbb{R}^K$  represent a  $d$ -dimensional feature space and  $K$ -dimensional sensitive group space, respectively, and  $\mathcal{Y} = \{0,1\}$  denotes the binary label space. For an individual represented by the pair  $(X,Z)$ ,  $X$  is a vector of features and  $Z$  is a vector of sensitive features (e.g., race, biological, age). We denote  $\mathcal{G} = \{g : \mathcal{X} \times \mathcal{Z} \mapsto \{0,1\}\}$  as the set
+
+of functions that define complex, potentially intersecting groups in  $\mathcal{X} \times \mathcal{Z}$ . For any  $g \in \mathcal{G}$ ,  $g(X,Z) = 1$  indicates that the individual  $(X,Z)$  belongs to group  $g$ . For example, let  $X_{1}$  be an individual's credit score and let  $Z_{1}$  and  $Z_{2}$  represent the individual's age and membership in the African American group. Then,  $g(X,Z) = \mathbf{1}\{X_{1} > 700 \wedge Z_{1} > 40 \wedge Z_{2} = 1\}$  specifies the group of all African Americans over the age of 40 with a credit score over 700. In this way, it is easy to define arbitrary overlapping groups defined by the intersection of basic attributes and other features. Finally, in our setting, a model is a function  $f: \mathcal{X} \to R$  that maps from the feature space to some discrete domain  $R \subseteq [0,1]$ . We denote its image as  $\operatorname{Im}(f) = \{f(X): X \in \mathcal{X}\}$  and assume that  $|\operatorname{Im}(f)| < \infty$ .
+
+Problem Setting. In this work, the primary objective is to assess whether a model  $f$  is fair with respect to a set of sensitive groups  $\mathcal{G}$  without having access to the functions in  $\mathcal{G}$  to determine group membership. Formally, and similar to previous work (Awasthi et al., 2021; Kallus et al., 2022), we consider a setting where we do not have access to samples  $(X,Z,Y)$  from the complete distribution  $\mathcal{D}$  – and thus we are unable to use the true set of grouping functions  $\mathcal{G}$  since their domain is supported  $\mathcal{X} \times \mathcal{Z}$ . Instead, we assume access to a sufficient number of samples  $(X,Y)$  from  $\mathcal{D}_{\mathcal{XY}}$ , the marginal distribution over  $\mathcal{X} \times \mathcal{Y}$ , which allows us to reliably evaluate the overall performance of the predictor  $f$  via its mean-squared error
+
+$$
+\operatorname {M S E} (f) = \underset {(X, Y) \sim \mathcal {D} _ {\mathcal {X} \mathcal {Y}}} {\mathbb {E}} [ (Y - f (X)) ^ {2} ]. \tag {1}
+$$
+
+Additionally, we assume a proxy developer that has access to samples  $(X,Z)$  from  $\mathcal{D}_{\mathcal{X}\mathcal{Z}}$ , the marginal distribution over  $\mathcal{X} \times \mathcal{Z}$ . They provide a set of learned proxy functions  $\hat{\mathcal{G}} = \{\hat{g} : \mathcal{X} \mapsto \{0,1\}\}$  for  $\mathcal{G}$  that only use features  $X$ , allowing us to determine proxy group membership. Moreover, via the proxy developer, we know how well any proxy  $\hat{g} \in \hat{\mathcal{G}}$  approximates its associated true  $g \in \mathcal{G}$  through its misclassification error,
+
+$$
+\operatorname {e r r} (\hat {g}) = \underset {(X, Z) \sim \mathcal {D} _ {\mathcal {X} \mathcal {Z}}} {\mathbb {P}} [ \hat {g} (X) \neq g (X, Z) ]. \tag {2}
+$$
+
+Through this setup, we are modeling real-world situations where we lack information about individuals' basic sensitive attributes, such as sex and race, preventing us from accurately identifying individuals' membership in complex intersecting groups (for example, white women over the age of 40). Instead, we rely on proxies to (approximately) represent all groups. Importantly, we do not make stringent, unverifiable assumptions about these proxy functions, unlike other studies (Prost et al., 2021; Awasthi et al., 2020; 2021); we only consider knowing their error rates,  $\mathbf{err}(\hat{g})$ .
+
+With our setting fully described, we now turn to our main objective: assessing the fairness of the model  $f$  with re
+
+spect to  $\mathcal{G}$ . In this work we focus on two fairness concepts—multiaccuracy and multicalibration—which we now formally define.
+
+# 3. Multiaccuracy and Multicalibration
+
+Multiaccuracy. Multiaccuracy (MA) is a notion of fairness originally introduced by (Kim et al., 2019; Hébert-Johnson et al., 2018). For any sensitive group  $g \in \mathcal{G}$ , MA evaluates the bias of a model  $f$ , conditional on membership in  $g$  via
+
+$$
+\mathsf {A E} _ {\mathcal {D}} (f, g) = \left| \mathbb {E} _ {\mathcal {D}} [ g (X, Z) (f (X) - Y) ] \right| \tag {3}
+$$
+
+and requires that  $\mathsf{A}\mathsf{E}_{\mathcal{D}}(f,g)$  be small for all groups  $g\in \mathcal{G}$ .
+
+Definition 3.1 (Multiaccuracy (Kim et al., 2019)). Fix a distribution  $\mathcal{D}$  and let  $\mathcal{G}$  be a set of groups. A model  $f$  is  $(\mathcal{G},\alpha)$ -multiaccurate if
+
+$$
+\mathsf {A E} _ {\mathcal {D}} ^ {\max } (f, \mathcal {G}) := \max  _ {g \in \mathcal {G}} \mathsf {A E} _ {\mathcal {D}} (f, g) \leq \alpha \tag {4}
+$$
+
+$(\mathcal{G}, \alpha)$ -MA requires that the predictions of  $f$  be approximately unbiased overall and on every group. Building on this, (Hébert-Johnson et al., 2018) introduced a stronger notion of group fairness known as multicalibration (MC), which demands unbiased and calibrated predictions. Central to evaluating MC is the expected calibration error (ECE) for a group  $g \in \mathcal{G}$
+
+$$
+\mathsf {E C E} _ {\mathcal {D}} (f, g) = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ g (X, Z) (f (X) - Y) | f (X) = v ] \right|
+$$
+
+where  $\mathcal{D}_f$  is the distribution of the predictions made by  $f$  under  $\mathcal{D}$ . MC requires that  $\mathsf{ECE}_{\mathcal{D}}(f,g)$  be small for all groups  $g\in \mathcal{G}$ .
+
+Definition 3.2 (Multicalibration (Hébert-Johnson et al., 2018)). Fix a distribution  $\mathcal{D}$  and let  $\mathcal{G}$  be a set of groups. A model  $f$  is  $(\mathcal{G},\alpha)$ -multicalibrated if
+
+$$
+\mathsf {E C E} _ {\mathcal {D}} ^ {\max } (f, \mathcal {G}) := \max  _ {g \in \mathcal {G}} \mathsf {E C E} _ {\mathcal {D}} (f, g) \leq \alpha . \tag {5}
+$$
+
+This is an  $l_{1}$  notion of MC as studied in (Gopalan et al., 2022). There also exist  $l_{2}$  (Globus-Harris et al., 2023b) and  $l_{\infty}$  (Hébert-Johnson et al., 2018) variants.  $(\mathcal{G},\alpha)$ -MC requires that  $f$ ’s predictions be approximately calibrated on all groups defined by  $\mathcal{G}$ . Note, MC is stronger than MA because intuitively MC requires MA on every level set of  $f$ .
+
+Having presented these definitions, the problem we face is now clear: Ideally, we would like to evaluate  $\mathsf{A}\mathsf{E}_{\mathcal{D}}^{\max}$  and  $\mathsf{E}\mathsf{C}\mathsf{E}_{\mathcal{D}}^{\max}$ . However, this requires access to samples  $(X,Z,Y)\sim \mathcal{D}$  and the functions  $\mathcal{G}$ , neither of which we assume to have. As alluded to before, no method exists that
+
+can guarantee—let alone, correct—that a predictor is  $(\mathcal{G},\alpha)$ -MA/MC in the absence of ground truth groups  $\mathcal{G}$ . Fortunately, in the following sections, we demonstrate that with proxies, one can successfully circumvent these limitations and still provide meaningful guarantees.
+
+# 4. Bounds on Multigroup Fairness Violations
+
+We will now demonstrate that is still possible to derive computable and useful upper bounds for the MA and MC violations of a model  $f$  across the true groups  $\mathcal{G}$ , even in the absence of true group information.
+
+Our first result provides computable upper bounds on  $\mathsf{A}\mathsf{E}_{\mathcal{D}}(f,g)$  and  $\mathsf{ECE}_D(f,g)$  for any group  $g$
+
+Lemma 4.1. Fix a distribution  $\mathcal{D}$  and model  $f$ . For any group  $g$  and its corresponding proxy  $\hat{g}$ ,
+
+$$
+\mathsf {A E} _ {\mathcal {D}} (f, g) \leq \mathsf {F} (f, \hat {g}) + \mathsf {A E} _ {\mathcal {D}} (f, \hat {g}) \tag {6}
+$$
+
+$$
+\mathsf {E C E} _ {\mathcal {D}} (f, g) \leq \mathsf {F} (f, \hat {g}) + \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) \tag {7}
+$$
+
+where
+
+$$
+\mathsf {F} (f, \hat {g}) = \min  \left(\operatorname {e r r} (\hat {g}), \sqrt {\operatorname {M S E} (f) \cdot \operatorname {e r r} (\hat {g})}\right). \tag {8}
+$$
+
+Furthermore the upper bounds are tight.
+
+A proof of this result is provided in Appendix B.1. We now make a few remarks. First, the upper bounds on the true MA/MC violations are tight: there exist non-trivial joint distributions over  $(f(X), g(X), \hat{g}(X), Y)$ —specifically, those for which  $\mathsf{err}(\hat{g}) > 0$  and  $\mathsf{MSE}(f) > 0$ —such that the bounds are attained with equality. Please see Appendix B.1 for a comprehensive discussion regarding these statements. Second, in our setting, both bounds can be directly evaluated because (1) we know  $\mathsf{err}(\hat{g})$  for all  $g \in \mathcal{G}$  via the proxy developer, who has access to  $\mathcal{G}$  and a sufficient number of samples  $(X, Z)$  from  $\mathcal{D}_{\mathcal{XZ}}$  to compute  $\mathsf{err}(\hat{g})$ ; and (2) we can reliably compute  $\mathsf{MSE}(f)$ ,  $\mathsf{AE}_{\mathcal{D}}(f, \hat{g})$ , and  $\mathsf{ECE}_{\mathcal{D}}(f, \hat{g})$  because we have access to a sufficient number of samples  $(X, Y)$  from  $\mathcal{D}_{\mathcal{XY}}$ .
+
+This result aligns with intuition, showing how the relationship between a proxy  $\hat{g}$  and the model  $f$  constrains the maximum possible values of  $\mathsf{AE}_{\mathcal{D}}(f,g)$  and  $\mathsf{ECE}_{\mathcal{D}}(f,g)$ . If the proxy  $\hat{g}$  is highly accurate in that it predicts the true group  $g$  better than  $f$  predicts the true label  $Y$ , i.e. if
+
+$$
+\operatorname {e r r} (\hat {g}) <   \operatorname {M S E} (f), \tag {9}
+$$
+
+then  $\mathsf{F}(f,\hat{g}) = \mathsf{err}(\hat{g})$ , and the true violations  $\mathsf{A}\mathsf{E}_{\mathcal{D}}(f,\hat{g})$  and  $\mathsf{ECE}_{\mathcal{D}}(f,\hat{g})$  are approximately bounded by their proxy estimates. Conversely, if  $f$  is highly accurate in predicting the label  $Y$  but the proxy is weaker, so that
+
+$$
+\operatorname {M S E} (f) <   \operatorname {e r r} (\hat {g}), \tag {10}
+$$
+
+# Algorithm 1 Multiaccuracy Regression
+
+1: Input: Initial model  $f$  and set of groups  $\mathcal{G}$  
+2: Solve
+
+$$
+\hat{f}\in \operatorname *{arg  min}_{\lambda \in \mathbb{R}^{|\mathcal{G}|}}\operatorname {MSE}(\hat{f})
+$$
+
+$$
+s. t. \hat {f} (X, Z) = f (X) + \sum_ {g \in \mathcal {G}} \lambda_ {g} \cdot g (X, Z)
+$$
+
+3: Return  $\hat{f}$
+
+then  $\mathsf{F}(f,\hat{g}) = \sqrt{\mathsf{MSE}(f)\cdot\mathsf{err}(\hat{g})}$ , and one can attain a better bound by considering a factor of  $\sqrt{\mathsf{MSE}(f)\cdot\mathsf{err}(\hat{g})}$  instead of  $\mathsf{err}(\hat{g})$ . Naturally, as  $\mathsf{MSE}(f)$  decreases, the maximum possible values of  $\mathsf{AE}_{\mathcal{D}}(f,\hat{g})$  and  $\mathsf{ECE}_{\mathcal{D}}(f,\hat{g})$  decreases as well.
+
+Most importantly, with this result, we can provide an upper bound for the true MA and MC violations of  $f$  across  $\mathcal{G}$ .
+
+Theorem 4.2. Fix a distribution  $\mathcal{D}$  and model  $f$ . Let  $\mathcal{G}$  be a set of true groups and  $\hat{\mathcal{G}}$  be its associated set of proxies. Then,  $f$  is  $(\mathcal{G}, \beta(f, \hat{\mathcal{G}}))$ -MA and  $(\mathcal{G}, \gamma(f, \hat{\mathcal{G}}))$ -MC where
+
+$$
+\beta (f, \hat {\mathcal {G}}) = \max  _ {\hat {g} \in \hat {\mathcal {G}}} \mathsf {F} (f, \hat {g}) + \mathsf {A E} _ {\mathcal {D}} (f, \hat {g}) \tag {11}
+$$
+
+$$
+\gamma (f, \hat {\mathcal {G}}) = \max  _ {\hat {g} \in \mathcal {G}} \mathsf {F} (f, \hat {g}) + \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) \tag {12}
+$$
+
+This result directly follows from Lemma 4.1, with a complete proof provided in Appendix B.2. It is particularly valuable because, even without directly evaluating the quantities of interest,  $\mathsf{AE}_{\mathcal{D}}^{\max}(f,\mathcal{G})$  and  $\mathsf{ECE}_{\mathcal{D}}^{\max}(f,\mathcal{G})$ , we can still evaluate these worst-case violations, which offers practical utility. For instance, if we need to ensure that  $f$  is  $(\mathcal{G},\alpha)$ -MC before deployment, we can proceed confidently even without direct access to  $\mathcal{G}$ , provided that  $\gamma (f,\hat{\mathcal{G}}) < \alpha$ . Conversely, if the worst-case violations are large, this suggests that  $f$  may potentially be significantly biased or uncalibrated for certain groups  $g$ .
+
+In a scenario where the worst-case violations are large, as model developers, we should pause deployment and ask: if it is the case that  $\beta (f,\hat{\mathcal{G}})$  or  $\gamma (f,\hat{\mathcal{G}})$  are large, can we reduce them such that we can provide better guarantees on the worst-case MA and MC violations of  $f$ ? We show that it is possible in the following section.
+
+# 5. Reducing Worst-Case Violations
+
+The results from the previous section allow us to upper bound the MA and MC violations using proxies. Now, we show that these violations can be provably reduced, yielding stronger worst-case guarantees. Recall that we have a fixed set of proxies  $\hat{\mathcal{G}}$ , a model  $f$ , and access to samples  $(X,Y)$ .
+
+# Algorithm 2 Multicalibration Boosting
+
+1: Input: Initial model  $f$ , set of groups  $G$ , and  $\alpha > 0$  
+2: Let  $m = \lceil \frac{1}{\alpha} \rceil, t = 0, f_0 := f$  
+3: while
+
+$$
+\max _ {g \in \mathcal {G}} \mathbb {P} [ g (X, Z) = 1 ] \cdot \mathbb {E} [ \Delta_ {v, g} ^ {2} | g (X, Z) = 1 ] > \alpha
+$$
+
+4: Set
+
+$$
+p _ {g, v} = \mathbb {P} [ g (X, Z) = 1, f _ {t} (X) = v ]
+$$
+
+$(v_{t},g_{t}) = \underset {v\in \left[\frac{1}{m}\right],g\in \mathcal{G}}{\arg \max}p_{g,v}\cdot \Delta_{v,g}^{2}\geq \alpha$
+
+$$
+S _ {v _ {t}, g _ {t}} = \{X \in \mathcal {X}: f _ {t} (X) = v, g _ {t} (X, Z) = 1 \}
+$$
+
+$$
+\tilde {v} _ {t} = \mathbb {E} [ Y | f _ {t} (X) = v _ {t}, g _ {t} (X, Z) = 1 ]
+$$
+
+$$
+v_{t}^{\prime} = \operatorname *{arg  min}_{v\in \left[ \begin{array}{c}1\\ m \end{array} \right]}|\tilde{v}_{t} - v|
+$$
+
+$$
+f _ {t + 1} (X) = \left\{ \begin{array}{l l} v _ {t} ^ {\prime}, & \text {i f X \in S _ {v _ {t} , g _ {t}}} \\ f _ {t} (X) & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+$$
+t = t + 1
+$$
+
+5: end while
+
+6: Let  $T \coloneqq t, \hat{f} \coloneqq f_T$  
+7: Return  $\hat{f}$
+
+Within the bounds  $\beta (f,\hat{\mathcal{G}})$  and  $\gamma (f,\hat{\mathcal{G}})$ , there are only two quantities we can modify by adjusting  $f$ : the mean squared error  $\mathrm{MSE}(f)$  and either  $\mathsf{A}\mathsf{E}_{\mathcal{D}}(f,\hat{g})$  or  $\mathsf{E}\mathsf{C}\mathsf{E}_{\mathcal{D}}(f,\hat{g})$ . Since we lack access to the true grouping functions  $\mathcal{G}$  and samples  $(X,Z)\sim \mathcal{D}_{\mathcal{XZ}}$ , we cannot reduce the proxy errors  $\mathsf{err}(\hat{g})$ . Thus, the question is, what modifications can be made to  $f$  such that the updated  $\hat{f}$  has smaller bounds? We answer this question for the bound on MC in the following theorem.
+
+Theorem 5.1. Fix a distribution  $\mathcal{D}$ , initial model  $f$ , and set of proxy groups  $\hat{\mathcal{G}}$ . If a model  $\hat{f}$  satisfies
+
+$$
+\mathsf {E C E} _ {\mathcal {D}} ^ {\max } (\hat {f}, \hat {\mathcal {G}}) <   \min  _ {\hat {g} \in \hat {\mathcal {G}}} \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) \tag {13}
+$$
+
+$$
+\operatorname {M S E} (\hat {f}) \leq \operatorname {M S E} (f) \tag {14}
+$$
+
+then, it will have a smaller worst-case MC violation, i.e.
+
+$$
+\gamma (\hat {f}, \hat {\mathcal {G}}) \leq \gamma (f, \hat {\mathcal {G}}). \tag {15}
+$$
+
+An identical result for MA is given in Appendix A and proofs for both results are provided in Appendix B.3. This results simply states that if we can obtain a new model  $\hat{f}$  that 1) is  $(\hat{\mathcal{G}},\alpha)$ -MC at level  $\alpha = \min_{\hat{g}\in \hat{\mathcal{G}}}\mathsf{ECE}_{\mathcal{D}}(f,\hat{g})$  and 2) has smaller MSE, then it is guaranteed to have a smaller worst-case violation. Fortunately, both objectives can be nearly achieved using Algorithm 1, proposed by Gopalan et al. (2022), and Algorithm 2, introduced by Roth (2022), which produce MC and MC predictors, respectively.
+
+<table><tr><td>Group</td><td>err(h)</td></tr><tr><td>Black Women</td><td>0.027</td></tr><tr><td>White Women</td><td>0.122</td></tr><tr><td>Asian</td><td>0.060</td></tr><tr><td>Seniors</td><td>0.000</td></tr><tr><td>Women</td><td>0.000</td></tr><tr><td>Multiracial</td><td>0.047</td></tr></table>
+
+Table 1. Proxy errors for ACSIncome  
+
+<table><tr><td>Group</td><td>err(h)</td></tr><tr><td>Black Women</td><td>0.005</td></tr><tr><td>White Women</td><td>0.046</td></tr><tr><td>Asian</td><td>0.000</td></tr><tr><td>Multiracial</td><td>0.000</td></tr><tr><td>Black Adults</td><td>0.000</td></tr><tr><td>Women</td><td>0.079</td></tr></table>
+
+Table 2. Proxy errors for ACSPubCov  
+
+<table><tr><td>Group</td><td>err(h)</td></tr><tr><td>Women</td><td>0.027</td></tr><tr><td>White</td><td>0.092</td></tr><tr><td>Asian</td><td>0.068</td></tr><tr><td>Black</td><td>0.039</td></tr><tr><td>Asian Men</td><td>0.039</td></tr><tr><td>Black Women</td><td>0.020</td></tr></table>
+
+Table 3. Proxy errors for CheXpert
+
+Algorithm 1 outlines a simple algorithm for MA. The following theorem establishes that the algorithm produces a model  $\hat{f}$  that satisfies MA while also guaranteeing an improvement or no deterioration in MSE.
+
+Theorem 5.2. Fix a distribution  $\mathcal{D}$ , predictor  $f$ , and set of groups  $\mathcal{G}$ . Algorithm 1 returns a model  $f$  that is  $(0, \mathcal{G})$ -MA. Moreover,
+
+$$
+\operatorname {M S E} (\hat {f}) \leq \operatorname {M S E} (f). \tag {16}
+$$
+
+A proof of this result can be found in (Detommaso et al., 2024), with finite-sample guarantees discussed in (Roth, 2022). Algorithm 1 updates the model  $\hat{f}$  by solving a standard linear regression problem, where the features are the predictions of the initial model  $f$  and grouping functions  $g$ .
+
+While Algorithm 1 solves a optimization problem to generate a MA model in a single step, Algorithm 2 is an iterative method to ensure MC. Algorithm 2 starts by checking if  $f$  is  $\alpha$ -MC via the group average squared calibration error
+
+$$
+\mathbb {E} \left[ \Delta_ {v, g} ^ {2} \mid g (X, Z) = 1 \right] \tag {17}
+$$
+
+where  $\Delta_{v,g} = \mathbb{E}[Y - f(X)|f(X) = v, g(X,Z) = 1]$ . If this exceeds  $\alpha$ , it identifies the conditioning event where the calibration error is the largest and refines  $f$ 's predictions. It iterates like this until convergence and this process returns a new model  $\hat{f}$  that is  $\alpha$ -MC and has an MSE that is close, potentially even lower, than the MSE of the initial model  $f$ .
+
+Theorem 5.3. Fix a distribution  $\mathcal{D}$ , predictor  $f$  and set of groups  $\mathcal{G}$ . Algorithm 2 stops after  $T < \frac{4}{\alpha^2}$  rounds and returns model  $\hat{f}$  that is  $(\sqrt{\alpha}, \mathcal{G})$ -MC. Moreover,
+
+$$
+\operatorname {M S E} (\hat {f}) \leq \operatorname {M S E} (f) + (1 - T) \frac {\alpha^ {2}}{4} + \alpha \tag {18}
+$$
+
+A proof, along with with finite-sample guarantees can be found in (Globus-Harris et al., 2023a; Roth, 2022).
+
+Having introduced these algorithms for obtaining multiaccurate and multicalibrated models  $\hat{f}$ , we now have a direct path to reducing worst-case violations. By applying Algorithm 1 or Algorithm 2 (at an appropriate level  $\alpha$ ) to our initial model  $f$  using the proxies  $\hat{\mathcal{G}}$  that is, enforcing multiaccuracy or multicalibration with respect to the proxies—we
+
+can systematically reduce worst-case violations on the true groups  $\mathcal{G}$ . This simple yet effective approach ensures allows us to still provide meaningful fairness guarantees and reliable predictions across sensitive subpopulations.
+
+# 6. Experimental Results
+
+We illustrate various aspects of our theoretical results on two tabular datasets, ACSIncome and ACSPublicCoverage (Ding et al., 2021), as well as on the CheXpert medical imaging dataset (Irvin et al., 2019). For the ACS datasets, we use a fixed  $10\%$  of the samples as the evaluation set. The remaining  $90\%$  of the data is split into training and validation sets, with  $60\%$  used for training the model  $f$  and proxies  $\hat{\mathcal{G}}$  and  $30\%$  for adjusting  $f$ . All reported results are averages over five train/validation splits on the evaluation set. For CheXpert, we use the splits provided by (Glocker et al., 2023) for training, calibration, and evaluation. The results and metrics are computed on the evaluation set. We report results for MC in the main body of the paper and defer those for MA to Appendices C and E because MC is a stronger notion that implies MA. The code necessary to reproduce these experiments is available at https://github.com/Sulam-Group/proxy_ta-mc.
+
+# 6.1. ACS Experiments
+
+For the two following tabular data experiments we use the ACS dataset, a larger version of the UCI Adult dataset. In particular, we use the 2018 California data, which contains approximately 200,000 samples. We follow Hansen et al. (2024) and define multiple sensitive groups  $\mathcal{G}$  using basic sensitive attributes  $Z$  (e.g., sex and race, which model developers aim not to discriminate towards), along with certain features  $X$  (e.g., age). Examples of groups  $g \in \mathcal{G}$  include white women and black adults. For both experiments, we simulate missing sensitive attributes by excluding some  $Z_{i}$  from the data we use to train our predictive model  $f$ . Instead, with an auxiliary dataset of samples  $(X,Z)$  and the true set of grouping functions  $\mathcal{G}$ , we obtain a set of proxy functions  $\hat{\mathcal{G}}$  to approximate  $\mathcal{G}$ .
+
+For our initial model  $f$ , we report the worst-case violations, which we can evaluate in our setting. To demonstrate that
+
+![](images/1d999dd4569f63b71a0b7b4d148be043a5f8f00337457df009dd9192574e0cab.jpg)  
+(a) ECE,  $\mathsf{ECE}^{\mathrm{max}}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACS Income. Here,  $f$  is a decision tree.
+
+enforcing MA and MC with respect to  $\hat{\mathcal{G}}$  provably reduces our upper bounds, we apply Algorithm 1 and Algorithm 2 to obtain an adjusted predictor  $f_{\mathrm{adj}}$  and report its worst-case violations as well. Additionally, for both the initial model  $f$  and adjusted model  $f_{\mathrm{adj}}$  we report the AE and ECE, with respect to the true groups  $g \in \mathcal{G}$ , along with their maximums,  $\mathsf{AE}^{\max}$  and  $\mathsf{ECE}^{\max}$ . Recall that in our setting, we cannot actually evaluate these quantities but we report them to showcase that they lie under our bounds, illustrating the validity of our theoretical results. For these tabular experiments, we model  $f$  with a logistic regression model, a decision tree, and Random Forest. We report the results for the decision tree in the following sections and defer the remainder to Appendices C and E.
+
+# 6.1.1.ACSINCOME
+
+For this experiment, we consider the task of predicting whether working adults living in California have a yearly income that exceeds $50,000. Examples of features  $X$  include occupation and education. To simulate missing sensitive attributes, we exclude the race attribute.
+
+In Table 1, we report the errors of the learned proxies  $\hat{\mathcal{G}}$  for specific groups. For groups that do not depend on race, such as seniors and women, their respective proxies  $\hat{g}$  are perfectly accurate, exhibiting zero misclassification error. However, for groups like multiracial and white women, the proxies exhibit some error, albeit small. This proves to be useful in providing meaningful guarantees on how multiaccurate and multicalibrated the model  $f$  may potentially be with respect to the true (but unobserved) groups  $\mathcal{G}$ .
+
+Figure 1a showcases the utility of our bounds. Notably, the worst-case violation (dotted red line) allows us to certify that the initial model is approximately 0.21-multicalibrated with respect to the true groups  $\mathcal{G}$ . This is indeed practically useful as it enables practitioners to obtain a certificate on the MC violation without having access to the true sensitive group information. Additionally, the right-hand graph in Figure 1a
+
+![](images/ff8e9b063032c322025b3c5df2197b28b4cd62ac599f16cadda656b4f80d200c.jpg)  
+(b) ECE,  $\mathsf{ECE}^{\mathrm{max}}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and updated model  $f_{\mathrm{adj}}$  on ACSPubcov. Here,  $f$  is a decision tree.
+
+![](images/468eea2b18eb88880a7bff53f6d865692d8328a1b7ccf392e77c8bbb32308e3e.jpg)
+
+highlights the benefit of applying Algorithm 2 and multicalibrating the initial model  $f$  with respect to the proxy groups  $\hat{\mathcal{G}}$ . After adjusting  $f$ , the upper bound decreases, allowing us to certify that the resulting model  $f_{\mathrm{adj}}$  is approximately 0.13-multicalibrated—a substantial improvement of  $38\%$ .
+
+# 6.1.2. ACSPUBLICCOVERAGE (ACSPUBCOV)
+
+In this experiment, we consider the task of predicting whether low-income individuals (< $30,000), not eligible for Medicare, have coverage from public health insurance. Examples of features X include age, education, income, and more. To simulate missing sensitive attributes, we exclude the sex attribute.
+
+In Table 2, we report the errors of the learned proxies  $\hat{\mathcal{G}}$  for specific groups. Notably, for groups independent of sex, such as asian and multiracial, the proxies  $\hat{g}$  are perfectly accurate, exhibiting zero misclassification error. However, for groups like black women and white women, the proxies exhibit some error, though they are small. This arises because, although the proxies functions  $\hat{g}$  are not explicit functions of sex attribute, there exists a feature, fertility, that indicates whether an individual has given birth within the past 12 months and serves as a good predictor of sex. This is a prime example of a real-world setting where, even though sensitive attributes may be missing, strong proxies can still enable us to determine true sensitive group membership with high accuracy.
+
+In Figure 1b, we show the results of applying Algorithm 2 to multicalibrate the initial model  $f$  with respect to the proxies  $\hat{\mathcal{G}}$ . Notably, our approach allows us to certify that the initial model is approximately 0.25-MC with respect to the true groups  $\mathcal{G}$  despite not having access to them. This result highlights the utility of our method, as it enables practitioners to obtain performance guarantees without needing the true group information. Furthermore, after applying Algorithm 2 to  $f$ , the resulting model  $f_{\mathrm{adj}}$  is certified to be approximately 0.09-MC, thereby providing a stronger guarantee.
+
+![](images/59d40d744fc50e5c71cad9d42705fb9b04fe3b60e146fae6f0844d236face253.jpg)  
+(a) ECE,  $\mathsf{ECE}^{\mathrm{max}}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on CheXpert. Here  $f$  is a decision tree trained on embeddings of a DenseNet-121 model pretrained on ImageNet.
+
+![](images/7baaa7f00537c605d2e30a0229f6102e5b3bbc2395b9a2569841e7592ded0df2.jpg)
+
+![](images/078c235ad564a34575270adc78cb8f88c0c123b45f82a93594fa70ebdaf13e55.jpg)  
+(b) ECE,  $\mathsf{ECE}^{\mathrm{max}}$  (dotted red line), and worst case violations (dotted blue line) of the original, unadjusted, logistic regression model (left) and end-to-end trained DenseNet-121 model (right).
+
+![](images/36ff54c109282627919e5ed015cabfcb498a042feadec6b859a6c77568c89e79.jpg)
+
+# 6.2. CheXpert
+
+CheXpert is a large public dataset for chest radiograph interpretation, with labeled annotations for 14 observations (positive, negative, or unlabeled) including cardiomegaly, atelectasis, and several others. The dataset contains self-reported sensitive attributes including race, sex, and age. Following the set up of Glocker et al. (2023), we work with a sample containing a total of 127,118 chest X-ray scans and consider the task of predicting the presence of pleural effusion in the X-rays.
+
+We consider all 14 groups that can be made from conjunctions of sex and race. Examples of groups  $g \in \mathcal{G}$  include black men, asian women, white women, etc. In this example, we assume that we do not have direct knowledge of patient's self-reported sex or race when training or evaluating our model  $f$  (as is common for privacy reasons). Instead, with an auxiliary dataset with samples  $(X,Z)$  we use the X-rays to learn proxy functions for sex and race. We then use them to construct proxies for all conjunctions as well. In Table 3, we report the proxy errors for specific groups.
+
+We consider three different models for  $f$ . The first is a decision tree classifier trained on features extracted from a DenseNet-121 model (Huang et al., 2017) pretrained on ImageNet (Deng et al., 2009). The second is a linear model (Breiman, 2001) trained on the same features. The third is a DenseNet-121 trained end-to-end on the X-rays.
+
+Figure 2a illustrates the results for the decision tree model. Before any adjustments, our worst-case violation serves as an early warning that the model  $f$  may be significantly uncalibrated on certain groups, with a violation as large as  $\alpha \approx 0.42$ . In a medical setting like this, such a finding is crucial, as it indicates that our predictions could be either overly confident or underconfident on sensitive groups. On the other hand, the right-hand graph of Figure 2a demonstrates the practical benefit of applying Algorithm 2 to multicali-
+
+brate the initial model  $f$  with respect to our highly accurate proxies  $\hat{\mathcal{G}}$ . After a straightforward adjustment, the upper bound on the worst-case violation decreases significantly, certifying that the adjusted model  $f_{\mathrm{adj}}$  is approximately 0.13-MC with respect to the true groups.
+
+Figure 2b presents the results for the logistic regression and fully-trained DenseNet models. In these cases, the worst-case violations for both models indicate that they are guaranteed to be approximately 0.11 and 0.12-multicalibrated with respect to the true groups. Notably, both models are approximately 0.03-multicalibrated with respect to the proxies. Thus, further adjustments provide negligible improvements.
+
+# 7. Conclusion
+
+In this work, we address the challenge of measuring multiaccuracy and multicalibration with respect to sensitive groups when sensitive group data is missing or unobserved. By leveraging proxy-sensitive attributes, we derive actionable upper bounds on true the multiaccuracy and multicalibration violations, offering a principled approach to assessing worst-case fairness violations. Furthermore, we demonstrate that adjusting models to be multiaccurate or multicalibrated with respect to proxy-sensitive attributes can significantly reduce these upper bounds, thereby providing useful guarantees on multiaccuracy and multicalibration violations for the true, but unknown, sensitive groups.
+
+Through empirical validation on real-world datasets, we show that multiaccuracy and multicalibration can be approximated even in the absence of complete sensitive group data. These findings highlight the practicality of using proxies to assess and enforce fairness in high-stakes decision-making contexts, where access to demographic information is often restricted. In particular, we illustrate the practical benefit of enforcing multiaccuracy and multicalibration with respect to proxies, providing practitioners with a simple and effective tool to improve fairness in their models.
+
+Naturally, multiaccuracy and multicalibration may not be the most appropriate fairness metrics across all settings. Nonetheless, whenever these notions are relevant, our methods offer, for the first time, the possibility to provide certificates and strengthen them without requiring access to ground truth group data. Lastly, note that without our recommendations of correcting for worst-case fairness with proxies, models trained on this data can inadvertently learn these sensitive attributes indirectly and base decisions on them, leading to potential negative outcomes. Our results and methodology prevent this by providing a principled approach to adjust models and reduce worst-case multiaccuracy and multicalibration violations.
+
+# Impact Statement
+
+In this work, we propose an effective solution to a technical problem: estimating and controlling for multiaccuracy and multicalibration using proxies. While proxies can be controversial and pose risks—such as reinforcing discrimination or compromising privacy—predictive models often learn sensitive attributes indirectly, leading to unintended harm. When used carefully, proxies can help mitigate these risks, as we demonstrate in this work. However, deploying proxies in real-world scenarios requires a careful evaluation of trade-offs through discussions with policymakers, domain experts, and other stakeholders. Ultimately, proxies should be employed responsibly and solely for assessing and promoting fairness.
+
+# Acknowledgements
+
+This work was supported by NIH award R01CA287422.
+
+# References
+
+Angwin, J., Larson, J., Mattu, S., and Kirchner, L. Machine bias. In *Ethics of data and analytics*, pp. 254-264. Auerbach Publications, 2022.  
+Awasthi, P., Kleindessner, M., and Morgenstern, J. Equalized odds postprocessing under imperfect group information. In International Conference on Artificial Intelligence and Statistics, pp. 1770-1780. PMLR, 2020.  
+Awasthi, P., Beutel, A., Kleindessner, M., Morgenstern, J., and Wang, X. Evaluating fairness of machine learning models under uncertain and incomplete information. In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, pp. 206-214, 2021.  
+Bharti, B., Yi, P., and Sulam, J. Estimating and controlling for equalized odds via sensitive attribute predictors. Advances in neural information processing systems, 36, 2024.
+
+Breiman, L. Random forests. Machine learning, 45(1): 5-32, 2001.  
+Brown, D. P., Knapp, C., Baker, K., and Kaufmann, M. Using bayesian imputation to assess racial and ethnic disparities in pediatric performance measures. Health services research, 51(3):1095-1108, 2016.  
+Calders, T., Kamiran, F., and Pechenizkiy, M. Building classifiers with independency constraints. In 2009 IEEE international conference on data mining workshops, pp. 13-18. IEEE, 2009.  
+Chen, J., Kallus, N., Mao, X., Svacha, G., and Udell, M. Fairness under unawareness: Assessing disparity when protected class is unobserved. In Proceedings of the conference on fairness, accountability, and transparency, pp. 339-348, 2019.  
+Commission, E. Proposal for a regulation laying down harmonised rules on artificial intelligence (artificial intelligence act), 2021. URL https://digital-strategy.ec.europa.eu/en/policies/regulatory-framework-ai.  
+Dastin, J. Amazon scraps secret ai recruiting tool that showed bias against women. In Ethics of data and analytics, pp. 296-299. Auerbach Publications, 2022.  
+Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., and Fei-Fei, L. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition (CVPR), pp. 248-255. IEEE, 2009.  
+Detommaso, G., Bertran, M., Fogliato, R., and Roth, A. Multicalibration for confidence scoring in llms. In Proceedings of the 41st International Conference on Machine Learning, ICML'24, 2024.  
+Diana, E., Gill, W., Kearns, M., Kenthapadi, K., Roth, A., and Sharifi-Malvajerdi, S. Multiaccurate proxies for downstream fairness. 2022 ACM Conference on Fairness, Accountability, and Transparency, 2022.  
+Ding, F., Hardt, M., Miller, J., and Schmidt, L. Retiring adult: New datasets for fair machine learning. Advances in neural information processing systems, 34:6478-6490, 2021.  
+Freire, M. N. and de Castro, L. N. e-recruitment recommender systems: a systematic review. Knowledge and Information Systems, 63:1-20, 2021.  
+Fremont, A., Weissman, J. S., Hoch, E., and Elliott, M. N. When race/ethnicity data are lacking: using advanced indirect estimation methods to measure disparities. *Rand health quarterly*, 6(1), 2016.
+
+Garin, S., Parekh, V., Sulam, J., et al. Medical imaging data science competitions should report dataset demographics and evaluate for bias [published online ahead of print april 3, 2023]. Nat Med.  
+Globus-Harris, I., Gupta, V., Jung, C., Kearns, M., Morgenstern, J., and Roth, A. Multicalibrated regression for downstream fairness. In Proceedings of the 2023 AAAI/ACM Conference on AI, Ethics, and Society, pp. 259-286, 2023a.  
+Globus-Harris, I., Harrison, D., Kearns, M., Roth, A., and Sorrell, J. Multicalibration as boosting for regression. arXiv preprint arXiv:2301.13767, 2023b.  
+Glocker, B., Jones, C., Bernhardt, M., and Winzeck, S. Algorithmic encoding of protected characteristics in chest x-ray disease detection models. *EBioMedicine*, 89, 2023.  
+Gopalan, P., Kalai, A. T., Reingold, O., Sharan, V., and Wieder, U. Omnipredictors. In Braverman, M. (ed.), 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), volume 215 of Leibniz International Proceedings in Informatics (LIPIs). Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2022.  
+Gupta, M., Cotter, A., Fard, M. M., and Wang, S. Proxy fairness. arXiv preprint arXiv:1806.11212, 2018.  
+Hansen, D., Devic, S., Nakkiran, P., and Sharan, V. When is multicalibration post-processing necessary? In Advances in Neural Information Processing Systems, 2024.  
+Hardt, M., Price, E., and Srebro, N. Equality of opportunity in supervised learning. Advances in neural information processing systems, 29, 2016.  
+Hébert-Johnson, U., Kim, M., Reingold, O., and Rothblum, G. Multicalibration: Calibration for the (computationally-identifiable) masses. In International Conference on Machine Learning, pp. 1939-1948. PMLR, 2018.  
+Holstein, K., Wortman Vaughan, J., Daumé III, H., Dudik, M., and Wallach, H. Improving fairness in machine learning systems: What do industry practitioners need? In Proceedings of the 2019 CHI conference on human factors in computing systems, pp. 1-16, 2019.  
+Huang, G., Liu, Z., Van Der Maaten, L., and Weinberger, K. Q. Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4700-4708, 2017.  
+Imai, K. and Khanna, K. Improving ecological inference by predicting individual ethnicity from voter registration records. Political Analysis, 24(2):263-272, 2016.
+
+Irvin, J., Rajpurkar, P., Ko, M., Yu, Y., Ciurea-Ilcus, S., Chute, C., Marklund, H., Haghgoo, B., Ball, R., Shpanskaya, K., et al. Chexpert: A large chest radiograph dataset with uncertainty labels and expert comparison. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pp. 590-597, 2019.  
+Kallus, N., Mao, X., and Zhou, A. Assessing algorithmic fairness with unobserved protected class using data combination. Management Science, 68(3):1959-1981, 2022.  
+Kim, J., Harper, A., McCormack, V., Sung, H., Houssami, N., Morgan, E., Mutebi, M., Garvey, G., Soerjomataram, I., and Fidler-Benaoudia, M. M. Global patterns and trends in breast cancer incidence and mortality across 185 countries. Nature Medicine, pp. 1-9, 2025.  
+Kim, M. P., Ghorbani, A., and Zou, J. Multiaccuracy: Black-box post-processing for fairness in classification. In Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society, pp. 247-254, 2019.  
+Lahoti, P., Beutel, A., Chen, J., Lee, K., Prost, F., Thain, N., Wang, X., and Chi, E. Fairness without demographics through adversarially reweighted learning. Advances in neural information processing systems, 33:728-740, 2020.  
+Li, D., Bharti, B., Wei, J., Sulam, J., and Yi, P. H. Sex imbalance produces biased deep learning models for knee osteoarthritis detection. Canadian Association of Radiologists Journal, 74(1):219-221, 2023.  
+Obermeyer, Z. and Mullainathan, S. Dissecting racial bias in an algorithm that guides health decisions for 70 million people. In Proceedings of the conference on fairness, accountability, and transparency, pp. 89-89, 2019.  
+of Science, W. H. O. and Policy, T. The blueprint for an ai bill of rights, 2022. URL https://www.whitehouse.gov/ospp/ ai-bill-of-rights/.  
+Prost, F., Awasthi, P., Blumm, N., Kumthekar, A., Potter, T., Wei, L., Wang, X., Chi, E. H., Chen, J., and Beutel, A. Measuring model fairness under noisy covariates: A theoretical perspective. In Proceedings of the 2021 AAAI/ACM Conference on AI, Ethics, and Society, pp. 873-883, 2021.  
+Roth, A. Uncertain: Modern Topics in Uncertainty Estimation. 2022.  
+Rudin, C., Wang, C., and Coker, B. The age of secrecy and unfairness in recidivism prediction. Harvard Data Science Review, 2(1):1, 2020.
+
+Shailaja, K., Seetharamulu, B., and Jabbar, M. Machine learning in healthcare: A review. In 2018 Second international conference on electronics, communication and aerospace technology (ICECA), pp. 910-914. IEEE, 2018.  
+Thomas, L., Crook, J., and Edelman, D. Credit scoring and its applications. SIAM, 2017.  
+United Nations Educational, S. and (UNESCO), C. O. Recommendation on the ethics of artificial intelligence, 2021.  
+Wang, S., Guo, W., Narasimhan, H., Cotter, A., Gupta, M., and Jordan, M. Robust optimization for fairness with noisy protected groups. Advances in neural information processing systems, 33:5190-5203, 2020.  
+Weissman, J. S. and Hasnain-Wynia, R. Advancing health care equity through improved data collection. New England Journal of Medicine, 364(24):2276-2277, 2011.  
+Yi, P. H., Bachina, P., Bharti, B., Garin, S. P., Kanhere, A., Kulkarni, P., Li, D., Parekh, V. S., Santomartino, S. M., Moy, L., et al. Pitfalls and best practices in evaluation of ai algorithmic biases in radiology. Radiology, 315(2): e241674, 2025.  
+Zafar, M. B., Valera, I., Gomez Rodriguez, M., and Gummadi, K. P. Fairness beyond disparate treatment & disparate impact: Learning classification without disparate mistreatment. In Proceedings of the 26th international conference on world wide web, pp. 1171-1180, 2017.  
+Zhang, Y. Assessing fair lending risks using race/ethnicity proxies. Management Science, 64(1):178-197, 2018.  
+Zhao, T., Dai, E., Shu, K., and Wang, S. Towards fair classifiers without sensitive attributes: Exploring biases in related features. 2022.  
+Zhu, Z., Yao, Y., Sun, J., Li, H., and Liu, Y. Weak proxies are sufficient and preferable for fairness with missing sensitive attributes. In International Conference on Machine Learning, pp. 43258-43288. PMLR, 2023.
+
+# A. Additional Theoretical Results
+
+Here we present a version of Theorem 5.1 for multiaccuracy.
+
+Theorem A.1. Fix a distribution  $\mathcal{D}$ , initial model  $f$ , and set of proxy groups  $\hat{\mathcal{G}}$ . If a model  $\hat{f}$  satisfies
+
+$$
+\mathsf {A E} ^ {\max } (\hat {f}, \hat {\mathcal {G}}) <   \min  _ {\hat {g} \in \hat {\mathcal {G}}} \mathsf {A E} _ {\mathcal {D}} (f, \hat {g}) \tag {19}
+$$
+
+$$
+\operatorname {M S E} (\hat {f}) \leq \operatorname {M S E} (f) \tag {20}
+$$
+
+then, it will have a smaller worst-case MA violation, i.e.  $\beta (\hat{f},\hat{\mathcal{G}})\leq \beta (f,\hat{\mathcal{G}})$
+
+A proof of this result is provided in Appendix B.3.
+
+# B. Proofs
+
+# B.1. Proof of Lemma 4.1
+
+Proof. Throughout these proofs, denote
+
+$$
+\mu_ {i, j} ^ {g} = \mathbb {P} [ g (X, Z) = i, \hat {g} (X) = j ] \quad \text {a n d} \quad \mu_ {i, j} ^ {g} (v) = \mathbb {P} [ g (X, Z) = i, \hat {g} (X) = j \mid f (X) = v ]. \tag {21}
+$$
+
+We begin with proving the result for multiaccuracy.
+
+Step 1: Establishing the Multiaccuracy Bound
+
+Fix a distribution  $\mathcal{D}$  and predictor  $f$ . Consider any group  $g \in \mathcal{G}$  and its corresponding proxy  $\hat{g} \in \hat{\mathcal{G}}$ . Then,
+
+$$
+\begin{array}{l} \mathsf {A E} _ {\mathcal {D}} (f, g) = \left| \mathbb {E} [ g (X, Z) (f (X) - Y) ] \right| (22) \\ = \left| \mathbb {E} [ g (X, Z) (f (X) - Y) ] - \mathbb {E} [ \hat {g} (X) (f (X) - Y) ] + \mathbb {E} [ \hat {g} (X) (f (X) - Y) ] \right| (23) \\ \leq \left| \mathbb {E} [ g (X, Z) (f (X) - Y) ] - \mathbb {E} [ \hat {g} (X) (f (X) - Y) ] \right| + \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) ] \right| (24) \\ = \left| \mathbb {E} [ g (X, Z) (f (X) - Y) - \hat {g} (X) (f (X) - Y) ] \right| + \mathbf {A E} _ {\mathcal {D}} (f, \hat {g}) (25) \\ = \left| \mathbb {E} \left[ \left(g (X, Z) - \hat {g} (X)\right) \cdot (f (X) - Y) \right] \right| + \mathbf {A} \mathbb {E} _ {\mathcal {D}} (f, \hat {g}) (26) \\ \leq \mathbb {E} [ | g (X, Z) - \hat {g} (X) | \cdot | f (X) - Y) | ] + \mathbf {A} \mathbb {E} _ {\mathcal {D}} (f, \hat {g}) (27) \\ \leq \min  \left(\sqrt {\mathbb {E} [ | g (X , Z) - \hat {g} (X) | ^ {2} ] \cdot \mathbb {E} [ | f (X) - Y | ^ {2} ]}, \mathbb {E} [ | g (X, Z) - \hat {g} (X) | ]\right) + \mathbf {A E} _ {\mathcal {D}} (f, \hat {g}) (28) \\ = \min  \left(\sqrt {\operatorname {M S E} (f) \cdot \operatorname {e r r} (\hat {g})}, \operatorname {e r r} (\hat {g})\right) + \mathsf {A E} _ {\mathcal {D}} (f, \hat {g}). (29) \\ \end{array}
+$$
+
+Here we applied the triangle inequality in line 24, Jensen's inequality in line 27, and Holder's inequality in line 28.
+
+Step 2: Tightness of the Multiaccuracy Bound
+
+We now show these bounds are tight. To be precise, we will prove that there exists a joint distribution over the random variables  $(f(X),Y,g(X,Z),\hat{g} (X))$  for which these bounds hold with equality.
+
+Consider a group  $g \in \mathcal{G}$  and its corresponding proxy  $\hat{g} \in \hat{\mathcal{G}}$ . First, consider the scenario where  $\mathrm{MSE}(f) \leq \mathrm{err}(\hat{g})$  so that by the first result of Lemma 4.1 we have
+
+$$
+\mathsf {A E} _ {\mathcal {D}} (f, g) \leq \mathsf {A E} _ {\mathcal {D}} (f, \hat {g}) + \sqrt {\operatorname {e r r} (\hat {g})} \cdot \sqrt {\operatorname {M S E} (f)}. \tag {30}
+$$
+
+Consider the following data generating process:
+
+- Conditioned on the event  $\{g(X,Z) = \hat{g} (X)\}$ , one has  $f(X) = Y$
+
+- Conditioned on the event  $\{g(X,Z) = 1, \hat{g}(X) = 0\}$ , one has that  $f(X) = \frac{\sqrt{\mathrm{MSE}(f)}}{\sqrt{\mathrm{err}(\hat{g})}}$  and  $Y = 0$ ,  
+-  $\mu_{0,1}^{g} = 0$  so that  $\operatorname{err}(\hat{g}) = \mu_{1,0}^{g}$ .
+
+Then,
+
+$$
+\begin{array}{l} \mathbb {E} [ g (X, Z) (f (X) - Y) ] = \mathbb {E} [ f (X) - Y \mid g (X, Z) = 1, \hat {g} (X) = 1 ] \mu_ {1, 1} ^ {g} + \mathbb {E} [ f (X) - Y \mid g (X, Z) = 1, \hat {g} (X) = 0 ] \mu_ {1, 0} ^ {g} (31) \\ = \sqrt {\operatorname {e r r} (\hat {g})} \cdot \sqrt {\operatorname {M S E} (f)}. (32) \\ \end{array}
+$$
+
+Further,
+
+$$
+\begin{array}{l} \mathbb {E} [ \hat {g} (X) (f (X) - Y) ] = \mathbb {E} [ f (X) - Y \mid g (X, Z) = 1, \hat {g} (X) = 1 ] \mu_ {1, 1} ^ {g} + \mathbb {E} [ f (X) - Y \mid g (X, Z) = 0, \hat {g} (X) = 1 ] \mu_ {0, 1} ^ {g} (33) \\ = 0. (34) \\ \end{array}
+$$
+
+As a result,
+
+$$
+\begin{array}{l} \left| \mathbb {E} [ g (X, Z) (f (X) - Y) ] \right| = \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) ] + \sqrt {\operatorname {e r r} (\hat {g})} \cdot \sqrt {\operatorname {M S E} (f)} \right| (35) \\ = \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) ] \right| + \sqrt {\operatorname {e r r} (\hat {g})} \cdot \sqrt {\operatorname {M S E} (f)}. (36) \\ \end{array}
+$$
+
+Now, consider the scenario where  $\mathsf{MSE}(f) > \mathsf{err}(\hat{g})$  so that by the first result of Lemma 4.1 we have
+
+$$
+\mathsf {A E} _ {\mathcal {D}} (f, g) \leq \mathsf {A E} _ {\mathcal {D}} (f, \hat {g}) + \operatorname {e r r} (\hat {g}). \tag {37}
+$$
+
+Consider the following data generating process:
+
+- Conditional on the event  $\{g(X,Z) = \hat{g} (X)\}$ ,  $f(X)\geq Y$  
+- Conditional on the event  $\{g(X,Z) = 1, \hat{g}(X) = 0\}$ ,  $f(X) = 1$  and  $Y = 0$ .  
+$\mu_{0,1}^{g} = 0$  so that  $\mathsf{err}(\hat{g}) = \mu_{1,0}^{g}$
+
+Then,
+
+$$
+\begin{array}{l} \mathbb {E} [ g (X, Z) (f (X) - Y) ] = \mathbb {E} [ f (X) - Y \mid g (X, Z) = 1, \hat {g} (X) = 1 ] \mu_ {1, 1} ^ {g} + \mathbb {E} [ f (X) - Y \mid g (X, Z) = 1, \hat {g} (X) = 0 ] \mu_ {1, 0} ^ {g} (38) \\ = \mathbb {E} [ f (X) - Y \mid g (X, Z) = 1, \hat {g} (X) = 1 ] \mu_ {1, 1} ^ {g} + \operatorname {e r r} (\hat {g}). (39) \\ \end{array}
+$$
+
+Further,
+
+$$
+\begin{array}{l} \mathbb {E} [ \hat {g} (X) (f (X) - Y) ] = \mathbb {E} [ f (X) - Y \mid g (X, Z) = 1, \hat {g} (X) = 1 ] \mu_ {1, 1} ^ {g} + \mathbb {E} [ f (X) - Y \mid g (X, Z) = 0, \hat {g} (X) = 1 ] \mu_ {0, 1} ^ {g} (40) \\ = \mathbb {E} [ f (X) - Y \mid g (X, Z) = 1, \hat {g} (X) = 1 ] \mu_ {1, 1} ^ {g} \geq 0. (41) \\ \end{array}
+$$
+
+In passing, note that requiring  $f(X) \geq Y$  above is much more than needed, but we take this for simplicity. Moving on, and as a result,
+
+$$
+\left| \mathbb {E} [ g (X, Z) (f (X) - Y) ] \right| = \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) ] + \operatorname {e r r} (\hat {g}) \right| = \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) ] \right| + \operatorname {e r r} (\hat {g}). \tag {42}
+$$
+
+We now prove the result for multicalibration.
+
+Step 1: Establishing the Multicalibration Bound
+
+Fix a distribution  $\mathcal{D}$  and predictor  $f$ . Consider any group  $g \in \mathcal{G}$  and its corresponding proxy  $\hat{g} \in \hat{\mathcal{G}}$ . Then,
+
+$$
+\begin{array}{l} \mathsf {E C E} _ {\mathcal {D}} (f, g) = \mathbb {E} \left[ \left| \mathbb {E} [ g (X, Z) (f (X) - Y) | f (X) = v ] \right| \right] (43) \\ = \mathbb {E} \left[ \left| \mathbb {E} [ g (X, Z) (f (X) - Y) | f (X) = v ] \right| - \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) | f (X) = v ] \right| \right. (44) \\ + \left. \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) | f (X) = v ] \right| \right] (45) \\ = \mathbb {E} \left[ \left| \mathbb {E} [ g (X, Z) (f (X) - Y) | f (X) = v ] \right| - \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) | f (X) = v ] \right| \right] + \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) (46) \\ \leq \mathbb {E} \left[ \left| \mathbb {E} [ g (X, Z) (f (X) - Y) | f (X) = v ] - \mathbb {E} [ \hat {g} (X) (f (X) - Y) | f (X) = v ] \right| \right] + \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) (47) \\ = \mathbb {E} \left[ \left| \mathbb {E} [ g (X, Z) (f (X) - Y) - \hat {g} (X) (f (X) - Y) | f (X) = v ] \right| \right] + \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) (48) \\ \leq \mathbb {E} \left[ \mathbb {E} \left[ \left| g (X, Z) (f (X) - Y) - \hat {g} (X) (f (X) - Y) \right| | f (X) = v \right] \right] + \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) (49) \\ = \mathbb {E} \left[ \left| g (X, Z) (f (X) - Y) - \hat {g} (X) (f (X) - Y) \right| \right] + \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) (50) \\ = \mathbb {E} \left[ \left| (g (X, Z) - \hat {g} (X)) \cdot (f (X) - Y) \right| \right] + \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) (51) \\ \leq \min  \left(\sqrt {\mathbb {E} [ | g (X , Z) - \hat {g} (X) | ^ {2} ] \cdot \mathbb {E} [ | f (X) - Y | ^ {2} ]}, \mathbb {E} [ | g (X, Z) - \hat {g} (X) | ]\right) + \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) (52) \\ = \min  \left(\sqrt {\operatorname {M S E} (f) \cdot \operatorname {e r r} (g , \hat {g})}, \operatorname {e r r} (\hat {g})\right) + \operatorname {E C E} _ {\mathcal {D}} (f, \hat {g}) (53) \\ \end{array}
+$$
+
+Here we applied the reverse triangle inequality in (47), Jensen's inequality in line (49), and Holder's inequality in line (52).
+
+Step 2: Tightness of the Multicalibration Bound
+
+We now show these bounds are tight. To be precise, we will prove that there exists a joint distribution over the random variables  $(f(X),Y,g(X,Z),\hat{g} (X))$  for which these bounds hold with equality.
+
+Consider a group  $g \in \mathcal{G}$  and its corresponding proxy  $\hat{g} \in \hat{\mathcal{G}}$ . First, consider the scenario where  $\mathrm{MSE}(f) \leq \mathrm{err}(\hat{g})$  so that by the first result of Lemma 4.1 we have
+
+$$
+\mathrm {E C E} _ {\mathcal {D}} (f, g) \leq \mathrm {E C E} _ {\mathcal {D}} (f, \hat {g}) + \sqrt {\operatorname {e r r} (\hat {g})} \cdot \sqrt {\operatorname {M S E} (f)}. \tag {54}
+$$
+
+Consider the same data generating process used to establish the MA bound,  $\mathsf{AE}_{\mathcal{D}}(f,g)\leq \mathsf{AE}_{\mathcal{D}}(f,\hat{g}) + \sqrt{\mathsf{err}(\hat{g})}\cdot \sqrt{\mathsf{MSE}(f)}$  Then,
+
+$$
+\begin{array}{l} \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ g (X, Z) (f (X) - Y) | f (X) = v ] \right| = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 1, \hat {g} (X) = 1 ] \cdot \mu_ {1, 1} ^ {g} (v) \right. (55) \\ + \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 1, \hat {g} (X) = 0 ] \cdot \mu_ {1, 0} ^ {g} (v) \Bigg | (56) \\ = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \frac {\sqrt {\operatorname {M S E} (f)}}{\sqrt {\operatorname {e r r} (\hat {g})}} \cdot \mu_ {1, 0} ^ {g} (v) \right| (57) \\ = \frac {\sqrt {\operatorname {M S E} (f)}}{\sqrt {\operatorname {e r r} (\hat {g})}} \cdot \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mu_ {1, 0} ^ {g} (v) \right| (58) \\ = \frac {\sqrt {\operatorname {M S E} (f)}}{\sqrt {\operatorname {e r r} (\hat {g})}} \cdot \mu_ {1, 0} ^ {g} = \sqrt {\operatorname {M S E} (f)} \sqrt {\operatorname {e r r} (\hat {g})}. (59) \\ \end{array}
+$$
+
+Further,
+
+$$
+\begin{array}{l} \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) | f (X) = v ] \right| = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 1, \hat {g} (X) = 1 ] \cdot \mu_ {1, 1} ^ {g} (v) \right. (60) \\ + \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 0, \hat {g} (X) = 1 ] \cdot \mu_ {0, 1} ^ {g} (v) \Bigg | (61) \\ = 0. (62) \\ \end{array}
+$$
+
+As a result,
+
+$$
+\underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ g (X, Z) (f (X) - Y) | f (X) = v ] \right| = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) | f (X) = v ] \right| + \sqrt {\operatorname {M S E} (f)} \sqrt {\operatorname {e r r} (\hat {g})}. \tag {63}
+$$
+
+Now, consider the scenario where  $\mathsf{MSE}(f) > \mathsf{err}(\hat{g})$  so that by the first result of Lemma 4.1 we have
+
+$$
+\operatorname {E C E} _ {\mathcal {D}} (f, g) \leq \operatorname {E C E} _ {\mathcal {D}} (f, \hat {g}) + \operatorname {e r r} (\hat {g}). \tag {64}
+$$
+
+Consider the same data generating process used to establish the MA bound,  $\mathsf{AE}_{\mathcal{D}}(f,g)\leq \mathsf{AE}_{\mathcal{D}}(f,\hat{g}) + \mathsf{err}(\hat{g})$  . Then,
+
+$$
+\begin{array}{l} \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ g (X, Z) (f (X) - Y) | f (X) = v ] \right| = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 1, \hat {g} (X) = 1 ] \cdot \mu_ {1, 1} ^ {g} (v) \right. (65) \\ + \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 1, \hat {g} (X) = 0 ] \cdot \mu_ {1, 0} ^ {g} (v) \Bigg | (66) \\ = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 1, \hat {g} (X) = 1 ] \cdot \mu_ {1, 1} ^ {g} (v) + \mu_ {1, 0} ^ {g} (v) \right| (67) \\ = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 1, \hat {g} (X) = 1 ] \cdot \mu_ {1, 1} ^ {g} (v) \right| (68) \\ + \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left[ \mu_ {1, 0} ^ {g} (v) \right] (69) \\ = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 1, \hat {g} (X) = 1 ] \cdot \mu_ {1, 1} ^ {g} (v) \right| + \mu_ {1, 0} ^ {g}. (70) \\ \end{array}
+$$
+
+Further,
+
+$$
+\begin{array}{l} \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) | f (X) = v ] \right| = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 1, \hat {g} (X) = 1 ] \cdot \mu_ {1, 1} ^ {g} (v) \right. (71) \\ + \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 0, \hat {g} (X) = 1 ] \cdot \mu_ {0, 1} ^ {g} (v) \Bigg | (72) \\ = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ f (X) - Y \mid f (X) = v, g (X, Z) = 1, \hat {g} (X) = 1 ] \cdot \mu_ {1, 1} ^ {g} (v) \right| (73) \\ \end{array}
+$$
+
+As a result,
+
+$$
+\underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ g (X, Z) (f (X) - Y) | f (X) = v ] \right| = \underset {v \sim \mathcal {D} _ {f}} {\mathbb {E}} \left| \mathbb {E} [ \hat {g} (X) (f (X) - Y) | f (X) = v ] \right| + \operatorname {e r r} (\hat {g}). \tag {74}
+$$
+
+The following bounds are tight in that we show there exists a distribution for which the bound is attained. The bounds can nonetheless be improved by assuming complete access to the marginal distributions over  $(f(X),Y,\hat{g} (X))$  and  $(g(X,Z),\hat{g} (X))$ . In this case, one could derive tight bounds that agree with this observed information with techniques used in(Kallus et al., 2022; Bharti et al., 2024) that rely on the Fréchet inequalities.
+
+# B.2. Proof of Theorem 4.2
+
+Proof. We prove the result for multiaccuracy. Recall Lemma 4.1, which states that for any group  $g$  and its proxy  $\hat{g}$
+
+$$
+\mathsf {A E} _ {\mathcal {D}} (f, g) \leq \mathsf {F} (f, \hat {g}) + \mathsf {A E} _ {\mathcal {D}} (f, \hat {g}) \tag {75}
+$$
+
+Thus,  $\forall g\in \mathcal{G}$
+
+$$
+\mathsf {A E} _ {\mathcal {D}} (f, g) \leq \beta (f, \hat {\mathcal {G}}) := \max  _ {\hat {g} \in \hat {\mathcal {G}}} \mathsf {F} (f, \hat {g}) + \mathsf {A E} _ {\mathcal {D}} (f, \hat {g}). \tag {76}
+$$
+
+This proves that  $f$  is  $(\mathcal{G},\beta (f,\hat{\mathcal{G}}))$ -multiaccurate. The proof for multicalibration follows an identical argument.
+
+# B.3. Proof of Theorem 5.1
+
+Proof. Fix a distribution  $\mathcal{D}$ , model  $f$ , set of groups  $\mathcal{G}$  and its corresponding proxies  $\hat{\mathcal{G}}$ . Recall by Theorem 4.2 that  $f$  is  $(\mathcal{G}, \gamma(f, \cdot)$ -multicalibrated where
+
+$$
+\gamma (f, \hat {\mathcal {G}}) = \max  _ {\hat {g} \in \mathcal {G}} \min  \left(\operatorname {e r r} (\hat {g}), \sqrt {\operatorname {M S E} (f) \cdot \operatorname {e r r} (\hat {g})}\right) + \operatorname {E C E} _ {\mathcal {D}} (f, \hat {g}) \tag {77}
+$$
+
+First, note that for  $\mathsf{MSE}(f) > \mathsf{err}(\hat{g})$ , the term
+
+$$
+\min  \left(\operatorname {e r r} (\hat {g}), \sqrt {\operatorname {M S E} (f) \cdot \operatorname {e r r} (\hat {g})}\right) \tag {78}
+$$
+
+is constant with respect to  $\mathrm{MSE}(f)$  and for  $\mathrm{MSE}(f) \leq \mathrm{err}(\hat{g})$  it increases as  $\mathrm{MSE}(f)$  increases.
+
+Now, suppose another model  $\hat{f}$  satisfies the following
+
+$$
+\mathsf {E C E} ^ {\max } (\hat {f}, \hat {\mathcal {G}}) <   \min  _ {\hat {g} \in \hat {\mathcal {G}}} \mathsf {E C E} _ {\mathcal {D}} (f, \hat {g}) \tag {79}
+$$
+
+$$
+\operatorname {M S E} (\hat {f}) \leq \operatorname {M S E} (f). \tag {80}
+$$
+
+Then,
+
+$$
+\begin{array}{l} \gamma (\hat {f}, \hat {\mathcal {G}}) = \max  _ {\hat {g} \in \mathcal {G}} \min  \left(\operatorname {e r r} (\hat {g}), \sqrt {\operatorname {M S E} (\hat {f}) \cdot \operatorname {e r r} (\hat {g})}\right) + \operatorname {E C E} _ {\mathcal {D}} (\hat {f}, \hat {g}) (81) \\ \leq \max  _ {\hat {g} \in \mathcal {G}} \min  \left(\operatorname {e r r} (\hat {g}), \sqrt {\operatorname {M S E} (f) \cdot \operatorname {e r r} (\hat {g})}\right) + \operatorname {E C E} _ {\mathcal {D}} (\hat {f}, \hat {g}) (82) \\ \leq \max  _ {\hat {g} \in \hat {\mathcal {G}}} \min  \left(\operatorname {e r r} (\hat {g}), \sqrt {\operatorname {M S E} (f) \cdot \operatorname {e r r} (\hat {g})}\right) + \operatorname {E C E} ^ {\max } (\hat {f}, \hat {\mathcal {G}}) (83) \\ \leq \max  _ {\hat {g} \in \mathcal {G}} \min  \left(\operatorname {e r r} (\hat {g}), \sqrt {\operatorname {M S E} (f) \cdot \operatorname {e r r} (\hat {g})}\right) + \min  _ {\hat {g} \in \hat {\mathcal {G}}} \operatorname {E C E} _ {\mathcal {D}} (f, \hat {g}) (84) \\ \leq \max  _ {\hat {g} \in \mathcal {G}} \min  \left(\operatorname {e r r} (\hat {g}), \sqrt {\operatorname {M S E} (f) \cdot \operatorname {e r r} (\hat {g})}\right) + \operatorname {E C E} _ {\mathcal {D}} (f, \hat {g}) (85) \\ = \gamma (f, \hat {\mathcal {G}}). (86) \\ \end{array}
+$$
+
+The proof of the multiaccuracy result in Theorem A.1 follows an identical argument.
+
+![](images/250e13c0d9fbccc21883809a1970c52c4eff770812de1a09c2dd6cfa72397305.jpg)
+
+# C. Additional Experiment Details
+
+# C.1. ACS Experiments
+
+Models. In the ACSIncome and PubCov experiments, we use Random Forests as the proxy models  $\hat{g}$ , and train three types of models  $f$ : logistic regression, decision tree, and Random Forest.
+
+Results. The sensitive groups we use along with their proxy errors are reported in Tables 4 and 5. All MA related results for the different models  $f$  are presented in Appendices E.1 and E.2. Note, all of the models are MA with respect to  $\mathcal{G}$  and  $\hat{\mathcal{G}}$ . As a result, adjusting with respect to the proxies provides no benefit. All MC related results for the different models  $f$  are presented in Appendices E.4 and E.5. Note, the logistic regression and Random Forest models are highly multicalibrated with respect to  $\mathcal{G}$  and  $\hat{\mathcal{G}}$ . As a result, adjusting provides no benefit. On the other hand, the decision tree is grossly uncalibrated with respect to some proxies. As a result, we see a benefit in multicalibrating with respect to the proxies.
+
+# C.2. CheXpert
+
+Models. In the CheXpert experiment, we follow Glocker et al. (2023) and train a DenseNet-121 model for the to predict race and sex. For the models  $f$ , we use three types. The first is a decision tree classifier trained on features extracted from a DenseNet-121 model (Huang et al., 2017) pretrained on ImageNet (Deng et al., 2009). The second is a linear model (Breiman, 2001) trained on the same features. The third is a DenseNet-121 model trained end-to-end on the raw X-ray images.
+
+Results. The groups used along with proxy errors are reported in Table 6. All multiaccuracy related results for the different types of models  $f$  are presented in Appendix E.3. Note, all of the models are multiaccurate with respect to  $\mathcal{G}$  and  $\hat{\mathcal{G}}$ . As a result, adjusting provides no benefit. All multicalibration related results for the different types of models  $f$  are presented in Appendix E.6. Note, the logistic regression and fully trained DenseNet-121 models are highly multicalibrated with respect to  $\mathcal{G}$  and  $\hat{\mathcal{G}}$ . As a result, adjusting provides no benefit. On the other hand, the decision tree is grossly uncalibrated with respect to some proxies. As a result, we see a benefit in multicalibrating with respect to the proxies.
+
+# D. Additional Tables
+
+<table><tr><td>Group</td><td>err(h)</td></tr><tr><td>Black Adults</td><td>0.044</td></tr><tr><td>Black Women</td><td>0.027</td></tr><tr><td>Women</td><td>0.000</td></tr><tr><td>Never Married</td><td>0.000</td></tr><tr><td>American Indian</td><td>0.007</td></tr><tr><td>Seniors</td><td>0.000</td></tr><tr><td>White Women</td><td>0.123</td></tr><tr><td>Multiracial</td><td>0.047</td></tr><tr><td>White Children</td><td>0.002</td></tr><tr><td>Asian</td><td>0.060</td></tr></table>
+
+Table 4. Sensitive groups and the proxy errors used in the ACSIncome experiment  
+
+<table><tr><td>Group</td><td>err(h)</td></tr><tr><td>Black Adults</td><td>0.044</td></tr><tr><td>Black Women</td><td>0.027</td></tr><tr><td>Women</td><td>0.000</td></tr><tr><td>Never Married</td><td>0.000</td></tr><tr><td>American Indian</td><td>0.007</td></tr><tr><td>White Women</td><td>0.123</td></tr><tr><td>Multiracial</td><td>0.047</td></tr><tr><td>White Children</td><td>0.002</td></tr><tr><td>Asian</td><td>0.060</td></tr></table>
+
+Table 5. Sensitive groups and the proxy errors used in the ACSPubCov experiment  
+
+<table><tr><td>Group</td><td>err(ˆg)</td></tr><tr><td>Men</td><td>0.027</td></tr><tr><td>Women</td><td>0.027</td></tr><tr><td>White</td><td>0.920</td></tr><tr><td>Asian</td><td>0.068</td></tr><tr><td>Black</td><td>0.039</td></tr><tr><td>Asian Men</td><td>0.039</td></tr><tr><td>Asian Women</td><td>0.034</td></tr><tr><td>Black Men</td><td>0.021</td></tr><tr><td>Black Women</td><td>0.020</td></tr><tr><td>White Men</td><td>0.067</td></tr><tr><td>White Women</td><td>0.062</td></tr></table>
+
+Table 6. Sensitive groups and the proxy errors used in the CheXpert experiment
+
+# E. Additional Figures
+
+# E.1. Multiaccuracy results for ACSIncome
+
+![](images/6f51876a4fd8db775f707001f531f074634721b82681eed91462369233a66791.jpg)  
+Figure B.1.  $\mathsf{AE}(f,g)$ ,  $\mathsf{AE}^{\max}(f,g)$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSIncome. Here,  $f$  is a logistic regression.
+
+![](images/6c94a71486f304aab461073a4f20d75b8d0140350bbfde306d98b8d67b339299.jpg)
+
+![](images/cec8c9fbdee8e9ec41d367db70daab4cd4e1d5c7806bc3064aa46af7fa1c238c.jpg)  
+Figure B.2.  $\mathsf{AE}(f,g)$ ,  $\mathsf{AE}^{\max}(f,g)$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSIncome. Here,  $f$  is a decision tree.
+
+![](images/1a49c6f9009bf8c47d02f83b50eba0b282db37862cb441c6d37af16b9ff2cb41.jpg)
+
+![](images/9f6652b13bf34d713c018dc2ec8ffe17838c0fe8336a9366dfea9ca0aa24c8ca.jpg)  
+Figure B.3.  $\mathsf{AE}(f,g)$ ,  $\mathsf{AE}^{\max}(f,g)$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSIncome. Here,  $f$  is a Random Forest.
+
+![](images/902a54639c7ac23e9f0b67a8e5d4230a82ddd6b03cb157156ee72623ce12bd00.jpg)
+
+# E.2. Multiaccuracy results for ACSPubCov
+
+![](images/d703bccbf81e2557d53844c35b6b1e29678d3c287b6c3876f400a6531581fce6.jpg)  
+Figure B.4.  $\mathsf{AE}(f,g)$ ,  $\mathsf{AE}^{\max}(f,g)$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSPubCov. Here,  $f$  is a logistic regression.
+
+![](images/5b277bc3d552aa14310777baeeace2a09cb8752a71e770eac6f45a32e2a7d1a5.jpg)
+
+![](images/8ecc3ee45aff372241329e63bdebd05147557bf37f63aef862a5c4773e86a27f.jpg)  
+Figure B.5.  $\mathsf{AE}(f,g)$ ,  $\mathsf{AE}^{\max}(f,g)$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSPubCov. Here,  $f$  is a decision tree.
+
+![](images/ec60e6bdeb7429add162af91368eed882218d4bc791f5dd4a9ea974bfcd6dfe3.jpg)
+
+![](images/1add008adf0cea64ee495ab5e45d47d39b16e7eeda955ef8ea38cd12d2129eab.jpg)  
+Figure B.6.  $\mathsf{AE}(f,g)$ ,  $\mathsf{AE}^{\max}(f,g)$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSPubCov. Here,  $f$  is a Random Forest.
+
+![](images/1946e81f0e175915279dd65aa6bca3a5af1bb95cb541a8cb99fba40916bb6047.jpg)
+
+# E.3. Multiaccuracy results for CheXpert
+
+![](images/3d6c8475c6500d0a5f764cda52d669d9dbd4df4aaf386af8fb74a01721cacdec.jpg)  
+Figure B.7.  $\mathsf{AE}(f,g)$ ,  $\mathsf{AE}^{\max}(f,g)$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on CheXpert. Here,  $f$  is a logistic regression.
+
+![](images/ac8750b69cca07d6b9c2bddad1c1f46eddffbcc1e0af2a5b617221040b4f5c2b.jpg)
+
+![](images/91044a4acba54c8a4e09015278fae714a81a0e6db5f26952ffd02999e5258e3b.jpg)  
+Figure B.8.  $\mathsf{AE}(f,g)$ ,  $\mathsf{AE}^{\max}(f,g)$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on CheXpert. Here,  $f$  is a decision tree.
+
+![](images/2bdb07e6087c128114923ab63dfc18fd3af94635577aac27d7a3f182ea22c569.jpg)
+
+![](images/d3c5b0cb92bc290ccddd1bbb1cd55216ca87b7b1f9231d896eb209baf31c5f1a.jpg)  
+Figure B.9.  $\mathsf{AE}(f,g)$ ,  $\mathsf{AE}^{\max}(f,g)$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on CheXpert. Here,  $f$  is a DenseNet-121 model.
+
+![](images/32d74024296ab2a1e82c41ce37752dc6a9c0ba009692a5ec032bb7c494a44df7.jpg)
+
+# E.4. Multicalibration results for ACSIncome
+
+![](images/6fdb5d46110559032e6e0dd292aa4a992adeb863a2d248ef539dab7ef359ac7d.jpg)  
+Figure B.10. ECE,  $\mathsf{ECE}^{\max}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSIncome. Here,  $f$  is a logistic regression.
+
+![](images/80865f6036111538764b127e61947059f3887bd9ce0d1d4768a85e6574984884.jpg)
+
+![](images/c2bf2bae5aad42ff9d8c909e50976d792c1d8ed5cd94528ce387ac4541ab58f2.jpg)  
+Figure B.11. ECE,  $\mathsf{ECE}^{\max}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSIncome. Here,  $f$  is a decision tree.
+
+![](images/88c4cebf175bb97802113b1a37201b9ce923f9239c772105f7192d68d6b7f049.jpg)
+
+![](images/706db2a637d8f67c93685d4db775f153a49b0f5b0eca8cfc5516b4705f310b27.jpg)  
+Figure B.12. ECE,  $\mathrm{ECE}^{\max}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSIncome. Here,  $f$  is a Random Forest.
+
+![](images/6a584fd1a73db247dc8081a0016d1f52225d2cc6e4a485c05661c5609c846799.jpg)
+
+# E.5. Multicalibration results for ACSPubCov
+
+![](images/9ae2caffde7fe1696cdba232cd334eb052bb1c42937389bc76d8478ab851521b.jpg)  
+Figure B.13. ECE,  $\mathsf{ECE}^{\max}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSPubCov. Here,  $f$  is a logistic regression.
+
+![](images/541991c72f4fa574609f65e1606e766ee862b593d9a1ca654f96bf104f53bd21.jpg)
+
+![](images/9a34e01f00f2a31bb9e56c3dcb235723e3d30b618027770740ec09431cabd6c4.jpg)  
+Figure B.14. ECE,  $\mathsf{ECE}^{\mathrm{max}}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSPubCov. Here,  $f$  is a decision tree.
+
+![](images/e12c5d0d2c543a5052d706c614333b44d8856c18033e6d9ba39d6c1893fc71b4.jpg)
+
+![](images/145a52732a21706073775588e068d24490f5e9b78acbefbda8b7aafe4e711669.jpg)  
+Figure B.15. ECE,  $\mathrm{ECE}^{\max}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on ACSPubCov. Here,  $f$  is a Random Forest.
+
+![](images/04014fe8a13af784d0c88ceed28b73734cd119064a857dcd5466d1942a6daa6e.jpg)
+
+# E.6. Multicalibration results for CheXpert
+
+![](images/7259a704981cf0fc7c9b6f3b20ec096ec762ed150957eedd6bb871edf7e18260.jpg)  
+Figure B.16. ECE,  $\mathsf{ECE}^{\max}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on CheXpert. Here,  $f$  is a logistic regression.
+
+![](images/e77df5229a421c252f157fb542acaf0675c2ce8c8f5f672ca5645fa42d5989ee.jpg)
+
+![](images/8962e9c6a493081522b6fe675bb71874caea2569427727b7aef1e4aec34259f5.jpg)  
+Figure B.17. ECE,  $\mathsf{ECE}^{\max}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on CheXpert. Here,  $f$  is a decision tree.
+
+![](images/cfd880513e387bb4e06763ce7857af71fe1c929e466a6a987c594ddc8ae9766c.jpg)
+
+![](images/42fb641a59833ad7c358b6e6b7b3cdc08aca13d03359fcfca944503842a09834.jpg)  
+Figure B.18. ECE,  $\mathsf{ECE}^{\mathrm{max}}$  (dotted red line), and worst case violations (dotted blue line) of the original model  $f$  and adjusted model  $f_{\mathrm{adj}}$  on CheXpert. Here,  $f$  is a DenseNet-121 model.
+
+![](images/fef488ee57526c5734c0cb13b6ced5f20e71426cc13ad109059b7cc7ca1f3125.jpg)

2025/Multiaccuracy and Multicalibration via Proxy Groups/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a92e686eea60effc04b72ddc335478fc72653da42b6ccb889c43345419cdd5f7
+size 1453595

2025/Multidimensional Adaptive Coefficient for Inference Trajectory Optimization in Flow and Diffusion/a77e6807-0eb0-428d-bfb7-8ed224b510c2_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d41717b2fadf66f9878cf4e7ae2967ffbd61c04c1bff62a63d6b2f43ce5584e9
+size 16647433

2025/Multidimensional Adaptive Coefficient for Inference Trajectory Optimization in Flow and Diffusion/full.md

@@ -0,0 +1,695 @@
+# Multidimensional Adaptive Coefficient for Inference Trajectory Optimization in Flow and Diffusion
+
+Dohoon Lee  $^{12}$  Jaehyun Park  $^{1}$  Hyunwoo J. Kim  $^{3}$  Kyogu Lee  $^{1,2,4}$
+
+# Abstract
+
+Flow and diffusion models have demonstrated strong performance and training stability across various tasks but lack two critical properties of simulation-based methods: freedom of dimensionality and adaptability to different inference trajectories. To address this limitation, we propose the Multidimensional Adaptive Coefficient (MAC), a plug-in module for flow and diffusion models that extends conventional unidimensional coefficients to multidimensional ones and enables inference trajectory-wise adaptation. MAC is trained via simulation-based feedback through adversarial refinement. Empirical results across diverse frameworks and datasets demonstrate that MAC enhances generative quality with high training efficiency. Consequently, our work offers a new perspective on inference trajectory optimality, encouraging future research to move beyond vector field design and to leverage training-efficient, simulation-based optimization.
+
+# 1. Introduction
+
+Compared to simulation-based methods like NeuralODE (Chen et al., 2018), flow and diffusion modeling (Sohl-Dickstein et al., 2015; Lipman et al., 2023) demonstrates remarkable performance and training stability across various tasks and has become a standard approach for generation tasks. However, it trades off two critical properties that simulation-based methods offer, which could enhance the quality of transportation: freedom of dimensionality and adaptability with respect to different inference trajectories. While simulation-based methodologies possess these prop
+
+<sup>1</sup>Music and Audio Research Group, Department of Intelligence and Information, Seoul National University, Seoul, Korea <sup>2</sup>Interdisciplinary Program in Artificial Intelligence, Seoul National University, Seoul, Korea <sup>3</sup>School of Computing, KAIST, Daejeon, Korea <sup>4</sup>Artificial Intelligence Institute, Seoul National University, Seoul, Korea. Correspondence to: Dohoon Lee <do-hoonlee.research@gmail.com>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+![](images/76222aa8708a97e2d22050e6bed2b30da35710f866c77ef3135babc55ab0f945.jpg)  
+(a)  $\alpha$
+
+![](images/8298f69082bb1b0bfc254babf3f3fc91e39afae6d83b9cb14b1619dfb350d231.jpg)  
+(b)  $\gamma$
+
+![](images/79be83754564f9c019ff5575a99b9155db9e0ad4b0cab98d4d1072f25c169873.jpg)  
+(c)  $\gamma_{\phi}$  (MAC)  
+Figure 1: Comparison of  $\alpha$ ,  $\gamma$ , and  $\gamma_{\phi}$ . As shown, employing  $\gamma_{\phi}$  (MAC) expands the search space for trajectory optimization by enabling adaptive, curved trajectories and time steps that are distinct for each sample.
+
+erties, they exhibit limitations in quality and training efficiency compared to flow and diffusion. We integrate these two properties of simulation-based methods into flow and diffusion models to enhance performance while preserving training efficiency, effectively combining their advantages.
+
+To achieve this, we introduce the Multidimensional Adaptive Coefficient (MAC). As described by Albergo et al. (2023), the trajectory with  $x_0 \sim \rho_0$  and  $x_1 \sim \rho_1$  in flow and diffusion for  $t \in [0,T]$  can be written as  $x(t) = \alpha_0(t)x_0 + \alpha_1(t)x_1$ ,  $x_0,x_1 \in \mathbb{R}^d$ , where, conventionally, the coefficients  $\alpha_0(t),\alpha_1(t) \in \mathbb{R}$  are unidimensional. We extend this by introducing a Multidimensional Coefficient  $\gamma_0(t),\gamma_1(t) \in \mathbb{R}^{d\times d}$ , which allows for different time scheduling across all data dimensions. Based on this, we introduce MAC  $\gamma_{\phi}(t,\mathbf{x}_{\theta ,\phi}^{S})$ , parameterized by  $\phi$ , to adapt to different inference trajectories  $\mathbf{x}_{\theta ,\phi}^{S}$ . To optimize MAC, we use it to construct a flow- or diffusion-based differential equation solver  $G_{\theta ,\phi}$  based on a flow or diffusion model  $H_{\theta}$  and adversarially optimize  $\theta$  and  $\phi$  using a discriminator  $D_{\psi}$ . By optimizing MAC, we achieve full multidimensionality and adaptability in the inference trajectory of flow and diffusion models. This can be applied to any type of framework as a plug-in module to enhance inference performance.
+
+![](images/8156b6c66909c5515dab9331765cc54c8c24f7edb93f3aee2c47f8e5a62b50f2.jpg)  
+(a) Inference without MAC
+
+![](images/0c5c28d7f7de93c3deddd75d2aefe665f2b9fe85317ff7392e638f191dbd830e.jpg)  
+(b) Inference with MAC  
+Figure 2: Comparison of inference trajectory optimization with and without MAC, given the same vector field estimated by  $H_{\theta}$ , indicated by gray arrows in the background. (a) Without MAC, inference trajectories strictly follow the directions defined by the vector field  $H_{\theta}$ , allowing adjustments only in step sizes, which must remain consistent across all samples. (b) With MAC, the search space (shaded in blue) expands, enabling flexible adjustments in both trajectory directions and step sizes, adaptively optimized for each individual sample. Thus, even when a suboptimal vector field is given, MAC effectively corrects errors by identifying optimal inference plans that maximize final transportation quality.
+
+For a given vector field defined by  $H_{\theta}$ , optimizing MAC corresponds to inference trajectory optimization. In particular, optimal inference plans are computed and refined offline via simulation, and subsequently deployed during inference without incurring additional optimization costs. Before MAC, it was challenging to induce significant performance gains through inference trajectory optimization alone, due to the absence of multidimensionality and adaptability. In contrast, as shown in Figure 2, MAC substantially expands the search space for optimization by introducing dimension-wise adaptive control, thereby enhancing the effectiveness of inference trajectory optimization.
+
+Our experiments span various frameworks (DDPM (Ho et al., 2020), FM (Lipman et al., 2023), EDM (Karras et al., 2022), SI (Albergo et al., 2023)) across multiple datasets (CIFAR-10 (Krizhevsky & Hinton, 2009), FFHQ (Karras et al., 2019), AFHQ (Choi et al., 2020), and ImageNet (Deng et al., 2009)). Notably, inference using MAC yields consistent performance improvements, achieving state-of-the-art results in CIFAR-10 conditional generation. These results are obtained with high training efficiency, suggesting that we successfully integrate the advantages of simulation-based and simulation-free dynamics.
+
+Consequently, we propose exploring the optimality of inference trajectories through training-efficient simulation in flow- and diffusion-based generative models. Prior approaches primarily focused on shaping the vector field, often relying on predefined, simulation-free notions of optimality, such as straightness (Liu et al., 2023; Tong et al., 2024). However, the inference trajectory is governed not solely by the vector field, but by the interplay between the vector field and the inference plan. From this perspective, simulation-free vector field training deviates from true transportation optimality, which should be defined by the final quality of
+
+transportation. Our approach addresses this limitation by introducing training-efficient simulation, thereby making simulation-based inference trajectory optimization practically feasible. In summary, our main contributions are:
+
+1. We introduce the concept of a Multidimensional Adaptive Coefficient (MAC) in flow and diffusion, laying the foundation for achieving both freedom of dimensionality and adaptability in inference trajectories, easily implemented as a plug-in module for any framework.  
+2. We propose a highly efficient training strategy for inference trajectory optimization by leveraging MAC in combination with adversarial training.  
+3. We suggest exploring the notion of trajectory optimality beyond predefined properties, emphasizing evaluation based on the final transportation quality.
+
+# 2. Related Works
+
+Trajectory Optimizations in Flow and Diffusion Various trajectory optimization approaches predefine optimality before training using simulation-free objectives. Methods such as Liu et al. (2023) and Tong et al. (2024) define straightness as the optimality criterion and optimize trajectories by maintaining consistency between  $(x_0, x_1)$  when training flow and diffusion models, aligning with an Optimal Transport (OT) perspective. Other approaches, including Lee et al. (2023); Park et al. (2024); Kim et al. (2025), also adopt the OT perspective as a predefined criterion for tuning the vector field. Singhal et al. (2023) defines optimality through a fixed sequence of diffusion steps aimed at reducing inference complexity. Bartosh et al. (2024) introduces a neural flow model that implicitly sets trajectory optimality within the diffusion process. Kapuński et al. (2024) optimizes trajectories by defining data-dependent geodesics. There are also methods that refine trajectories after training,
+
+such as Albergo et al. (2024), which defines optimality by minimizing trajectory length under the Wasserstein-2 metric, focusing on a shortest-distance criterion.
+
+Despite these diverse perspectives on trajectory optimality, there are two significant differences between these methods and ours. First, we assess optimality solely based on the final transportation quality through simulation, which is a crucial factor in generative modeling. Second, none of the existing methods achieve full flexibility in inference trajectory design on two fronts: multidimensionality and adaptability with respect to different inference trajectories.
+
+# Few-Step Generation via Distillation and Fine-Tuning
+
+Existing approaches can be broadly categorized into non-adversarial and adversarial methods. Non-adversarial methods, such as Geng et al. (2023); Berthelot et al. (2023); Yin et al. (2024); Song & Dhariwal (2024), focus on 1-step distillation techniques without adversarial objectives. These approaches aim for few-step generation by leveraging distributional losses and equilibrium models, effectively distilling the diffusion process without involving a discriminator. Conversely, adversarial approaches such as Wang et al. (2023); Lu et al. (2023); Luo et al. (2023); Xu et al. (2024), integrate a discriminator within diffusion models via a GAN-based framework. Kim et al. (2024) also generalizes Song et al. (2023) to enable efficient sampling using a discriminator.
+
+While our method shares similarities in achieving few-step generation by leveraging a pre-trained diffusion model, a key difference is that prior works optimize only the vector field, omitting the inference plan. In contrast, our method jointly optimizes both the vector field and the inference plan, thereby enabling inference trajectory optimization.
+
+# 3. Methodology
+
+We consider the task of transporting between two distributions  $x_0 \sim \rho_0$  and  $x_1 \sim \rho_1$ , where  $x_0, x_1 \in \mathbb{R}^d$ . Following Albergo et al. (2023), for  $t \in [0, T]$ , the trajectory  $x(t)$  is defined as:
+
+$$
+x (t) = \alpha_ {0} (t) x _ {0} + \alpha_ {1} (t) x _ {1}, \quad v (t, x (t)) = \dot {x} (t), \tag {1}
+$$
+
+where  $\alpha(t) = [\alpha_0(t), \alpha_1(t)] \in \mathbb{R}^2$  represents the unidimensional coefficients, and  $\dot{x}(t)$  denotes the derivative of  $x(t)$  with respect to  $t$ . The diffusion model  $H_{\theta}$  estimates the vector field  $v$  as follows:
+
+$$
+\begin{array}{l} v _ {\theta} (t, x (t)) = \dot {\alpha} _ {0} (t) \hat {x} _ {0, \theta} + \dot {\alpha} _ {1} (t) \hat {x} _ {1, \theta}, \tag {2} \\ H _ {\theta} (t, x (t)) = [ \hat {x} _ {0, \theta}, \hat {x} _ {1, \theta} ]. \\ \end{array}
+$$
+
+For example, Song et al. (2021) predict the score value  $\nabla_{x}\log p(x(t);t) = -\hat{x}_{1,\theta} / \alpha_{1}(t)$  to obtain the vector field. There are also flow-based methods, such as Lipman et al. (2023), that do not explicitly target  $\hat{x}_{0,\theta}$  or  $\hat{x}_{1,\theta}$ , but instead
+
+directly model the vector field  $H_{\theta}(t,x(t)) = v_{\theta}(t,x(t))$  with the flow model  $H_{\theta}$ . All these methods achieve generative modeling by numerically solving an ODE or SDE using the predicted vector field. We use DDPM (Ho et al., 2020), FM (Lipman et al., 2023), EDM (Karras et al., 2022), and SI (Albergo et al., 2023), as summarized in Appendix A.
+
+# 3.1. Multidimensional Adaptive Coefficient
+
+To introduce multidimensionality into the coefficient, we first define the multidimensional coefficient,  $\gamma(t) = [\gamma_0(t), \gamma_1(t)] \in \mathbb{R}^{d \times d \times 2}$ , which generalizes the unidimensional coefficient by extending it to higher dimensions.
+
+Definition 3.1. (Multidimensional Coefficient) Given a trajectory defined by  $x(t) = \gamma_0(t)x_0 + \gamma_1(t)x_1$ , where  $x_0, x_1 \in \mathbb{R}^d$ , the multidimensional coefficient is defined as:
+
+$$
+\gamma (t) = [ \gamma_ {0} (t), \gamma_ {1} (t) ]: [ 0, T ] \to \mathbb {R} ^ {d \times d \times 2},
+$$
+
+subject to the following conditions:  $\gamma_0(0) = I$ ,  $\gamma_1(0) = 0_{d\times d}$ ,  $\gamma_1(T) = TI$ , and  $\gamma (t)\in C^1 ([0,T],\mathbb{R}^{d\times d\times 2})$
+
+$\gamma (t)\in C^{1}([0,T],\mathbb{R}^{d\times d\times 2})$  indicates that  $\gamma$  is continuously differentiable to the first order with respect to  $t$  on the interval  $[0,T]$ . The boundary conditions ensure that  $x(t)$  becomes  $x_0$  and  $x_{T}$  for  $t = 0$  and  $t = T$ , respectively, which is necessary for transportation. The unidimensional coefficient  $\alpha$  can be regarded as a special case of the multidimensional coefficient  $\gamma$ , where  $\gamma$  is a scalar matrix for a given  $t$ .
+
+For adaptability with respect to different inference trajectories, we define a multidimensional coefficient  $\gamma_{\phi}$ , parameterized by  $\phi$ , allowing it to adapt to different inference trajectories  $x_{\theta,\phi}(t)$  over the inference times  $\tau = \{t_0, \dots, t_N\}$ :
+
+Definition 3.2. (Multidimensional Adaptive Coefficient (MAC)) Let  $S = \{t^{(1)},\ldots ,t^{(\ell)}\} \subseteq [0,T]$  be an arbitrary set of inference times, and define the corresponding inference trajectory as  $\mathbf{x}_{\theta ,\phi}^{\mathcal{S}} = [x_{\theta ,\phi}(t^{(1)}),\dots,x_{\theta ,\phi}(t^{(\ell)})]\in \mathbb{R}^{d\times \ell}$ . Then, the MAC  $\gamma_{\phi}(t,\mathbf{x}_{\theta ,\phi}^{\mathcal{S}}) = [\gamma_{0,\phi}(t,\mathbf{x}_{\theta ,\phi}^{\mathcal{S}}),\gamma_{1,\phi}(t,\mathbf{x}_{\theta ,\phi}^{\mathcal{S}})]$  is defined as:
+
+$$
+\gamma_ {\phi} (t, \mathbf {x} _ {\theta , \phi} ^ {\mathcal {S}}): [ 0, T ] \times \mathbb {R} ^ {d \times \ell} \to \mathbb {R} ^ {d \times d \times 2},
+$$
+
+and is parameterized by  $\phi$ . Boundary conditions and smoothness follow Definition 3.1.
+
+Efficient Computation of MAC We adopt a diagonal matrix for  $\gamma_{\phi}$ , which significantly reduces the output dimensionality and the size of the neural network for MAC, while retaining sufficient capacity to model non-linear trajectories. Accordingly, we consider  $\gamma(t) \in \mathbb{R}^{d \times 2}$  instead of  $\gamma(t) \in \mathbb{R}^{d \times d \times 2}$  (details in Appendix B). Furthermore, we evaluate the  $\phi$ -parameterized part of  $\gamma_{\phi}$  only once at  $t = T$  and reuse the resulting function throughout the entire inference schedule  $\tau$ . This allows us to compute  $\gamma_{\phi}$  over the full schedule with a single forward pass through the  $\phi$ -network.
+
+# 3.2. Inference Trajectory Optimization
+
+Consider a flow- and diffusion-based differential equation solver  $G_{\theta, \phi}$  with fixed configurations (e.g., NFE, integration schemes), where  $\theta$  represents the vector field from the pretrained flow or diffusion model parameters, and  $\phi$  is the parameter for MAC. We propose two optimization problems. The first is:
+
+$$
+\phi^ {*} = \underset {\phi} {\arg \min } \mathbb {D} \left(\rho_ {0}, \hat {\rho} _ {0, \theta , \phi}\right), \tag {3}
+$$
+
+where  $\hat{\rho}_{0,\theta,\phi}$  denotes the generated distribution from  $G_{\theta,\phi}$ , and  $\mathbb{D}$  is a divergence metric. Solving Equation 3 corresponds to inference trajectory optimization with a fixed vector field. This setting allows us to isolate the impact of optimizing MAC and clearly demonstrate its effectiveness.
+
+To achieve higher transportation performance, we optimize the full inference trajectory by jointly optimizing the vector field  $\theta$  and the inference plan  $\phi$ , as follows:
+
+$$
+\theta^ {*}, \phi^ {*} = \underset {\theta , \phi} {\arg \min } \mathbb {D} \left(\rho_ {0}, \hat {\rho} _ {0, \theta , \phi}\right). \tag {4}
+$$
+
+For low-dimensional datasets, the divergence terms above can be computed directly using the Wasserstein distance. For high-dimensional datasets, following Goodfellow et al. (2014), we minimize Equations 3 and 4 by using the following min-max problem:
+
+$$
+\left. \underset {\theta , \phi} {\min } \underset {\psi} {\max } \mathbb {E} _ {x _ {0}} \left[ \log D _ {\psi} \left(x _ {0}\right) \right] + \mathbb {E} _ {x _ {T}} \left[ \log \left(1 - D _ {\psi} \left(G _ {\theta , \phi} (\tau , x _ {T})\right)\right) \right], \right. \tag {5}
+$$
+
+where  $D_{\psi}$  represents the discriminator, and  $x_{T} \sim \rho_{T}$  is the starting point of the inference trajectory. As shown,  $\theta$  and  $\phi$  in  $G_{\theta, \phi}$  aim to deceive  $D_{\psi}$ . Based on Equation 5, we use the simulation-based adversarial objective exclusively for  $\phi$  (details in Section 3.2.3). Optionally, before trajectory optimization, we pre-train the flow and diffusion model  $H_{\theta}$  using various  $\gamma$  sampled from a well-designed hypothesis set  $\Gamma_h$  (details in Section 3.2.2).
+
+# 3.2.1. HYPOTHESIS SET FOR MAC
+
+We define the multidimensional coefficient set  $\Gamma$  as:
+
+$$
+\Gamma = \left\{\begin{array}{l}\gamma (t): [ 0, T ] \rightarrow \mathbb {R} ^ {d \times 2}, \quad \text {w h e r e}\\\gamma (t) = \left[ \gamma_ {0} (t), \gamma_ {1} (t) \right], \quad \gamma (t) \in C ^ {1} ([ 0, T ], \mathbb {R} ^ {d \times 2}),\\\gamma_ {0} (0) = \mathbf {1} _ {d}, \gamma_ {1} (0) = \mathbf {0} _ {d}, \gamma_ {1} (T) = T \mathbf {1} _ {d}\end{array}\right\}. \tag {6}
+$$
+
+To design the hypothesis set  $\Gamma_h \subseteq \Gamma$  for exploring  $\gamma_{\phi}$ , we consider three main properties. First, the hypothesis set should be broad enough to include the optimal coefficient  $\gamma_{\phi^*}$  while avoiding unnecessary complexity to reduce the burden on the flow and diffusion model. As shown in Figure 3, some multidimensional coefficients exhibit excessive
+
+![](images/3393915109bf4b1aa3c881d51833a1c145016407b400a1063698d06a299a6d84.jpg)  
+(a)
+
+![](images/3ad6cb763f608c8beb9fb4eb97bdfe31879b0b25f66d0ebb6a0ae6bd56a5d477.jpg)  
+(b)
+
+![](images/8be172604345676981ff833fa817e72a4f8d88285f590f707fe6b08dcc8880a8.jpg)  
+(c)
+
+![](images/a3db4046007e067d98d8911a0d28c494a6e6bb1e961a5c058350ef8e422f61d0.jpg)  
+(d)  
+Figure 3: Crude coefficients: (a) Oscillatory behavior in  $t$  due to high-frequency components; (b) High adjacent pixel differences. Refined coefficients: (c) Constrained multidimensionality for larger  $t$  in pre-training; (d) Unconstrained multidimensionality for adversarial training.
+
+high-frequency components in  $t$  and across different dimensions. Given the vast size of the coefficient set, it is crucial to exclude such crude coefficients using appropriate constraints and define a well-designed hypothesis set for  $\gamma_{\phi}$  to explore. Second, the computation of  $\gamma_{\phi}$  via the parameter  $\phi$  should require low NFE. Lastly, for pre-training the flow or diffusion models, the cost of sampling random  $\gamma \sim \Gamma_h$  should be low. Considering these factors, we choose to model the weights of sinusoids—akin to a Fourier expansion—by parameter  $\phi$ , as in Albergo et al. (2024). Our chosen design is:
+
+$$
+\Gamma_ {h} = \left\{\begin{array}{l}\gamma_ {\phi} (t, x _ {T}): [ 0, T ] \times \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {d \times 2}, \quad \text {w h e r e}\\\gamma_ {\phi} (t, x _ {T}) = \left[ \gamma_ {0, \phi} (t, x _ {T}), \gamma_ {1, \phi} (t, x _ {T}) \right],\\\gamma_ {0, \phi} (t, x _ {T}) = \mathcal {F} _ {0} \left(b _ {m} (t), w _ {\phi} \left(x _ {T}\right)\right),\\\gamma_ {1, \phi} (t, x _ {T}) = \mathcal {F} _ {1} \left(b _ {m} (t), w _ {\phi} \left(x _ {T}\right)\right), \quad \phi \in \mathcal {P}\end{array}\right\}, \tag {7}
+$$
+
+where  $\mathcal{P}$  represents the parameter set for  $\phi$ . The function  $\mathcal{F}$  constructs the weighted sinusoidal expansion using basis functions  $b_{m}(t)$  and their weights  $w_{\phi}$ , defined as:
+
+$$
+b _ {m} (t) = \sin (\pi m (t / T) ^ {1 / q}) \in \mathbb {R}, \quad q \in \mathbb {R}, \tag {8}
+$$
+
+$$
+w _ {\phi} \left(x _ {T}\right) = s \operatorname {L P F} \circ \tanh  \left(\mathrm {N N} _ {\phi} \left(x _ {T}\right)\right), \quad s \in \mathbb {R}, \tag {9}
+$$
+
+where  $w_{\phi}(x_T) \in \mathbb{R}^{d \times M \times 2}$  represents the multidimensional weights for the sinusoids, modeled by the neural network  $\mathrm{NN}_{\phi}$ . LPF refers to low-pass filtering, implemented as convolution with a Gaussian kernel, applied across different  $d$  dimensions to exclude high frequencies. This parameterization always satisfies  $\Gamma_h \subseteq \Gamma$ . For image generation, we employ a U-Net  $U_{\phi}$  for  $\mathrm{NN}_{\phi}$ , and condition it as  $U_{\phi}(x_T, c)$  for conditional generation. Details are provided in Appendix C.
+
+This parameterization satisfies all three properties mentioned above. First, high-frequency components in  $t$  and  $d$  can be excluded by controlling hyperparameters such as  $s$  and LPF configurations. Second,  $\gamma_{\phi}$  can be computed with 1 NFE using  $\mathrm{NN}_{\phi}$ . Lastly, because the tanh in Equation 9 bounds the output of  $\mathrm{NN}_{\phi}$  to  $(-1,1)$ , we can efficiently sample a random multidimensional coefficient  $\gamma(t,u) \sim \Gamma_h$  by directly sampling sinusoidal weights  $w$  from a uniform distribution rather than evaluating  $\mathrm{NN}_{\phi}$  during pre-training. Specifically,  $w(u) = s\mathrm{LPF} \circ u$ ,  $u \sim \mathcal{U}(-1,1) \in \mathbb{R}^{d \times M \times 2}$ .
+
+# 3.2.2. OPTIONAL  $\gamma$ -PRE-TRAINING FOR  $H_{\theta}$
+
+To ensure that  $H_{\theta}$  can better accommodate  $\gamma$ , we introduce a pre-training procedure. The loss functions for pre-training  $H_{\theta}$  follow the original flow and diffusion losses, except that  $\gamma$  is used instead of  $\alpha$ , and coefficient conditioning is incorporated. For example, in the case of DDPM, the loss function is given by  $\mathcal{L}_{\theta}^{\mathrm{pre}} = \mathbb{E}_{t,x_0,x_1,u}\| H_\theta (t,x(t),\gamma (t,u)) - x_0\| _2^2$ . Detailed versions of the loss functions for each flow and diffusion framework are provided in Appendix E.
+
+Coefficient Conditioning When adopting existing pretrained models for  $H_{\theta}$ , we apply  $H_{\theta}(\bar{\gamma}_{\phi}(t,x_T),x(t))$  during trajectory optimization, where  $\bar{\gamma}_{\phi}(t,x_{T})\in \mathbb{R}$  denotes the scalar average of  $\gamma_{\phi}(t,x_{T})$  over all dimensions. However, to allow  $\gamma_{\phi}$  to receive precise gradients through  $H_{\theta}$ , we incorporate  $\gamma_{\phi}(t,x_{T})$  directly into the model during inference trajectory optimization by using  $H_{\theta}(t,x(t),\gamma_{\phi}(t,x_{T}))$ . To support this without modifying the structure of  $H_{\theta}$ , we concatenate  $\gamma_{\phi}(t,x_{T})$  with  $x(t)$  along the channel axis as input to  $H_{\theta}$ . We also pre-train  $H_{\theta}$  with coefficient conditioning in the form  $H_{\theta}(t,x(t),\gamma (t,u))$ , consistent with its usage during inference trajectory optimization.
+
+$\Gamma_h$  for Pre-Training Given that a large  $\Gamma_h$  of coefficients for pre-training can burden  $H_{\theta}$  and potentially degrade performance, we use a smaller  $\Gamma_h$  during pre-training and fully enable multidimensionality across  $t$  during adversarial training, as shown in Figure 3 (c) and (d). Specifically, we use  $\gamma$  with high multidimensionality near  $t = 0$  and low multidimensionality at larger  $t$  by configuring the LPF during the pre-training of  $H_{\theta}$ . For adversarial training, we fully enable multidimensionality for  $\gamma_{\phi}$  across the entire range of  $t$ . Details are provided in Appendix C.
+
+Optionality of Pre-Training We emphasize that this pretraining stage is optional, as MAC can effectively leverage existing models pre-trained with  $\alpha$  without substantially compromising performance. While using  $\gamma$  for pre-training increases the probability of generating multidimensional interpolated values  $x(t)$ , multidimensional interpolation inherently arises even with a standard coefficient  $\alpha$ , since  $x_{1} \sim \rho_{1} = \mathcal{N}(0, I) \in \mathbb{R}^{d}$  implies that each dimension  $x_{1,i}$  is independently drawn from  $\mathcal{N}(0,1)$ . Consequently, for large-scale datasets where training from scratch is computa
+
+tionally prohibitive, MAC remains compatible with existing pre-trained models, ensuring practical applicability.
+
+# 3.2.3. ADVERSARIAL OPTIMIZATION
+
+For Equation 5, we use the hinge loss (Lim & Ye, 2017) with the StyleGAN-XL (Sauer et al., 2022) discriminator for  $D_{\psi}$ , as used in Kim et al. (2024). Details for the flow-and diffusion-based differential equation solver  $G_{\theta, \phi}$  are provided in Appendix D. We use separate loss functions for  $\theta$  and  $\phi$ , applying the simulation-based objective exclusively to  $\phi$  as follows:
+
+$$
+\begin{array}{l} \mathcal {L} _ {\phi} = - \mathbb {E} _ {x _ {T}} \left[ D _ {\psi} \left(G _ {\theta , \phi} (\tau , x _ {T})\right) \right], \\ \mathcal {L} _ {\psi} = \mathbb {E} _ {x _ {0}} [ \max  (0, 1 - D _ {\psi} (x _ {0})) ] \tag {10} \\ + \mathbb {E} _ {x _ {T}} [ \max  (0, 1 + D _ {\psi} (G _ {\theta , \phi} (\tau , x _ {T}))) ], \\ \end{array}
+$$
+
+where  $\mathcal{L}_{\phi}$  and  $\mathcal{L}_{\psi}$  indicate that gradients are computed with respect to  $\phi$  and  $\psi$ , respectively. Through these loss functions,  $\phi$  is optimized to determine better coefficients via simulation through  $G_{\theta,\phi}$ . For  $\theta$ , we use the adversarial loss defined as:
+
+$$
+\mathcal {L} _ {\theta} = - \mathbb {E} _ {t, x _ {0}, x _ {1}, z} \left[ D _ {\psi} \left(H _ {\theta} (t, x (t), \gamma_ {\phi} (t, z))\right) \right], \tag {11}
+$$
+
+$$
+x (t) = \gamma_ {0, \phi} (t, z) \odot x _ {0} + \gamma_ {1, \phi} (t, z) \odot x _ {1},
+$$
+
+where  $z\sim \rho_T$  (sampled like  $x_{T}$  , used only to train  $H_{\theta})$ $x_0\sim \rho_0,x_1\sim \rho_1,t\sim \tau$  and  $H_{\theta}(\bar{\gamma}_{\phi}(t,z),x(t))$  is used when coefficient conditioning is disabled. Given that  $H_{\theta}$  approximates  $x_0$  , it can be adversarially optimized with  $D_{\psi}$  Since  $H_{\theta}$  only needs to handle coefficients from  $\gamma_{\phi}$  rather than the full hypothesis set  $\Gamma_h$  , it is trained exclusively using  $\gamma_{\phi}$  , thereby reducing its computational burden, as  $\gamma_{\phi}$  is significantly smaller than  $\Gamma_h$  . The final loss terms are:
+
+$$
+\mathcal {L} _ {\phi , \psi} ^ {\mathrm {a d v}} = \mathcal {L} _ {\phi} + \mathcal {L} _ {\psi}, \tag {12}
+$$
+
+$$
+\mathcal {L} _ {\theta , \phi , \psi} ^ {\mathrm {a d v}} = \mathcal {L} _ {\theta} + \mathcal {L} _ {\phi} + \mathcal {L} _ {\psi}. \tag {13}
+$$
+
+Here, Equation 12 corresponds to Equation 3, and Equation 13 corresponds to Equation 4. The procedure corresponding to Equation 13 is summarized in Algorithm 1.
+
+Algorithm 1  $\gamma$ -Pre-Training and Adversarial Optimization  
+Input:  $\rho_0,\rho_1,\rho_T,H_\theta ,\gamma_\phi ,G_{\theta ,\phi},D_\psi ,\mathcal{T},\tau$    
+if  $\gamma$  -pretraining then repeat Sample  $x_0\sim \rho_0$ $x_{1}\sim \rho_{1}$ $u\sim \mathcal{U}(-1,1)$ $t\sim \mathcal{T}$ $[\hat{x}_{0,\theta},\hat{x}_{1,\theta}]\gets H_{\theta}(t,x(t),\gamma (t,u))$  Compute  $\mathcal{L}_{\theta}^{\mathrm{pre}}$  ; Update  $\theta$  until  $\theta$  converges   
+end if   
+repeat Sample  $x_0\sim \rho_0$ $x_{1}\sim \rho_{1}$ $x_{T}\sim \rho_{T}$ $z\sim \rho_T$ $t\sim \tau$ $\hat{x}_{0,\theta ,\phi}\leftarrow \left\{ \begin{array}{ll}H_{\theta}(t,x(t),\gamma_{\phi}(t,z)) & \text{with}\gamma \text{-pretraining}\\ H_{\theta}(\bar{\gamma}_{\phi}(t,z),x(t)) & \text{w/o}\gamma \text{-pretraining} \end{array} \right.$ $x_{\theta ,\phi}(t_N)\gets G_{\theta ,\phi}(\tau ,x_T)$  Compute  $\mathcal{L}_{\theta ,\phi ,\psi}^{\mathrm{adv}}$  ; Update  $\theta ,\phi ,\psi$    
+until  $\theta$ $\phi$  ，  $\psi$  converge
+
+Table 1:  ${\mathcal{W}}_{2}$  distance  $\left( \downarrow \right)$  for 2-dimensional transportation results. The best performance is highlighted.  
+
+<table><tr><td rowspan="2">Method \ NFE</td><td colspan="2">Gaussian to 8 Gaussians</td><td colspan="2">Gaussian to Moons</td><td colspan="2">8 Gaussians to Moons</td><td colspan="2">Moons to 8 Gaussians</td></tr><tr><td>5</td><td>10</td><td>5</td><td>10</td><td>5</td><td>10</td><td>5</td><td>10</td></tr><tr><td>SIα</td><td>0.763±0.040</td><td>0.673±0.055</td><td>0.882±0.035</td><td>0.643±0.060</td><td>0.981±0.112</td><td>0.649±0.165</td><td>1.271±0.185</td><td>0.998±0.203</td></tr><tr><td>SIγ + opt φ</td><td>0.721±0.082</td><td>0.452±0.033</td><td>0.682±0.093</td><td>0.359±0.098</td><td>0.924±0.235</td><td>0.311±0.051</td><td>0.908±0.109</td><td>0.500±0.072</td></tr><tr><td>SIFTα</td><td>0.457±0.021</td><td>0.440±0.052</td><td>0.245±0.023</td><td>0.217±0.019</td><td>0.321±0.064</td><td>0.318±0.068</td><td>0.488±0.050</td><td>0.492±0.056</td></tr><tr><td>SIFTγ + opt φ</td><td>0.399±0.017</td><td>0.415±0.016</td><td>0.230±0.015</td><td>0.188±0.006</td><td>0.258±0.015</td><td>0.221±0.014</td><td>0.421±0.012</td><td>0.407±0.031</td></tr></table>
+
+![](images/1e1cf4e5c6ccd85cd7a46d9fe2fabb6d3890e2958a1dc601df2b0f1148dabfcd.jpg)  
+(a)  $\mathrm{SI}_{\alpha}$
+
+![](images/bea61f59dea0925db5a179333e50403a28290d679a687543404b3c211d5f7d0d.jpg)  
+(b)  $\mathrm{SI}_{\gamma} + \mathrm{opt}\phi$
+
+![](images/a1872f38e123afe9930453aa96d4356cdac3a9ecf4b9a20c90cca3ebc0145ad5.jpg)  
+(c)  $\mathrm{SI}_{\alpha}^{\mathrm{OT}}$  
+Figure 4: Comparison of inference trajectories from 8 Gaussians to Moons.
+
+![](images/0272b193053bdbd490d56ff5a8bc524f98f21dbb2af032a23a255e6a1ebbdeae.jpg)  
+(d)  $\mathrm{SI}_{\gamma}^{\mathrm{OT}} + \mathrm{opt}\phi$
+
+# 4. Experiments
+
+# 4.1. 2-Dimensional Transportation: Optimizing Only  $\phi$
+
+We conduct experiments on 2-dimensional synthetic datasets used by Tong et al. (2024) with SI (Albergo et al., 2023). To validate the benefits of using MAC  $\gamma_{\phi}$ ,
+
+we train  $\phi$  exclusively while freezing  $\theta$ , as in Equation 3, ensuring a fair comparison with baseline methods. We use the Wasserstein loss function  $\mathcal{L}_{\phi} = \mathcal{W}_2(x_0,G_{\theta ,\phi}(\tau ,x_T))$  instead of a discriminator. To further evaluate whether MAC improves transportation even when optimality is defined as a straight trajectory, we test additional con
+
+figurations of minibatch pairing for pre-training  $(x_0, x_1)$ : random pairing and OT pairing (Tong et al., 2024). The minibatch-OT method encourages the flow and diffusion
+
+![](images/d22b87b023d719dc677ab3fc133028129f5a43f82711fb16936e4c053d8f2d83.jpg)  
+Figure 5:  $x_{T} \sim \rho_{T}$  (black) and  $x_{0} \sim \rho_{0}$  (blue).
+
+Table 2: FID (↓) of models pre-trained with  $\alpha$  and  $\gamma$ .  
+
+<table><tr><td rowspan="2">Method\NFE</td><td colspan="3">CIFAR-10</td><td colspan="3">ImageNet-32</td></tr><tr><td>100</td><td>150</td><td>200</td><td>100</td><td>150</td><td>200</td></tr><tr><td>SIα</td><td>4.75</td><td>4.51</td><td>4.30</td><td>8.08</td><td>7.79</td><td>7.63</td></tr><tr><td>SIγ</td><td>3.98</td><td>3.74</td><td>3.63</td><td>6.33</td><td>6.21</td><td>6.20</td></tr><tr><td>FMα</td><td>4.52</td><td>4.23</td><td>4.07</td><td>7.78</td><td>7.53</td><td>7.38</td></tr><tr><td>FMγ</td><td>3.59</td><td>3.42</td><td>3.42</td><td>6.18</td><td>6.03</td><td>6.01</td></tr><tr><td>DDPMα</td><td>6.64</td><td>4.84</td><td>4.10</td><td>8.13</td><td>7.40</td><td>7.14</td></tr><tr><td>DDPMγ</td><td>4.73</td><td>4.11</td><td>3.83</td><td>6.84</td><td>6.51</td><td>6.42</td></tr></table>
+
+Table 3: FID after adversarial optimization for varying NFE on CIFAR-10.  
+
+<table><tr><td>Method \ NFE</td><td>4 (+)</td><td>6 (+)</td><td>8 (+)</td><td>10 (+)</td></tr><tr><td>SIγ + adv φ</td><td>20.59</td><td>6.62</td><td>4.85</td><td>4.14</td></tr><tr><td>FMγ + adv φ</td><td>16.42</td><td>8.17</td><td>6.56</td><td>6.13</td></tr><tr><td>DDPMγ + adv φ</td><td>72.64</td><td>20.13</td><td>13.72</td><td>10.04</td></tr></table>
+
+model to learn a straight trajectory by pairing  $x_0$  and  $x_1$  as an OT solution within a minibatch during pre-training, where optimality for the inference trajectory is defined as straight. Details are provided in Appendix E.1.
+
+As shown in Table 1, using MAC consistently achieves the best results, even for models trained with minibatch-OT. This suggests that a straight trajectory is not always optimal, even in OT-trained models, and that MAC can adaptively discover better inference trajectories to correct errors. Figure 4 further illustrates how MAC adjusts the trajectory direction to optimize transportation, resulting in a path that is not straight. A comparison of (c) with (d) reveals a distinct piecewise linear trajectory in (d), indicating that a nonstraight trajectory achieves superior performance. These results empirically suggest that the optimality of the inference trajectory should be defined and explored in terms of transportation quality, rather than by a predefined property.
+
+Table 4: FID after adversarial optimization with different conditionings of  ${\gamma }_{\phi }\left( {t,\cdot }\right)$  on CIFAR-10 (10 (+) NFE).  
+
+<table><tr><td>Method \ Conditioning</td><td>1d</td><td>z ~ ρT</td><td>xt ~ ρT</td></tr><tr><td>SIγ + adv φ</td><td>7.84</td><td>6.48</td><td>4.14</td></tr><tr><td>FMγ + adv φ</td><td>9.20</td><td>9.06</td><td>6.13</td></tr><tr><td>DDPMγ + adv φ</td><td>26.09</td><td>23.31</td><td>10.04</td></tr></table>
+
+Table 5: Performance comparisons on CIFAR-10, FFHQ, AFHQ, and ImageNet. * indicates metrics computed by us.  
+
+<table><tr><td>Model</td><td>NFE</td><td>CIFAR-10 uncond. FID (↓)</td><td>CIFAR-10 cond. FID (↓)</td></tr><tr><td>GAN</td><td></td><td></td><td></td></tr><tr><td>StyleGAN-Ada (Karras et al., 2020)</td><td>1</td><td>2.92</td><td>2.42</td></tr><tr><td>StyleGAN-XL (Sauer et al., 2022)</td><td>1</td><td>-</td><td>1.85</td></tr><tr><td>Diffusion Model</td><td></td><td></td><td></td></tr><tr><td>DDPMα (Ho et al., 2020)</td><td>1000</td><td>3.17</td><td>-</td></tr><tr><td>Score SDEα (Song et al., 2021)</td><td>2000</td><td>2.38</td><td>2.20</td></tr><tr><td>EDMα (Karras et al., 2022)</td><td>35</td><td>1.98</td><td>1.79</td></tr><tr><td>Rectified Flow (Distillation)</td><td></td><td></td><td></td></tr><tr><td>2-Rectified Flowα (Liu et al., 2023)</td><td>1</td><td>4.85</td><td>-</td></tr><tr><td>2-Rectified Flowα++ (Lee et al., 2024)</td><td>1</td><td>3.07</td><td>-</td></tr><tr><td></td><td>2</td><td>2.40</td><td>-</td></tr><tr><td>Consistency Model (Distillation)</td><td></td><td></td><td></td></tr><tr><td>CDα (Song et al., 2023)</td><td>1</td><td>3.55</td><td>-</td></tr><tr><td></td><td>2</td><td>2.93</td><td>-</td></tr><tr><td>CDα + adv θ (Lu et al., 2023)</td><td>1</td><td>2.65</td><td>-</td></tr><tr><td>CTMα (Kim et al., 2024)</td><td>1</td><td>5.19</td><td>-</td></tr><tr><td>CTMα + adv θ (Kim et al., 2024)</td><td>1</td><td>1.98</td><td>1.73</td></tr><tr><td></td><td>2</td><td>1.87</td><td>1.63</td></tr><tr><td></td><td>5</td><td>1.86*</td><td>1.98*</td></tr><tr><td></td><td>6</td><td>1.93*</td><td>2.04*</td></tr><tr><td>Inference Trajectory Optimization</td><td></td><td></td><td></td></tr><tr><td>EDMγ + adv θ, φ (Ours)</td><td>5 (+)</td><td>1.69*</td><td>1.37*</td></tr></table>
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">NFE</td><td colspan="2">ImageNet-64 cond.</td></tr><tr><td>FID (↓)</td><td>FDINov2 (↓)</td></tr><tr><td>GAN</td><td></td><td></td><td></td></tr><tr><td>StyleGAN-XL (Sauer et al., 2022)</td><td>1</td><td>2.09</td><td>-</td></tr><tr><td>Diffusion Model</td><td></td><td></td><td></td></tr><tr><td>ADMα (Dhariwal &amp; Nichol, 2021)</td><td>250</td><td>2.07</td><td>-</td></tr><tr><td>EDMα (Karras et al., 2022)</td><td>79</td><td>2.44</td><td>-</td></tr><tr><td>Consistency Model (Distillation)</td><td></td><td></td><td></td></tr><tr><td rowspan="2">CDα (Song et al., 2023)</td><td>1</td><td>6.20</td><td>-</td></tr><tr><td>2</td><td>4.70</td><td>-</td></tr><tr><td rowspan="4">CTMα + adv θ (Kim et al., 2024)</td><td>1</td><td>1.92</td><td>160.8*</td></tr><tr><td>2</td><td>1.73</td><td>157.7*</td></tr><tr><td>5</td><td>3.02*</td><td>195.0*</td></tr><tr><td>6</td><td>3.17*</td><td>205.1*</td></tr><tr><td>Inference Trajectory Optimization</td><td></td><td></td><td></td></tr><tr><td>EDMα + adv θ, φ (Ours)</td><td>5 (+)</td><td>1.48*</td><td>70.2*</td></tr><tr><td>Model</td><td>NFE</td><td>FFHQ-64 FID (↓)</td><td>AFHQ-64 FID (↓)</td></tr><tr><td>Diffusion Model</td><td></td><td></td><td></td></tr><tr><td>EDMα (Karras et al., 2022)</td><td>79</td><td>2.39</td><td>1.96</td></tr><tr><td>Rectified Flow (Distillation)</td><td></td><td></td><td></td></tr><tr><td rowspan="2">2-Rectified Flowα++ (Lee et al., 2024)</td><td>1</td><td>5.21</td><td>4.11</td></tr><tr><td>2</td><td>4.26</td><td>3.12</td></tr><tr><td>Inference Trajectory Optimization</td><td></td><td></td><td></td></tr><tr><td>EDMγ + adv θ, φ (Ours)</td><td>5 (+)</td><td>2.27*</td><td>2.04*</td></tr></table>
+
+![](images/6991a6b1b1f0bb5140ad74b1c62263b5038586932f067a51680e18280dde5994.jpg)  
+(a) CIFAR-10 uncond.
+
+![](images/7b421af1fc99e94ba4cc5fefbc460767ec9f12b9561e49fbf7ac5310a1b45939.jpg)  
+(b) CIFAR-10 cond.  
+Figure 6:  $\mathrm{EDM}_{\gamma} + \mathrm{adv}\theta$  and  $\theta ,\phi$
+
+Table 6: FID from the ablation study on CIFAR-10. Experiments used 500k images, compared to 1500k in Table 5.  
+
+<table><tr><td rowspan="2">Configuration \ NFE</td><td colspan="2">Unconditional</td><td colspan="2">Conditional</td></tr><tr><td>5 (Euler)</td><td>35 (Heun)</td><td>5 (Euler)</td><td>35 (Heun)</td></tr><tr><td>EDMα</td><td>68.73</td><td>1.97</td><td>48.76</td><td>1.79</td></tr><tr><td>EDMγ</td><td>69.58</td><td>2.08</td><td>48.53</td><td>1.81</td></tr><tr><td>EDMγ + adv φ (no multi.)</td><td>33.55</td><td>-</td><td>25.56</td><td>-</td></tr><tr><td>EDMγ + adv φ</td><td>18.67</td><td>-</td><td>7.77</td><td>-</td></tr><tr><td>EDMγ + adv θ</td><td>2.28</td><td>-</td><td>2.14</td><td>-</td></tr><tr><td>EDMγ + adv θ, φ</td><td>1.81</td><td>-</td><td>1.42</td><td>-</td></tr></table>
+
+# 4.2. Image Generation: Optimizing Only  $\phi$
+
+To isolate the benefits of using MAC in image generation, we conduct experiments that optimize only  $\phi$ . We employ DDPM (Ho et al., 2020), FM (Lipman et al., 2023), and SI (Albergo et al., 2023) on the CIFAR-10 (Krizhevsky & Hinton, 2009) and ImageNet-32 (Deng et al., 2009) datasets. Detailed training configurations and additional results are presented in Appendix E.2.
+
+Pre-Training Before adversarial optimization, we compare the Fréchet Inception Distance (FID) (Heusel et al., 2017) of models trained with  $\alpha$  and  $\gamma$ . Interestingly, as shown in Table 2, models trained with  $\gamma$  achieve the lowest FID across all frameworks and NFE compared to models trained with  $\alpha$ . These results suggest that training flow and diffusion models with  $\gamma \sim \Gamma_h$  and coefficient labeling can improve the innate performance of the model, despite the increased complexity compared to  $\alpha$ .
+
+Adversarial Training We achieve approximately  $10 \times$  NFE efficiency across all frameworks. Specifically, in SI, we obtain FID scores of 4.14 and 7.06 on CIFAR-10 and ImageNet-32, respectively, using  $10(+)$  NFE (where  $(+)$  accounts for the computation of  $\gamma_{\phi}$ , given that the size of  $U_{\phi}$  is smaller than that of  $H_{\theta}$ ). These results demonstrate the compatibility of MAC across various frameworks. As shown in Table 3, MAC can also be optimized for different sampling configurations. Additionally, we validate the impact of using the starting point  $x_{T}$  as a condition for MAC. As shown in Table 4, incorporating  $x_{T}$  as an input to  $\gamma_{\phi}$  consistently improves performance across all frameworks, providing empirical evidence of MAC's adaptability to  $x_{T}$ .
+
+# 4.3. Image Generation: Optimizing  $\theta$  and  $\phi$
+
+We optimize  $\theta$  and  $\phi$  using Equation 13 on the CIFAR-10 (Krizhevsky & Hinton, 2009), FFHQ-64 (Karras et al., 2019), AFHQv2-64 (Choi et al., 2020), and ImageNet-64
+
+![](images/8fd58d794d581d01fd0a7ddacb3bc0379808efa7c4e2cc2ac3f71c87ed7d7fc4.jpg)  
+(a) CIFAR-10 uncond.
+
+![](images/6f937aad6c0629194604e40ccaeccca0ab14976290df318667e5ab44296feec6.jpg)  
+(b) CIFAR-10 cond.
+
+(Deng et al., 2009) datasets with EDM (Karras et al., 2022). As mentioned in Section 3.2.2, since pre-training  $H_{\theta}$  with  $\gamma$  is optional for practicality, we use the existing pre-trained model  $\mathrm{EDM}_{\alpha}$  for ImageNet. We measure FID and  $\mathrm{FD}_{\mathrm{DINOv2}}$  (Oquab et al., 2024). Details are provided in Appendix E.3.
+
+Impact of Trajectory Optimization As shown in Table 5, our approach generates high-quality samples across various datasets with only 5 (+) NFE, achieving a state-of-the-art result (FID = 1.37) on CIFAR-10 conditional generation. For a fair comparison in terms of NFE, we select CTM (Kim et al., 2024) due to its popularity, strong performance, and use of the same model architecture based on EDM and adversarial training with StyleGAN-XL discriminator. We then compute FID using 5 and 6 NFE on CIFAR-10 and ImageNet-64. As shown in Table 5, increasing the NFE of CTM does not significantly reduce the FID, and in some cases, the FID even increases. This indicates that MAC provides additional performance gains that cannot be achieved by distillation or vector field tuning methods alone, even with increased NFE.
+
+To empirically validate the benefit of MAC, we conduct ablation studies by training either  $\theta$  or  $\phi$  individually. As presented in Table 6, the results reveal that jointly training  $\theta$  and  $\phi$  yields the best performance. Figure 6 further illustrates that FID decreases more rapidly during joint training compared to training  $\theta$  alone. These findings suggest that performance improvements stem not only from the adversarial training of  $H_{\theta}$ , but also from the combined training of both  $H_{\theta}$  and  $\gamma_{\phi}$ , demonstrating the benefits of MAC.
+
+![](images/f59f6500aebe4bff313894669d8b8fe8d3605bd9e8c7187160b4782a1e02c454.jpg)  
+(a) CIFAR-10 uncond.
+
+![](images/e2aae6c380cf0e3d90437aeed02a2dc8ace023de9e1be8b920aa9f9581d66103.jpg)  
+(b) CIFAR-10 cond.
+
+![](images/af00e6ce9f802a46518e85e668c86b6459d651718ff33288d9acc8b4362cb561.jpg)  
+Figure 7: Training with different multidimensionalities. T/F indicate whether multidimensionality is applied along the channel, height, and width axes, respectively.  
+(c) FFHQ.  
+Figure 8: T-SNE visualization of various coefficients. Yellow:  $w = 0$  (corresponding to  $\alpha$ ); Purple:  $w = su$ ; Green:  $w = s\mathrm{LPF} \circ u$  (used for pre-training); Blue:  $w_{\phi} = s\mathrm{LPF} \circ \tanh(U_{\phi}(x_T, c))$  (trained).
+
+![](images/3f9aeb0607b268fc7eb701fb15ddb645ccb71529c8092958a650b0c41b997768.jpg)  
+(d) AFHQ.
+
+Impact of Multidimensionality of MAC To examine how the multidimensionality of MAC influences performance, we train  $\phi$  with different levels of multidimensionality by averaging  $\tanh(U_{\phi}(x_T))$  in Equation 9 across specific axes. For instance, to retain multidimensionality solely in the height axis ([F, T, F]), we use the same  $w_{\phi}$  across the channel and width axes by averaging along those dimensions. As shown in Figure 7, incorporating more axes consistently improves performance, indicating that MAC's multidimensionality positively impacts generation quality.
+
+Analysis of Trained MAC To analyze how MAC is trained, we plot t-SNE embeddings of four different sinusoidal weights  $w$  across multiple datasets, as shown in Figure 8. Notably, the trained  $w_{\phi}$  diverges from weights randomly sampled from the predefined hypothesis set and is far from  $\alpha$ . This suggests that during adversarial training,  $\gamma_{\phi}$  adaptively identifies optimal coefficients without heavily depending on the pre-trained distribution of  $\gamma \sim \Gamma_h$ . Interestingly,  $\gamma_{\phi}$  exhibits a sparser distribution in CIFAR-10 conditional generation than in the unconditional setting, with nearly identical  $w_{\phi}$  values for samples sharing the same label condition  $c$ . This indicates that the optimality of the coefficient depends more on the label condition  $c$  than on the starting point  $x_{T}$  of the differential equation.
+
+Training Efficiency Our method demonstrates notable training efficiency despite incorporating simulation-based training. As shown in Table 7, the number of training images required by our approach is significantly lower across all datasets—approximately 20 to 2000 times fewer than distillation-based methods such as CTM.
+
+Training times are 10, 2, 6, and 9 hours for CIFAR-10, FFHQ, AFHQ, and ImageNet, respectively, which are also comparatively low. The primary cost of simulation
+
+Table 7: kimg  $(\downarrow)$  for adversarial training.  
+
+<table><tr><td>Dataset</td><td>CTMα + adv θ</td><td>Ours</td></tr><tr><td>CIFAR-10</td><td>25.6k</td><td>1.5k</td></tr><tr><td>FFHQ</td><td>38.4k</td><td>1.0k</td></tr><tr><td>AFHQ</td><td>51.2k</td><td>1.0k</td></tr><tr><td>ImageNet</td><td>61.4k</td><td>30</td></tr></table>
+
+dynamics arises from VRAM requirements. However, this remains practically feasible, as strong performance can be achieved with just 5 (+) NFE. These results not only highlight the effectiveness of simulation-based optimality, but also showcase the strength of combining simulation-free and simulation-based methodologies—leveraging the advantages of each while mitigating their limitations.
+
+# 5. Conclusions
+
+We have introduced the Multidimensional Adaptive Coefficient (MAC), a plug-in module for flow- and diffusion-based generative models that enables both freedom of dimensionality and adaptability to different inference trajectories—properties previously found only in simulation-based generative modeling. Our method improves generative performance across various datasets and frameworks with high training efficiency. These results highlight the potential of inference trajectory optimization with MAC via simulation—a direction that has been underexplored compared to vector field design. We suggest broadening the notion of trajectory optimality beyond predefined criteria such as straightness, toward a more general view based on final transportation quality. We hope our work encourages further exploration.
+
+# 6. Future Directions
+
+First, the current implementation of  $\gamma_{\phi}$  is tied to a fixed NFE during trajectory optimization. Future work could explore adaptive control of NFE based on inference trajectories. Second, while  $\gamma_{\phi}$  currently conditions only on  $x_{T}$ , future extensions could, with appropriate design, condition on trajectories  $\mathbf{x}_{\theta ,\phi}^S$  over diverse inference times and incorporate other contextual signals. Third, optimizing MAC under alternative objectives beyond distributional divergence may broaden its applicability to domains where task-specific criteria are more relevant. Finally, interpreting MAC selection as a form of trajectory-level policy learning may inspire novel approaches, particularly when combined with ideas from control theory or reinforcement learning.
+
+# Impact Statement
+
+Our research introduces the Multidimensional Adaptive Coefficient (MAC), enhancing the generative quality and efficiency of flow and diffusion models. The broader implications of this advancement are twofold. On the positive side, MAC improves the accessibility and practicality of generative models, potentially benefiting diverse applications such as medical imaging, drug discovery, and creative industries by reducing computational costs and enhancing output quality. However, improved generative technologies also carry potential risks, including the misuse in creating deepfakes or synthetic biological threats. It is essential for future work to develop robust safeguards and detection methods to mitigate such harmful applications. Continued interdisciplinary efforts in governance, ethics, and technical controls will be critical to ensuring the beneficial integration of our contributions into society.
+
+# Acknowledgement
+
+This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) [NO.RS-2021-II211343, Artificial Intelligence Graduate School Program (Seoul National University)]
+
+# References
+
+Albergo, M. S., Boffi, N. M., and Vanden-Eijnden, E. Stochastic interpolants: A unifying framework for flows and diffusions. arXiv preprint arXiv:2303.08797, 2023.  
+Albergo, M. S., Boffi, N. M., Lindsey, M., and Vanden-Eijnden, E. Multimarginal generative modeling with stochastic interpolants. In The Twelfth International Conference on Learning Representations, 2024.  
+Bartosh, G., Vetrov, D., and Naesseth, C. A. Neural flow diffusion models: Learnable forward process for improved diffusion modelling. In Globerson, A., Mackey, L., Belgrave, D., Fan, A., Paquet, U., Tomczak, J., and Zhang, C. (eds.), Advances in Neural Information Processing Systems, volume 37, pp. 73952-73985. Curran Associates, Inc., 2024.  
+Berthelot, D., Autef, A., Lin, J., Yap, D. A., Zhai, S., Hu, S., Zheng, D., Talbott, W., and Gu, E. Tract: Denoising diffusion models with transitive closure time-distillation. arXiv preprint arXiv:2303.04248, 2023.  
+Chen, R. T. Q., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. In Bengio, S., Wallach, H., Larochelle, H., Grauman, K., Cesa-Bianchi, N., and Garnett, R. (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.
+
+Choi, Y., Uh, Y., Yoo, J., and Ha, J.-W. Stargan v2: Diverse image synthesis for multiple domains. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.  
+Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., and Fei-Fei, L. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248-255. IEEE, 2009.  
+Dhariwal, P. and Nichol, A. Diffusion models beat gans on image synthesis. In Ranzato, M., Beygelzimer, A., Dauphin, Y., Liang, P., and Vaughan, J. W. (eds.), Advances in Neural Information Processing Systems, volume 34, pp. 8780-8794. Curran Associates, Inc., 2021.  
+Geng, Z., Pokle, A., and Kolter, J. Z. One-step diffusion distillation via deep equilibrium models. In Oh, A., Naumann, T., Globerson, A., Saenko, K., Hardt, M., and Levine, S. (eds.), Advances in Neural Information Processing Systems, volume 36, pp. 41914-41931. Curran Associates, Inc., 2023.  
+Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y. Generative adversarial nets. In Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N., and Weinberger, K. (eds.), Advances in Neural Information Processing Systems, volume 27. Curran Associates, Inc., 2014.  
+Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., and Hochreiter, S. Gans trained by a two time-scale update rule converge to a local nash equilibrium. In Guyon, I., Luxburg, U. V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., and Garnett, R. (eds.), Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.  
+Ho, J., Jain, A., and Abbeel, P. Denoising diffusion probabilistic models. In Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., and Lin, H. (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 6840-6851. Curran Associates, Inc., 2020.  
+Kapušniak, K., Potapchik, P., Reu, T., Zhang, L., Tong, A., Bronstein, M., Bose, A. J., and Di Giovanni, F. Metric flow matching for smooth interpolations on the data manifold. In Globerson, A., Mackey, L., Belgrave, D., Fan, A., Paquet, U., Tomczak, J., and Zhang, C. (eds.), Advances in Neural Information Processing Systems, volume 37, pp. 135011-135042. Curran Associates, Inc., 2024.  
+Karras, T., Laine, S., and Aila, T. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.
+
+Karras, T., Aittala, M., Hellsten, J., Laine, S., Lehtinen, J., and Aila, T. Training generative adversarial networks with limited data. In Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., and Lin, H. (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 12104-12114. Curran Associates, Inc., 2020.  
+Karras, T., Aittala, M., Aila, T., and Laine, S. Elucidating the design space of diffusion-based generative models. In Koyejo, S., Mohamed, S., Agarwal, A., Belgrave, D., Cho, K., and Oh, A. (eds.), Advances in Neural Information Processing Systems, volume 35, pp. 26565-26577. Curran Associates, Inc., 2022.  
+Karras, T., Aittala, M., Lehtinen, J., Hellsten, J., Aila, T., and Laine, S. Analyzing and improving the training dynamics of diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 24174-24184, June 2024.  
+Kim, B., Hsieh, Y.-G., Klein, M., marco cuturi, Ye, J. C., Kawar, B., and Thornton, J. Simple reflow: Improved techniques for fast flow models. In The Thirteenth International Conference on Learning Representations, 2025.  
+Kim, D., Lai, C.-H., Liao, W.-H., Murata, N., Takida, Y., Uesaka, T., He, Y., Mitsufuji, Y., and Ermon, S. Consistency trajectory models: Learning probability flow ODE trajectory of diffusion. In The Twelfth International Conference on Learning Representations, 2024.  
+Krizhevsky, A. and Hinton, G. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.  
+Lee, S., Kim, B., and Ye, J. C. Minimizing trajectory curvature of ODE-based generative models. In Krause, A., Brunskill, E., Cho, K., Engelhardt, B., Sabato, S., and Scarlett, J. (eds.), Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pp. 18957-18973. PMLR, 23-29 Jul 2023.  
+Lee, S., Lin, Z., and Fanti, G. Improving the training of rectified flows. In Globerson, A., Mackey, L., Belgrave, D., Fan, A., Paquet, U., Tomczak, J., and Zhang, C. (eds.), Advances in Neural Information Processing Systems, volume 37, pp. 63082-63109. Curran Associates, Inc., 2024.  
+Lim, J. H. and Ye, J. C. Geometric gan, 2017.  
+Lipman, Y., Chen, R. T. Q., Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling. In The Eleventh International Conference on Learning Representations, 2023.
+
+Liu, X., Gong, C., and qiang liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. In The Eleventh International Conference on Learning Representations, 2023.  
+Lu, H., Lu, Y., Jiang, D., Szabados, S. R., Sun, S., and Yu, Y. CM-GAN: Stabilizing GAN training with consistency models. In ICML 2023 Workshop on Structured Probabilistic Inference & Generative Modeling, 2023.  
+Luo, W., Hu, T., Zhang, S., Sun, J., Li, Z., and Zhang, Z. Diff-instruct: A universal approach for transferring knowledge from pre-trained diffusion models. In Oh, A., Naumann, T., Globerson, A., Saenko, K., Hardt, M., and Levine, S. (eds.), Advances in Neural Information Processing Systems, volume 36, pp. 76525-76546. Curran Associates, Inc., 2023.  
+Oquab, M., Darcet, T., Moutakanni, T., Vo, H. V., Szafraniec, M., Khalidov, V., Fernandez, P., HAZIZA, D., Massa, F., El-Nouby, A., Assran, M., Ballas, N., Galuba, W., Howes, R., Huang, P.-Y., Li, S.-W., Misra, I., Rabbat, M., Sharma, V., Synnaeve, G., Xu, H., Jegou, H., Mairal, J., Labatut, P., Joulin, A., and Bojanowski, P. DINoV2: Learning robust visual features without supervision. Transactions on Machine Learning Research, 2024. ISSN 2835-8856. Featured Certification.  
+Park, D., Lee, S., Kim, S., Lee, T., Hong, Y., and Kim, H. J. Constant acceleration flow. In Globerson, A., Mackey, L., Belgrave, D., Fan, A., Paquet, U., Tomczak, J., and Zhang, C. (eds.), Advances in Neural Information Processing Systems, volume 37, pp. 90030-90060. Curran Associates, Inc., 2024.  
+Ronneberger, O., Fischer, P., and Brox, T. U-net: Convolutional networks for biomedical image segmentation. In Medical image computing and computer-assisted intervention-MICCAI 2015: 18th international conference, Munich, Germany, October 5-9, 2015, proceedings, part III 18, pp. 234-241. Springer, 2015.  
+Sauer, A., Schwarz, K., and Geiger, A. Stylegan-xl: Scaling stylegan to large diverse datasets. In ACM SIGGRAPH 2022 Conference Proceedings, SIGGRAPH '22, New York, NY, USA, 2022. Association for Computing Machinery. ISBN 9781450393379. doi: 10.1145/3528233.3530738.  
+Singhal, R., Goldstein, M., and Ranganath, R. Where to diffuse, how to diffuse, and how to get back: Automated learning for multivariate diffusions. In The Eleventh International Conference on Learning Representations, 2023.  
+Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N., and Ganguli, S. Deep unsupervised learning using nonequilibrium thermodynamics. In Bach, F. and Blei, D. (eds.),
+
+Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pp. 2256-2265, Lille, France, 07-09 Jul 2015. PMLR.  
+Song, Y. and Dhariwal, P. Improved techniques for training consistency models. In The Twelfth International Conference on Learning Representations, 2024.  
+Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., and Poole, B. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021.  
+Song, Y., Dhariwal, P., Chen, M., and Sutskever, I. Consistency models. In Krause, A., Brunskill, E., Cho, K., Engelhardt, B., Sabato, S., and Scarlett, J. (eds.), Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pp. 32211-32252. PMLR, 23-29 Jul 2023.  
+Tong, A., FATRAS, K., Malkin, N., Huguet, G., Zhang, Y., Rector-Brooks, J., Wolf, G., and Bengio, Y. Improving and generalizing flow-based generative models with minibatch optimal transport. Transactions on Machine Learning Research, 2024. ISSN 2835-8856. Expert Certification.  
+Wang, Z., Zheng, H., He, P., Chen, W., and Zhou, M. Diffusion-GAN: Training GANs with diffusion. In The Eleventh International Conference on Learning Representations, 2023.  
+Xu, Y., Zhao, Y., Xiao, Z., and Hou, T. Ufogen: You forward once large scale text-to-image generation via diffusion gans. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8196-8206, June 2024.  
+Yin, T., Gharbi, M., Zhang, R., Shechtman, E., Durand, F., Freeman, W. T., and Park, T. One-step diffusion with distribution matching distillation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6613-6623, 2024.
+
+# Outline of Appendix
+
+Appendix A provides an overview of flow and diffusion frameworks. Appendix B provides justification for the use of diagonal matrices in the design of  $\gamma_{\phi}$ . Appendix C describes the hypothesis set for MAC. Appendix D presents details of the differential equation solver  $G_{\theta, \phi}$ . Appendix E outlines experimental setups, including datasets, training configurations, and additional results. Appendix F explains evaluation metrics, and Appendix G provides visualizations of generated samples and inference trajectories.
+
+# A. Flow and Diffusion Frameworks
+
+During inference, all frameworks transport samples between  $\rho_T$  and  $\rho_0$ , either in the forward or reverse direction.
+
+Denoising Diffusion Probabilistic Models (DDPM) DDPM (Ho et al., 2020) is one of the most widely used diffusion-based generative frameworks. Given  $x_0 \sim \rho_0 = \mathcal{N}(0, I)$  and  $x_1 \sim \rho_1$  as data, the VP diffusion coefficient is:
+
+$$
+\alpha (t) = [ \alpha_ {0} (t), \alpha_ {1} (t) ] = \left[ \sqrt {1 - b _ {1 - t} ^ {2}}, b _ {1 - t} \right], \quad b _ {t} = e ^ {- \frac {1}{2} \int_ {0} ^ {t} c (s) d s}, \quad c (s) = c _ {\min } + s (c _ {\max } - c _ {\min }), \quad T = 1, \tag {14}
+$$
+
+where  $c_{\mathrm{min}} = 0.1$  and  $c_{\mathrm{max}} = 20$ . DDPM minimizes the loss  $\mathcal{L}_{\theta} = \mathbb{E}_{t,x_0,x_1}\left[\| H_{\theta}(t,x(t)) - x_0\| _2^2\right]$ , where  $t\sim \mathcal{U}(0,1)$ .
+
+Flow Matching (FM) FM (Lipman et al., 2023) introduces a simple yet effective objective for generation. Given  $x_0 \sim \mathcal{N}(0, I)$  and  $x_1 \sim \rho_1$  as data, it employs the following coefficient:
+
+$$
+\alpha (t) = \left[ \alpha_ {0} (t), \alpha_ {1} (t) \right] = \left[ 1 - \left(1 - \sigma_ {\min }\right) t, t \right], \quad T = 1. \tag {15}
+$$
+
+FM directly models  $v$  using the objective  $\mathcal{L}_{\theta} = \mathbb{E}_{t,x_0,x_1}\left[\| H_{\theta}(t,x(t)) - (x_1 - (1 - \sigma_{\min})x_0)\| _2^2\right]$ , where  $t\sim \mathcal{U}(0,1)$ .
+
+Elucidating Diffusion Model (EDM) EDM (Karras et al., 2022) refines and stabilizes diffusion model training. Given  $x_0 \sim \rho_0$  as data and  $x_1 \sim \rho_1 = \mathcal{N}(0, I)$ , the coefficient is defined as:
+
+$$
+\alpha (t) = \left[ \alpha_ {0} (t), \alpha_ {1} (t) \right] = [ 1, t ], \quad T = 8 0. \tag {16}
+$$
+
+The loss function is:
+
+$$
+\mathcal {L} _ {\theta} = \mathbb {E} _ {t, x _ {0}, x _ {1}} \left[ \lambda (t) c _ {\text {o u t}} (t) ^ {2} \left\| F _ {\theta} \left(c _ {\text {n o i s e}} (t), c _ {\text {i n}} (t) x (t)\right) - \frac {1}{c _ {\text {o u t}} (t)} \left(x _ {0} - c _ {\text {s k i p}} (t) x (t)\right) \right\| _ {2} ^ {2} \right], \tag {17}
+$$
+
+where:
+
+$$
+c _ {\text {i n}} (t) = \frac {1}{\sqrt {\alpha_ {1} ^ {2} (t) + \sigma_ {\text {d a t a}} ^ {2}}}, \quad c _ {\text {o u t}} (t) = \frac {\alpha_ {1} (t) \cdot \sigma_ {\text {d a t a}}}{\sqrt {\sigma_ {\text {d a t a}} ^ {2} + \alpha_ {1} ^ {2} (t)}}, \quad c _ {\text {s k i p}} (t) = \frac {\sigma_ {\text {d a t a}} ^ {2}}{\alpha_ {1} ^ {2} (t) + \sigma_ {\text {d a t a}} ^ {2}}, \tag {18}
+$$
+
+$$
+c _ {\mathrm {n o i s e}} (t) = \frac {1}{4} \ln t, \quad \lambda (t) = \frac {\alpha_ {1} ^ {2} (t) + \sigma_ {\mathrm {d a t a}} ^ {2}}{(\alpha_ {1} (t) \cdot \sigma_ {\mathrm {d a t a}}) ^ {2}},
+$$
+
+with  $t$  sampled from  $\ln (t)\sim \mathcal{N}(-1.2,1.2^2)$ , and  $\sigma_{\mathrm{data}} = 0.5$ . EDM models  $H_{\theta}(t,x(t)) = c_{\mathrm{skip}}(t)x(t) + c_{\mathrm{out}}(t)F_{\theta}(c_{\mathrm{noise}}(t),c_{\mathrm{in}}(t)x(t)) = \hat{x}_{0,\theta}$ . During inference, EDM transports from  $\rho_T = \mathcal{N}(0,T^2 I)$  to  $\rho_0$ .
+
+Stochastic Interpolant (SI) SI (Albergo et al., 2023) facilitates transportation between arbitrary distributions  $\rho_0$  and  $\rho_{1}$ . The conventional coefficient design is:
+
+$$
+\alpha (t) = \left[ \alpha_ {0} (t), \alpha_ {1} (t) \right] = [ 1 - t, t ], \quad T = 1, \tag {19}
+$$
+
+SI models  $H_{\theta}(t,x(t)) = [H_{0,\theta}(t,x(t)),H_{1,\theta}(t,x(t))] = [\hat{x}_{0,\theta},\hat{x}_{1,\theta}].$  The loss function is:
+
+$$
+\mathcal {L} _ {k, \theta} = \int_ {0} ^ {1} \mathbb {E} _ {x _ {0}, x _ {1}} \left[ | H _ {k, \theta} (t, x (t)) | ^ {2} - 2 x _ {k} \cdot H _ {k, \theta} (t, x (t)) \right] d t, \quad k = 0, 1, \quad t \sim \mathcal {U} (0, 1). \tag {20}
+$$
+
+# B. Justification for Diagonal Matrices in MAC
+
+We argue that using diagonal matrices is sufficient to represent non-linear trajectories for inference trajectory optimization. Theoretically, the expressible space of the interpolated value  $x(t)$  remains  $x(t) \in \mathbb{R}^d$  for both the Hadamard product and matrix-vector multiplication. While the primary benefit of full matrix multiplication lies in its ability to model non-linear trajectories by incorporating cross-dimensional effects, we contend that a diagonal matrix is already adequate for this purpose—provided that the U-Net  $U_{\phi}$ , used to parameterize  $\gamma_{\phi}$ , possesses enough expressive power to generate diverse non-linear trajectories.
+
+Moreover, using diagonal matrices for the output of MAC significantly reduces model size compared to using full matrices. One alternative is to use a low-rank-plus-diagonal decomposition,  $\gamma_{\phi} = D + UV^{T}$ , where  $D = \gamma$  is a diagonal matrix, and  $U,V\in \mathbb{R}^{d\times r}$ , with  $r$  being a typically small rank (e.g.,  $r = 4$  or 8). However, even with a small  $r$ , this approach would still require a large number of output channels from  $U_{\phi}$ , significantly increasing model complexity. Specifically,  $U_{\phi}$  would require an output channel size of  $C\times (2r + 1)\times M\times 2$ , where  $C$  is the image channel size and  $M$  is the number of sinusoidal functions. For instance, with  $r = 4$ , this results in  $3\times 9\times 5\times 2 = 270$  output channels for  $U_{\phi}$  when  $NFE = 5$ , compared to only  $C\times M\times 2 = 30$  channels in the diagonal case.
+
+In conclusion, we believe that using a diagonal matrix, in conjunction with a sufficiently expressive U-Net  $U_{\phi}$ , provides ample capacity to model non-linear trajectories while keeping the model size efficient and manageable.
+
+# C. Hypothesis Set Design for MAC
+
+We design the hypothesis set and parameterization for  $\gamma_{\phi}$  as follows:
+
+$$
+\Gamma_ {h} = \left\{\begin{array}{l l}\gamma_ {\phi} (t, x _ {T}) = \left[ \gamma_ {0, \phi} (t, x _ {T}), \gamma_ {1, \phi} (t, x _ {T}) \right]: [ 0, T ] \times \mathbb {R} ^ {d} \rightarrow \mathbb {R} ^ {d \times 2},&\text {w h e r e}\\\gamma_ {0, \phi} (t, x _ {T}) = \mathcal {F} _ {0} \left(b _ {m} (t), w _ {\phi} \left(x _ {T}\right)\right) = T \frac {f _ {\phi} (t , x _ {T})}{f _ {\phi} (t , x _ {T}) + g _ {\phi} (t , x _ {T})},&\\\gamma_ {1, \phi} (t, x _ {T}) = \mathcal {F} _ {1} \left(b _ {m} (t), w _ {\phi} \left(x _ {T}\right)\right) = T \frac {g _ {\phi} (t , x _ {T})}{f _ {\phi} (t , x _ {T}) + g _ {\phi} (t , x _ {T})},&\phi \in \mathcal {P}\end{array}\right\}, \tag {21}
+$$
+
+This parameterization can vary depending on the flow and diffusion framework. For example, we use  $\gamma_{0,\phi}$  as described above for SI, but set  $\gamma_{0,\phi}(t,x_T) = \mathbf{1}_d$  for EDM to align with its original formulation. The functions  $f_{\phi}$  and  $g_{\phi}$  are defined as:
+
+$$
+f _ {\phi} (t, x _ {T}) = 1 - \frac {t}{T} + \left(\sum_ {m = 1} ^ {M} w _ {m, \phi} ^ {f} \left(x _ {T}\right) b _ {m} (t)\right) ^ {2}, \quad g _ {\phi} (t, x _ {T}) = \frac {t}{T} + \left(\sum_ {m = 1} ^ {M} w _ {m, \phi} ^ {g} \left(x _ {T}\right) b _ {m} (t)\right) ^ {2}, \tag {22}
+$$
+
+$$
+b _ {m} (t) = \sin \left(\pi m \left(\frac {t}{T}\right) ^ {1 / q}\right) \in \mathbb {R}, \quad w _ {\phi} \left(x _ {T}\right) = s \operatorname {L P F} \circ \tanh  \left(\mathrm {N N} _ {\phi} \left(x _ {T}\right)\right), \quad q \in \mathbb {R}, \quad s \in \mathbb {R}, \tag {23}
+$$
+
+where  $M$  and  $q$  are hyperparameters for the sinusoidal basis, and the scaling factor  $s$  controls the output range of  $w_{\phi}(x_T) \in (-s,s)$ . When  $s = 0.0$ ,  $\gamma$  reduces to  $\alpha$ . For image generation, we use a U-Net  $U_{\phi}$  as  $\mathrm{NN}_{\phi}$ . The outputs  $U_{\phi}(x_T) = [U_{f,\phi}(x_T), U_{g,\phi}(x_T)] \in \mathbb{R}^{C \times H \times W \times M \times 2}$ , where  $C$ ,  $H$ , and  $W$  denote the channel, height, and width of the image.  $U_{f,\phi}(x_T)$  and  $U_{g,\phi}(x_T)$  each have a channel shape of  $C \times M$ , structured along the channel axis.
+
+Design Choice of Sinusoidals The inference time schedule of EDM is defined as:
+
+$$
+t _ {i} = \left(t _ {\max } ^ {\frac {1}{q}} + \frac {i}{N - 1} \left(t _ {\min } ^ {\frac {1}{q}} - t _ {\max } ^ {\frac {1}{q}}\right)\right) ^ {q}, \quad t _ {\min } = 0. 0 0 2, t _ {\max } = 8 0, q = 7. \tag {24}
+$$
+
+As illustrated in Figure 9,  $b_{m}(t) = \sin (\pi m(t / T)^{1 / 7})$  effectively covers the entire EDM time schedule, whereas  $b_{m}(t) = \sin (\pi m(t / T))$  has minimal magnitude when  $t \leq 1$ . Since  $w_{\phi}$  is constrained to  $(-s,s)$ , the choice of sinusoidal basis significantly influences the controllability of  $\gamma_{\phi}$  during simulation. Accordingly, we use  $q = 1$  for DDPM, FM, and SI, and  $q = 7$  for EDM.
+
+In addition, to avoid aliasing in  $b_{m}(t)$ , we follow the Nyquist criterion, requiring the sampling rate  $f_{s} = N \geq 2f_{\max} = M$ . Based on this, we set  $M = N$  during adversarial training to ensure sufficient frequency resolution.
+
+![](images/6bb9e2b04df315f3285948434b2a4b4d73e454d04f07e456505c6daf8bf2e8b5.jpg)  
+(a)  $\sin (\pi m(t / T))$
+
+![](images/3c115ea647d8e81c75cc45e7493706a5346f5345ba01ac7419099f7fe240d291.jpg)  
+(b)  $\sin (\pi m(t / T)^{1 / 7})$  
+Figure 9: Comparison of  $b_{m}(t)$  for  $M = 10$ . The dotted line represents the EDM inference time schedule for  $N = 10$ .
+
+Low-Pass Filtering (LPF) For LPF, we apply a 2D convolution with a Gaussian kernel, where the kernel size is  $\frac{20 \times \text{resolution}}{32} - 1$  and the standard deviation of the Gaussian is  $\sigma = \frac{4.0 \times \text{resolution}}{32}$ . Here, resolution refers to the image height or width. To mitigate boundary effects introduced by LPF, we apply zero-padding of  $\frac{\text{kernel size} + 1}{2}$  on all sides of  $U_{\phi}$ 's input and crop the borders after convolution to restore the original shape. To preserve consistent scaling, we compute the minimum and maximum values before LPF for each batch and rescale the post-LPF output to match the original range. LPF is applied only for image generation experiments.
+
+Hypothesis Set for Pre-training and Adversarial Training Since using a large hypothesis set of coefficients during pre-training may overburden  $H_{\theta}$  and degrade performance, we adopt a smaller hypothesis set for pre-training and enable full multidimensionality across  $t$  only during adversarial training. For Section 4.2, LPF is used exclusively during pretraining and omitted during adversarial training. For Section 4.3, we configure the convolution group size for LPF during pre-training to 1, yielding an LPF output shape of  $[B, 1, \text{resolution}, \text{resolution}]$ . This enforces high multidimensionality for small  $t$  and low multidimensionality for large  $t$ . In contrast, during adversarial training, we use a convolution group size of  $[B, 2 \times 3 \times M, \text{resolution}, \text{resolution}]$ , producing an LPF output shape of  $[B, 2 \times 3 \times M, \text{resolution}, \text{resolution}]$ .
+
+# D. Details of the Differential Equation Solver
+
+The displacement of the trajectory  $x(t_{i + 1}) - x(t_i)$ , parameterized by  $v_{\theta ,\phi}$ , is expressed as:
+
+$$
+\begin{array}{l} \Delta t _ {i} v _ {\theta , \phi} \left(t _ {i}, x \left(t _ {i}\right), x _ {T}\right) = \Delta t _ {i} \dot {\gamma} _ {0, \phi} \left(t _ {i}, x _ {T}\right) \odot \hat {x} _ {0, \theta} + \Delta t _ {i} \dot {\gamma} _ {1, \phi} \left(t _ {i}, x _ {T}\right) \odot \hat {x} _ {1, \theta} \\ \approx \Delta \gamma_ {0, \phi} (t _ {i}, x _ {T}) \odot \hat {x} _ {0, \theta} + \Delta \gamma_ {1, \phi} (t _ {i}, x _ {T}) \odot \hat {x} _ {1, \theta}, \\ \end{array}
+$$
+
+where the time displacement  $\Delta t_{i} = t_{i + 1} - t_{i}$  is derived from the inference time schedule  $\tau = \{t_0,\dots ,t_N\}$ , with  $t_0 = T > \ldots >t_N = 0$ . The corresponding displacements for  $\gamma$  are:
+
+$$
+\Delta \gamma_ {0, \phi} (t _ {i}, x _ {T}) = \gamma_ {0, \phi} (t _ {i + 1}, x _ {T}) - \gamma_ {0, \phi} (t _ {i}, x _ {T}), \quad \Delta \gamma_ {1, \phi} (t _ {i}, x _ {T}) = \gamma_ {1, \phi} (t _ {i + 1}, x _ {T}) - \gamma_ {1, \phi} (t _ {i}, x _ {T}). \tag {26}
+$$
+
+This approach helps reduce numerical errors when solving differential equations under curved  $\gamma$ . Accordingly, the trajectory displacement for EDM is given by:
+
+$$
+\Delta t _ {i} v _ {\theta , \phi} \left(t _ {i}, x \left(t _ {i}\right), x _ {T}\right) = \left\{ \begin{array}{l l} \frac {\Delta \gamma_ {1 , \phi} \left(t _ {i} , x _ {T}\right)}{\gamma_ {1 , \phi} \left(t _ {i} , x _ {T}\right)} \odot \left(x \left(t _ {i}\right) - H _ {\theta} \left(t _ {i}, x \left(t _ {i}\right), \gamma_ {\phi} \left(t _ {i}, x _ {T}\right)\right)\right) & \text {w i t h} \gamma - \text {p r e - t r a i n i n g} \\ \frac {\Delta \gamma_ {1 , \phi} \left(t _ {i} , x _ {T}\right)}{\gamma_ {1 , \phi} \left(t _ {i} , x _ {T}\right)} \odot \left(x \left(t _ {i}\right) - H _ {\theta} \left(\bar {\gamma} _ {\phi} \left(t _ {i}, x _ {T}\right), x \left(t _ {i}\right)\right)\right) & \text {w / o} \gamma - \text {p r e - t r a i n i n g} \end{array} . \right. \tag {27}
+$$
+
+For SI:
+
+$$
+\Delta t _ {i} v _ {\theta , \phi} \left(t _ {i}, x \left(t _ {i}\right), x _ {T}\right) = \Delta \gamma_ {0, \phi} \left(t _ {i}, x _ {T}\right) \odot \hat {x} _ {0, \theta} + \Delta \gamma_ {1, \phi} \left(t _ {i}, x _ {T}\right) \odot \hat {x} _ {1, \theta}, \tag {28}
+$$
+
+$$
+[ \hat {x} _ {0, \theta}, \hat {x} _ {1, \theta} ] = \left\{ \begin{array}{l l} H _ {\theta} \left(t _ {i}, x \left(t _ {i}\right), \gamma_ {\phi} \left(t _ {i}, x _ {T}\right)\right) & \text {w i t h} \gamma - \text {p r e - t r a i n i n g} \\ H _ {\theta} \left(\bar {\gamma} _ {\phi} \left(t _ {i}, x _ {T}\right), x \left(t _ {i}\right)\right) & \text {w / o} \gamma - \text {p r e - t r a i n i n g} \end{array} . \right. \tag {29}
+$$
+
+The differential equation solver  $G_{\theta, \phi}$ , using Euler discretization, is then defined as:
+
+$$
+G _ {\theta , \phi} (\tau , x _ {T}, v _ {\theta , \phi}) = x _ {\theta , \phi} (t _ {N}) = x _ {T} + \sum_ {i = 0} ^ {N - 1} \Delta t _ {i} v _ {\theta , \phi} (t _ {i}, x _ {\theta , \phi} (t _ {i}), x _ {T}), \tag {30}
+$$
+
+$$
+x _ {\theta , \phi} (t _ {i + 1}) \leftarrow x _ {\theta , \phi} (t _ {i}) + \Delta t _ {i} v _ {\theta , \phi} (t _ {i}, x _ {\theta , \phi} (t _ {i}), x _ {T}).
+$$
+
+# E. Details for Experiments
+
+All experiments are conducted on NVIDIA RTX 3080Ti, RTX 4090, and RTX A6000 GPUs. Euler discretization is used for all inference procedures.
+
+# E.1. 2-Dimensional Transportation: Optimizing Only  $\phi$
+
+# E.1.1.  $\gamma$ -PRE-TRAINING
+
+We follow the implementation from Tong et al. (2024), employing a multilayer perceptron (MLP) consisting of four linear layers with 64 hidden units and SiLU activation functions. The Stochastic Interpolant (SI) model is trained with a batch size of 256 for 20,000 iterations. The loss function is defined as:
+
+$$
+\mathcal {L} _ {k} (\theta) = \int_ {0} ^ {1} \mathbb {E} _ {x _ {0}, x _ {1}, u} \left[ | H _ {k, \theta} (t, x (t), \gamma (t, u)) | ^ {2} - 2 x _ {k} \cdot H _ {k, \theta} (t, x (t), \gamma (t, u)) \right] d t, k = 0, 1, t \sim \mathcal {U} (0, 1), u \sim \mathcal {U} (- 1, 1) \in \mathbb {R} ^ {d \times M \times 2}. \tag {31}
+$$
+
+# E.1.2. OPTIMIZATION OF  $\phi$
+
+To train  $\gamma_{\phi}$ , we use a batch size of 1024 and run for 2000 iterations with  $s = 0.1$ . Each configuration is trained with three different random seeds, and we report the mean and standard deviation of the Wasserstein distance.
+
+# E.2. Image Generation: Optimizing Only  $\phi$
+
+# E.2.1. TRAINING CONFIGURATIONS
+
+Table 8: U-Net configurations for  $H_{\theta}$ .  
+
+<table><tr><td>Configuration</td><td>CIFAR-10</td><td>ImageNet-32</td></tr><tr><td>Channels</td><td>128</td><td>256</td></tr><tr><td>Depth</td><td>2</td><td>3</td></tr><tr><td>Channel multipliers</td><td>1,2,2,2</td><td>1,2,2,2</td></tr><tr><td>Attention heads</td><td>4</td><td>4</td></tr><tr><td>Head channel size</td><td>64</td><td>64</td></tr><tr><td>Attention resolution</td><td>16</td><td>16</td></tr><tr><td>Dropout</td><td>0.1</td><td>0.1</td></tr></table>
+
+We use the U-Net architecture from Dhariwal & Nichol (2021) for  $H_{\theta}$  and the U-Net from Ronneberger et al. (2015) for  $U_{\phi}$ . Configuration details for  $H_{\theta}$  are given in Table 8. For  $\gamma_{\phi}$ , we use a channel progression of [256, 512, 1024, 2048]. The discriminator  $D_{\psi}$  consists of four convolutional layers with 1024 channels, batch normalization, leaky ReLU activations, and a final sigmoid output.
+
+Table 9: Hyperparameters for pre-training and adversarial training.  
+
+<table><tr><td>Hyperparameter</td><td colspan="2">CIFAR-10</td><td colspan="2">ImageNet-32</td></tr><tr><td></td><td>Pre-training</td><td>Adversarial training</td><td>Pre-training</td><td>Adversarial training</td></tr><tr><td>Batch size</td><td>128</td><td>16</td><td>512</td><td>15</td></tr><tr><td>GPUs</td><td>1</td><td>1</td><td>4</td><td>1</td></tr><tr><td>Iterations</td><td>400k</td><td>200k</td><td>250k</td><td>200k</td></tr><tr><td>Peak LR</td><td>2e-4</td><td>2e-4</td><td>2e-4</td><td>2e-4</td></tr><tr><td>LR Scheduler</td><td>Poly decay</td><td>Poly decay</td><td>Poly decay</td><td>Poly decay</td></tr><tr><td>Warmup steps</td><td>5k</td><td>5k</td><td>5k</td><td>5k</td></tr><tr><td>Warmup steps for Dψ</td><td>-</td><td>20k</td><td>-</td><td>20k</td></tr></table>
+
+We use the Adam optimizer with  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$ , weight decay of 0.0, and  $\epsilon = 1 \times 10^{-8}$ , along with polynomial decay learning rate scheduling. An exponential moving average (EMA) with a decay rate of 0.999 is applied during all training phases. For adversarial training, we use the vanilla GAN loss from Equation 5 instead of hinge loss for simplicity. FID is evaluated every 10,000 steps, and we report the lowest value obtained. Hyperparameters are summarized in Table 9.
+
+# E.2.2. EXPERIMENTS FOR HYPOTHESIS SET HYPERPARAMETER TUNING
+
+Table 10: FID comparison between  $\alpha$  and  $\gamma$  under various configurations (e.g., s, LPF) across different NFE.  
+
+<table><tr><td rowspan="2">Method\NFE</td><td colspan="4">CIFAR-10</td><td colspan="4">ImageNet-32</td></tr><tr><td>10</td><td>100</td><td>150</td><td>200</td><td>10</td><td>100</td><td>150</td><td>200</td></tr><tr><td>SIα</td><td>14.43</td><td>4.75</td><td>4.51</td><td>4.30</td><td>17.72</td><td>8.08</td><td>7.79</td><td>7.63</td></tr><tr><td>SIγ(s=0.005)</td><td>14.59</td><td>3.98</td><td>3.74</td><td>3.63</td><td>17.41</td><td>6.33</td><td>6.21</td><td>6.20</td></tr><tr><td>SIγ(s=0.1, LPF)</td><td>15.44</td><td>3.77</td><td>3.68</td><td>3.75</td><td>17.86</td><td>6.63</td><td>6.47</td><td>6.44</td></tr><tr><td>FMα</td><td>13.70</td><td>4.52</td><td>4.23</td><td>4.07</td><td>16.92</td><td>7.78</td><td>7.53</td><td>7.38</td></tr><tr><td>FMγ(s=0.005)</td><td>13.81</td><td>3.59</td><td>3.42</td><td>3.42</td><td>16.85</td><td>6.18</td><td>6.03</td><td>6.01</td></tr><tr><td>FMγ(s=0.1, LPF)</td><td>15.13</td><td>3.64</td><td>3.57</td><td>3.64</td><td>17.52</td><td>6.40</td><td>6.27</td><td>6.31</td></tr><tr><td>DDPMα</td><td>98.47</td><td>6.64</td><td>4.84</td><td>4.10</td><td>111.54</td><td>8.13</td><td>7.40</td><td>7.14</td></tr><tr><td>DDPMγ(s=0.005)</td><td>74.44</td><td>3.77</td><td>5.96</td><td>7.84</td><td>139.69</td><td>7.67</td><td>12.37</td><td>11.70</td></tr><tr><td>DDPMγ(s=0.005,LPF)</td><td>72.23</td><td>4.73</td><td>4.11</td><td>3.83</td><td>135.48</td><td>6.84</td><td>6.51</td><td>6.42</td></tr><tr><td>DDPMγ(s=0.1, LPF)</td><td>71.80</td><td>4.46</td><td>6.32</td><td>12.60</td><td>142.99</td><td>6.70</td><td>8.69</td><td>10.91</td></tr></table>
+
+Table 11: FID for different NFE and Gaussian kernel  $\sigma$  in LPF,evaluated on CIFAR-10 with  ${\mathrm{{SI}}}_{\gamma \left( {s = {0.1},\mathrm{{LPF}}}\right) }$  .  
+
+<table><tr><td>σ\ NFE</td><td>10</td><td>20</td><td>30</td><td>40</td><td>50</td><td>100</td><td>150</td><td>200</td></tr><tr><td>0.1</td><td>14.89</td><td>8.05</td><td>6.49</td><td>5.45</td><td>6.53</td><td>9.59</td><td>10.67</td><td>11.20</td></tr><tr><td>1.0</td><td>14.92</td><td>8.47</td><td>5.32</td><td>4.45</td><td>4.68</td><td>6.06</td><td>7.01</td><td>7.50</td></tr><tr><td>2.0</td><td>16.25</td><td>9.56</td><td>7.56</td><td>6.06</td><td>4.72</td><td>3.77</td><td>3.95</td><td>4.17</td></tr><tr><td>4.0</td><td>15.44</td><td>9.13</td><td>7.39</td><td>6.36</td><td>5.59</td><td>3.77</td><td>3.68</td><td>3.75</td></tr></table>
+
+Table 12: FID after adversarial training with 10 NFE on CIFAR-10 for varying values of  $M$  .  
+
+<table><tr><td>Method \ M</td><td>5</td><td>10</td><td>15</td><td>20</td><td>25</td><td>30</td></tr><tr><td>SIγ(s=0.1, LPF)</td><td>6.89</td><td>4.14</td><td>4.42</td><td>5.32</td><td>6.11</td><td>5.74</td></tr><tr><td>FMγ(s=0.1, LPF)</td><td>5.93</td><td>6.13</td><td>6.70</td><td>6.18</td><td>5.97</td><td>6.42</td></tr><tr><td>DDPMγ(s=0.1, LPF)</td><td>10.15</td><td>10.04</td><td>9.60</td><td>9.04</td><td>8.94</td><td>9.19</td></tr></table>
+
+Table 13: FID after adversarial training of SI under various configurations on CIFAR-10 with 10 NFE.  
+
+<table><tr><td>Method \ M</td><td>5</td><td>10</td><td>15</td><td>20</td></tr><tr><td>s = 0.0</td><td>10.20</td><td>9.75</td><td>11.30</td><td>11.53</td></tr><tr><td>s = 0.005</td><td>6.60</td><td>4.79</td><td>4.45</td><td>5.28</td></tr><tr><td>s = 0.005, LPF</td><td>7.37</td><td>4.42</td><td>4.26</td><td>5.31</td></tr><tr><td>s = 0.1, LPF</td><td>7.21</td><td>4.14</td><td>5.59</td><td>5.32</td></tr></table>
+
+We conduct extensive experiments to tune hyperparameters for the hypothesis set. Tables 10 and 11 show the effects of different coefficients, LPF configurations, and other settings during pre-training. Tables 12 and 13 present results from adversarial training under various configurations. These results help identify optimal hyperparameter settings that enhance model performance in both pre-training and adversarial training stages.
+
+# E.3. Image Generation: Optimizing  $\theta$  and  $\phi$
+
+# E.3.1.  $\gamma$ -PRE-TRAINING
+
+Table 14: Hyperparameters used for pre-training EDM.  
+
+<table><tr><td>Hyperparameter</td><td>CIFAR-10</td><td>FFHQ &amp; AFHQ</td></tr><tr><td>Number of GPUs</td><td>8</td><td>8</td></tr><tr><td>Duration (Mimg)</td><td>200</td><td>200</td></tr><tr><td>Minibatch size</td><td>512</td><td>256</td></tr><tr><td>Learning rate</td><td>1e-3</td><td>2e-4</td></tr><tr><td>LR ramp-up (Mimg)</td><td>10</td><td>10</td></tr><tr><td>EMA half-life (Mimg)</td><td>0.5</td><td>0.5</td></tr><tr><td>Dropout probability</td><td>13%</td><td>5% (FFHQ) / 25% (AFHQ)</td></tr><tr><td>Channel multiplier</td><td>128</td><td>128</td></tr><tr><td>Channels per resolution</td><td>2-2-2</td><td>1-2-2-2</td></tr><tr><td>Augment probability</td><td>12%</td><td>15%</td></tr><tr><td>M</td><td>10</td><td>10</td></tr><tr><td>Low-pass filtering</td><td>True</td><td>True</td></tr><tr><td>s</td><td>0.05</td><td>0.05</td></tr></table>
+
+We adopt the codebase from Karras et al. (2022) and follow the EDM configuration, replacing  $\alpha$  with  $\gamma$  and applying coefficient conditioning. The pre-training loss function is:
+
+$$
+\mathcal {L} _ {\theta} = \mathbb {E} _ {t, x _ {0}, x _ {1}, u} \left[ \lambda (t, u) c _ {\text {o u t}} (t, u) ^ {2} \left\| F _ {\theta} \left(c _ {\text {n o i s e}} (t), c _ {\text {i n}} (t, u) x (t), c _ {\text {c o e f f}} (t, u)\right) - \frac {1}{c _ {\text {o u t}} (t , u)} \left(x _ {0} - c _ {\text {s k i p}} (t, u) x (t)\right) \right\| _ {2} ^ {2} \right], \tag {32}
+$$
+
+where:
+
+$$
+c _ {\text {i n}} (t, u) = \frac {1}{\sqrt {\gamma_ {1} ^ {2} (t , u) + \sigma_ {\text {d a t a}} ^ {2}}}, \quad c _ {\text {o u t}} (t, u) = \frac {\gamma_ {1} (t , u) \cdot \sigma_ {\text {d a t a}}}{\sqrt {\sigma_ {\text {d a t a}} ^ {2} + \gamma_ {1} ^ {2} (t , u)}}, \quad c _ {\text {s k i p}} (t, u) = \frac {\sigma_ {\text {d a t a}} ^ {2}}{\gamma_ {1} ^ {2} (t , u) + \sigma_ {\text {d a t a}} ^ {2}}, \tag {33}
+$$
+
+$$
+c _ {\mathrm {n o i s e}} (t) = \frac {1}{4} \ln t, \quad c _ {\mathrm {c o e f f}} (t, u) = \frac {1}{4} \ln \gamma_ {1} (t, u), \quad \lambda (t, u) = \frac {\gamma_ {1} ^ {2} (t , u) + \sigma_ {\mathrm {d a t a}} ^ {2}}{(\gamma_ {1} (t , u) \cdot \sigma_ {\mathrm {d a t a}}) ^ {2}},
+$$
+
+with  $\ln (t)\sim \mathcal{N}(-1.2,1.2^2)$ ,  $u\sim \mathcal{N}(-1,1)\in \mathbb{R}^{d\times M\times 2}$ , and  $\sigma_{\mathrm{data}} = 0.5$ . For coefficient conditioning (Section 3.2.2), we concatenate  $[c_{\mathrm{in}}(t,u)x(t),c_{\mathrm{coeff}}(t,u)]$  along the channel axis as input to  $F_{\theta}$ . We use the Adam optimizer with  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$ , and  $\epsilon = 1e - 8$ .
+
+# E.3.2. ADVERSARIAL TRAINING
+
+Table 15: Hyperparameters used for adversarial training.  
+
+<table><tr><td>Hyperparameter</td><td>CIFAR-10</td><td>FFHQ &amp; AFHQ</td><td>ImageNet</td></tr><tr><td>Number of GPUs</td><td>8</td><td>8</td><td>8</td></tr><tr><td>Training duration for γφ (kimg)</td><td>1500</td><td>1000</td><td>30</td></tr><tr><td>Minibatch size for vθ</td><td>512</td><td>256</td><td>256</td></tr><tr><td>Minibatch size for γφ</td><td>128</td><td>64</td><td>32</td></tr><tr><td>Minibatch size for Dψ</td><td>128</td><td>64</td><td>512</td></tr><tr><td>Learning rate for vθ</td><td>1e-5</td><td>1e-5</td><td>1e-5</td></tr><tr><td>Learning rate for γφ</td><td>1e-4</td><td>1e-4</td><td>1e-4</td></tr><tr><td>Learning rate for Dψ</td><td>1e-3</td><td>1e-3</td><td>1e-3</td></tr><tr><td>EMA half-life (kimg)</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Low-pass filtering</td><td>True</td><td>True</td><td>True</td></tr><tr><td>s</td><td>0.05</td><td>0.05</td><td>0.05</td></tr></table>
+
+For  $U_{\phi}$ , we adopt a U-Net architecture based on Song et al. (2021) with the following configuration: 256 channels, channel multipliers [1, 2, 4], a dimensionality multiplier of 4, 4 blocks, and an attention resolution of 16. The embedding layer for  $t$  is disabled. We disable dropout in both  $H_{\theta}$  and  $\gamma_{\phi}$  to ensure deterministic behavior of  $G_{\theta,\phi}$ . We use the Adam optimizer with  $\beta_{1} = 0.0$ ,  $\beta_{2} = 0.99$ , and  $\epsilon = 1e - 8$ . The sampling time  $t$  is drawn from  $\ln(t) \sim \mathcal{N}(-1.2, 1.2^2)$  and is quantized according to the inference time schedule  $\tau$ . For ablation studies, each configuration is trained for 500 kimg, corresponding to approximately 4000 iterations. When training only  $\phi$  (Table 6, Figure 7), LPF is not applied.
+
+# E.3.3. OPTIMIZED INFERENCE TRAJECTORIES DEVIATE FROM LINEARITY
+
+![](images/3c5d766724c3cb3ebd3d72e2d3f645a041d557bc6d5ef4891876af6a30c23e03.jpg)  
+Figure 10:  $L_{2}$  norm between the linear trajectory  $x_{\mathrm{lin}}(t)$  and the optimized trajectory  $x_{\theta, \phi}(t)$  over  $t$ .
+
+To confirm that the optimized inference trajectories in the image generation experiment are not linear lines, we compute the  $L_{2}$  norm between linear trajectories  $x_{\mathrm{lin}}(t) = \frac{t}{T} x_T + \left(1 - \frac{t}{T}\right)x_{\theta ,\phi}(t_N)$  and the optimized trajectories  $x_{\theta ,\phi}(t)$ . As shown in Figure 10, the optimized inference trajectories clearly deviate from the linear path. This result demonstrates that our method effectively discovers superior, non-linear trajectories in high-dimensional spaces, leading to enhanced performance.
+
+# F. Metrics Calculation
+
+For CIFAR-10, AFHQ, and FFHQ experiments, we follow the evaluation protocol and code provided by Karras et al. (2022) to compute the Fréchet Inception Distance (FID). For ImageNet-64 experiments, we use the ImageNet dataset from EDM (Karras et al., 2022) as the reference and calculate both FID and  $\mathrm{FD_{DINOv2}}$  using the code from EDM2 (Karras et al., 2024). All metrics are computed over 50,000 generated samples. Each experiment is run three times with different random seeds, and we report the minimum FID and  $\mathrm{FD_{DINOv2}}$  values observed.
+
+# G. Generated Samples
+
+![](images/4d0075cbbf5fa5804d3556e9dbfd30650aa0447df764e7f17c666688da4be37b.jpg)  
+Figure 11: Inference trajectories from  $\mathrm{EDM}_{\gamma} + \mathrm{adv}\theta ,\phi$  for conditional generation on CIFAR-10.
+
+![](images/aaad04e195f4031b9d0badb051743017dc9e0030ae30094a5ab1a73ffb868107.jpg)  
+FID 1.98  
+NFE 35  
+CIFAR-10 unconditional.
+
+![](images/0983fa1a1d9de03323e8c42e4648c9334842de374b45ad711d34979d45e3df46.jpg)  
+FID 1.69  
+NFE 5(+)
+
+![](images/cd13b7c522df69bc0de2618afede2f13c7cd45835232f792b4b21300d27a31e0.jpg)  
+FID 1.79  
+NFE 35  
+CIFAR-10 conditional.
+
+![](images/e6663aa38e81b52cb5049bfe93405b265e603dc33d8a31e5f799ac4e9c77d78b.jpg)  
+FID 1.37  
+NFE 5(+)
+
+![](images/cb838f207642052f8ce9bb960e56bebc34b014f51df9e325e1659a8e8c69b996.jpg)  
+FID 2.39  
+NFE 79  
+AFHQ-64.
+
+![](images/70aa35836084938d6be1ae214eeea05209889a39e81cc83680ef469e856ca441.jpg)  
+FID 2.27  
+NFE 5(+)
+
+![](images/2f3ed6069129d98b362d84438ef078fe3d0a6df2f5a7ea86dca69753514f003d.jpg)  
+FID 1.96  
+NFE 35  
+FFHQ-64.  
+Figure 12: Generated samples on CIFAR-10, FFHQ-64, and AFHQ-64 using  $\mathrm{EDM}_{\alpha}$  (left) and  $\mathrm{EDM}_{\gamma} + \mathrm{adv}\theta ,\phi$  (right).
+
+![](images/5766636cd31b13919415140253fc2631595b4ae26783028e326ddce19aa6b177.jpg)  
+FID 2.04  
+NFE 5(+)
+
+![](images/176d24266c9f1e36a23c4bd5dc21dde2787b591b91c133158afcadcacb8e98a6.jpg)  
+FID 1.48 FDINOV2 70.2 NFE 5(+)  
+Figure 13: Generated samples on ImageNet-64 using  $\mathrm{EDM}_{\alpha} + \mathrm{adv}\theta ,\phi$

2025/Multidimensional Adaptive Coefficient for Inference Trajectory Optimization in Flow and Diffusion/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8a0c9f44be16d5ef3e28880e35b1e47594554e208ed1df4b87fe425ae2cf3075
+size 2562560

2025/Multilayer Matrix Factorization via Dimension-Reducing Diffusion Variational Inference/e19e76fc-b602-4624-9f13-cc4f80943df0_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d3d17cc10deea54fc2845d4f6e0e174923bbe1fc4f1af6c273838fb9c38c48a6
+size 4200616

2025/Multilayer Matrix Factorization via Dimension-Reducing Diffusion Variational Inference/full.md

@@ -0,0 +1,807 @@
+# Multilayer Matrix Factorization via Dimension-Reducing Diffusion Variational Inference
+
+Junbin Liu<sup>1</sup> Farzan Farnia<sup>2</sup> Wing-Kin Ma<sup>1</sup>
+
+# Abstract
+
+Multilayer matrix factorization (MMF) has recently emerged as a generalized model of, and potentially a more expressive approach than, the classic matrix factorization. This paper considers MMF under a probabilistic formulation, and our focus is on inference methods under variational inference. The challenge in this context lies in determining a variational process that leads to a computationally efficient and accurate approximation of the maximum likelihood inference. One well-known example is the variational autoencoder (VAE), which uses neural networks for the variational process. In this work, we take insight from variational diffusion models in the context of generative models to develop variational inference for MMF. We propose a dimension-reducing diffusion process that results in a new way to interact with the layered structures of the MMF model. Experimental results demonstrate that the proposed diffusion variational inference method leads to improved performance scores compared to several existing methods, including the VAE.
+
+# 1. Introduction
+
+Over decades, matrix factorization (MF) methods have played a crucial role in a wide variety of problems such as dimensionality reduction, low-dimension representation learning, blind source separation (Hyvärinen et al., 2023), hyperspectral unmixing (Ma et al., 2013), topic modeling (Arora et al., 2012), community detection (Yang & Leskovec, 2013), to name a few. The broad interest of
+
+$^{1}$ Department of Electronic Engineering, the Chinese University of Hong Kong, Hong Kong SAR of China  $^{2}$ Department of Computer Science and Engineering, the Chinese University of Hong Kong, Hong Kong SAR of China. Correspondence to: Junbin Liu <liujunbin@link.cuhk.edu.hk>, Farzan Farnia <farnia@cse.cuhk.edu.hk>, Wing-Kin Ma <wkma@ee.cuhk.edu.hk>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+researchers in this subject has led to both diverse and substantial developments. In particular, we have seen different ways to exploit the hidden structures of the underlying matrix factors, such as statistical independence, sparsity, and non-negativity. Some of such methods are equipped with desirable results such as identifiability guarantees—that is, the guarantees of identifying the underlying ground-truth factors—which are essential in applications such as blind source separation; see, e.g., (Gillis, 2020; Khemakhem et al., 2020; Wu et al., 2021) and the references therein. MF is intimately linked with the notion of learning low-dimensional structures from higher-dimensional data. It is closely related to latent-variable component analysis such as independent component analysis (ICA).
+
+More recently, there has been interest in multilayer, and possibly nonlinear, MF (Trigeorgis et al., 2016; Zhao et al., 2017; Xue et al., 2017; Fan, 2021; De Handschutter & Gillis, 2023). Multilayer MF (MMF) is a more general model than the (two-factor) MF model, and it is anticipated that MMF should provide more powerful results. It was empirically shown that MMF can extract meaningful hierarchical features from data, which offers new insights in applications such as clustering; see, e.g., (Trigeorgis et al., 2016; De Handschutter et al., 2021) and the references therein. MMF is also related to nonlinear latent-variable component analysis (Khemakhem et al., 2020). In particular, if the nonlinear system is modeled by a neural network, we can see it as a multilayer system.
+
+One powerful approach to MF or MMF is to formulate the factorization model as a latent-variable model and treat the factorization problem as a probabilistic inference problem. In this direction, variational inference (VI) has been found to be promising in providing a practical way to handle complex models (Rezende et al., 2014; Ranganath et al., 2015). In particular, for MMF, variational autoencoders (VAEs) appear to be the only available solution in the literature so far; see (Khemakhem et al., 2020) and also (Li et al., 2024).
+
+Lately, diffusion models (DMs) (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2021; Luo, 2022) have caught tremendous attention in the context of generative models. They have been found to provide competitive performance in various generation tasks. There are several ways to derive
+
+and understand DMs: We can consider stochastic differential equations, seeking to reverse a diffusion process (Song et al., 2021); DMs can also be seen as an outcome of denoising score matching and Langevin dynamics for learning the data distribution and generating data (Song & Ermon, 2019). DMs can also be derived by formulating a latent-variable model and then by performing a specific (diffusion) type of VI (Ho et al., 2020; Kingma et al., 2021)—we are most attracted by this interpretation. Given the success of DMs, we consider this question: Can we take the VI in DMs and apply it to MMF?
+
+So far, and to our best knowledge, DM-based VI has not been considered for MMF. And DM-based VI cannot be directly applied to MMF. This is because the current DMs assume equal dimensions with the latent variables, while MMF has unequal latent-variable dimensions. In this paper, we explore the application of DM-based VI to MMF. We will propose a dimension-reducing (DR) variational diffusion model. The distinct characteristic is that we associate each layer of the DM with a layer of the MMF model, and we seek to use light-weight methods to deal with each layer. This is different from DMs and hierarchical VAEs (Ranganath et al., 2016; Sønderby et al., 2016; Vahdat & Kautz, 2020) for generative models, which would employ deep neural networks at each layer. From the proposed DR diffusion model, we will derive a VI scheme. Numerical results will be provided to demonstrate the performance of the proposed DR diffusion VI (DRD-VI).
+
+It is worth noting that, in the context of generative models, there are studies that consider dimensionality reduction for diffusion models. In (Rombach et al., 2022; Wang et al., 2023), the authors apply dimensionality reduction before the diffusion model. In (Jing et al., 2022; Zhang et al., 2023), the authors concatenate multiple diffusion models, and at each stage dimensionality reduction is applied. In the aforementioned studies, dimensionality reduction is done outside of the diffusion process. Our study differs in that we embed dimensionality reduction inside the diffusion model.
+
+In addition it is interesting to have a comparison with the hierarchical VAE (HVAE) approach (Sønderby et al., 2016), which, similar to the DM, is also capable of interacting with the layered structures of the model. The HVAE was not considered in the context of MMF, to the best of our knowledge, although in principle it is possible to do so. As noted earlier, the HVAE employs a deep network for each layer of the variation process. While this makes the variational process more powerful, it also makes the HVAE more difficult to train. Moreover, it was argued that the stochastic approximation in the HVAE may have larger variance as the number of layers is larger (Luo, 2022). In comparison, our proposed DRD-VI adopts light-weight operations at each layer of the variational process, which in turn makes the
+
+training easier. This is an advantage that has been noted in the context of generative models; see (Luo, 2022).
+
+# 2. Background
+
+# 2.1. Multilayer Matrix Factorization
+
+Consider the following problem. Let  $\pmb{y} \in \mathbb{R}^{M}$  denote a data point. It is modeled to follow a generative model
+
+$$
+\boldsymbol {y} = f _ {\boldsymbol {\theta}} (\boldsymbol {z}) + \boldsymbol {v}, \tag {1}
+$$
+
+where  $f_{\pmb{\theta}}:\mathbb{R}^{N}\to \mathbb{R}^{M}$  is a function parameterized by  $\pmb{\theta}$ ,  $\pmb {z}\in \mathbb{R}^N$  is the latent variable associated with  $\pmb{y}$ , whose dimension  $N$  is assumed to be less than the data dimension  $M$ ;  $\pmb{v}$  is noise and is modeled as  $\pmb{v}\sim \mathcal{N}(\pmb {0},\sigma^2\pmb {I})$ . Let  $\{\pmb{y}_1,\pmb{y}_2,\dots ,\pmb{y}_L\}$  be a given set of data points that follow the model in (1) and are independently distributed. Our goal is to estimate  $\pmb{\theta}$  from  $\{\pmb{y}_1,\dots ,\pmb{y}_L\}$  and then to estimate the latent variable  $\pmb{z}_n$  of each  $\pmb{y}_n$ .
+
+If  $f_{\theta}$  takes a linear form  $f_{\theta}(z) = Az$ , with  $\theta = \{A\}$ , then the problem can be seen as a matrix factorization problem. In particular, by letting  $Y = [y_1, \ldots, y_L]$  and  $z = [z_1, \ldots, z_L]$ , the problem is essentially to recover  $A$  and  $Z$  from  $Y$  such that  $Y \approx AZ$ . In recent years, we have seen interest in multilayer matrix factorization (MMF), which considers
+
+$$
+f _ {\boldsymbol {\theta}} (z) = \boldsymbol {A} _ {1} \boldsymbol {A} _ {2} \dots \boldsymbol {A} _ {T} z, \quad \boldsymbol {\theta} = \left\{\boldsymbol {A} _ {1}, \boldsymbol {A} _ {2}, \dots , \boldsymbol {A} _ {T} \right\},
+$$
+
+and leads to a multilayer factorization  $\mathbf{Y} \approx \mathbf{A}_1 \dots \mathbf{A}_T \mathbf{Z}$ . In MMF, we would impose structures at each layer. Let  $\mathbf{x}_t = \mathbf{A}_{t+1} \dots \mathbf{A}_T \mathbf{z}$ . In non-negative MMF, we constrain  $\mathbf{x}_t \geq \mathbf{0}$  (as well as  $z \geq \mathbf{0}$ ) (Trigeorgis et al., 2016). Such MMF was numerically demonstrated to provide meaningful results in learning attribute representations of images. We have also seen interest in the following model:
+
+$$
+f _ {\boldsymbol {\theta}} (\boldsymbol {z}) = \rho (\boldsymbol {A} _ {1} \rho (\boldsymbol {A} _ {2} \rho (\dots \rho (\boldsymbol {A} _ {T} \boldsymbol {z}) \dots))), \tag {2}
+$$
+
+where  $\rho$  is a component-wise nonlinear activation function. The function  $\rho$  is used to impose structures at each layer; e.g., if  $\rho$  is a ReLU function, we enforce non-negativity with each layer's output. The model in (2) can be viewed as a neural network, modeling a nonlinear relationship between  $y$  and  $z$ . In this sense, we are also dealing with a nonlinear latent-variable component analysis problem; see, e.g., (Khemakhem et al., 2020). Additionally, it was argued that nonlinear factorization  $f_{\theta}(z) = \rho(Az)$  provides an effective model for low-dimensional embedding of high-dimensional data (Saul, 2022).
+
+# 2.2. Variational Inference for MMF
+
+We consider a probabilistic framework for MMF. Consider the generative model in (1) and (2). Assume that the latent variable  $z$  follows a known distribution  $p(z)$ , called
+
+the latent prior. For example, in independent component analysis (ICA) the latent prior is chosen as a component-wise independent and non-Gaussian distribution. In simplex component analysis (SCA) (Wu et al., 2021), an important type of non-negative matrix factorization, the latent prior may be chosen as a simplex uniform distribution
+
+$$
+p (\boldsymbol {z}) = \mathbb {1} _ {\Delta} (\boldsymbol {z}) / Z, \tag {3}
+$$
+
+where  $\Delta = \{\pmb {z}\in \mathbb{R}^N\mid \pmb {z}\geq \mathbf{0},\mathbf{1}^\top \pmb {z} = 1\}$  is a unit simplex;  $\mathbb{1}_{\mathcal{X}}$  is an indicator function  $(\mathbb{1}_{\mathcal{X}}(\pmb {x}) = 1$  if  $\pmb {x}\in \mathcal{X}$  and  $\mathbb{1}_{\mathcal{X}}(\pmb {x}) = 0$  if  $\pmb {x}\notin \mathcal{X}$ ;  $Z$  is a normalizing constant. We can also consider a non-negative bounded uniform distribution
+
+$$
+p (\boldsymbol {z}) = \mathbb {1} _ {[ 0, 1 ] ^ {N}} (\boldsymbol {z}) \tag {4}
+$$
+
+for non-negative matrix factorization. The distribution of the data point  $\pmb{y}$  can be expressed as
+
+$$
+p _ {\boldsymbol {\theta}} (\boldsymbol {y}) = \int p _ {\boldsymbol {\theta}} (\boldsymbol {y} | \boldsymbol {z}) p (\boldsymbol {z}) d \boldsymbol {z} = \mathbb {E} _ {p (\boldsymbol {z})} [ p _ {\boldsymbol {\theta}} (\boldsymbol {y} | \boldsymbol {z}) ], \tag {5}
+$$
+
+where  $p_{\pmb{\theta}}(\pmb{y}|\pmb{z}) = \mathcal{N}(\pmb{y};f_{\pmb{\theta}}(\pmb{z}),\sigma^{2}\mathbf{I})$
+
+We want to estimate  $\pmb{\theta}$  from a given dataset  $\{\pmb{y}_1,\dots ,\pmb{y}_L\}$ . We pursue maximum-likelihood (ML) estimation. Let
+
+$$
+\mathcal {L} (\boldsymbol {\theta}; \boldsymbol {y}) = \log p _ {\boldsymbol {\theta}} (\boldsymbol {y}),
+$$
+
+denote the log-likelihood function for  $y$ . ML estimation determines  $\theta$  by solving
+
+$$
+\max  _ {\boldsymbol {\theta}} \sum_ {n = 1} ^ {L} \mathcal {L} (\boldsymbol {\theta}; \boldsymbol {y} _ {n}).
+$$
+
+The challenge with the ML estimation problem above is that the log-likelihood  $\mathcal{L}(\pmb{\theta};\pmb{y})$  has no known tractable expression in general; this is because (5) is a multi-dimensional integral that has no closed-form or explicit equation in general. One can approximate (5) by a stochastic (Monte Carlo sampling) approximation method, but such methods were often found to be computationally inefficient in practice.
+
+Recent research has considered variational inference (VI), together with stochastic approximation, as a more practical way to approximate the log-likelihood function. Let  $q_{\phi}(z|y)$  be some distribution function with parameter  $\phi$ , which will be called the variational distribution in the sequel. Consider the Jensen inequality
+
+$$
+\mathcal {L} (\boldsymbol {\theta}; \boldsymbol {y}) \geq \widehat {\mathcal {L}} (\boldsymbol {\theta}, \phi ; \boldsymbol {y}) = \mathbb {E} _ {q _ {\phi} (\boldsymbol {z} | \boldsymbol {y})} \left[ \log \frac {p _ {\boldsymbol {\theta}} (\boldsymbol {y} | \boldsymbol {z}) p (\boldsymbol {z})}{q _ {\phi} (\boldsymbol {z} | \boldsymbol {y})} \right].
+$$
+
+The function  $\widehat{\mathcal{L}}$  is called the evidence lower bound (ELBO). The idea is to choose a  $q_{\phi}$  such that  $\widehat{\mathcal{L}}$  would be computationally efficient to compute or approximate. We also
+
+hope that the choice of  $q_{\phi}$  would lead to a small gap between  $\mathcal{L}$  and  $\widehat{\mathcal{L}}$ , and thereby a good approximation of the log-likelihood function.
+
+Take the famous variational autoencoder (VAE) as an example. The latent prior is Gaussian, specifically,  $p(z) = \mathcal{N}(z;0,\mathbf{I})$ . The variational distribution is chosen as  $q_{\phi}(z|\mathbf{y}) = \mathcal{N}(z;\mu_{\phi}(\mathbf{y}),\mathrm{Diag}(\sigma_{\phi}^{2}(\mathbf{y})))$ , where  $\mu_{\phi}$  and  $\sigma_{\phi}$  are neural networks with parameter  $\phi$ . By the parameterization trick (see (Kingma & Welling, 2013) for details), it was found that  $\widehat{\mathcal{L}}$  can be efficiently handled by stochastic approximation. This VAE approach has been employed in ICA and SCA (Khemakhem et al., 2020; Li et al., 2024).
+
+Let us write down the VI problem:
+
+$$
+\max  _ {\boldsymbol {\theta}, \boldsymbol {\phi}} \frac {1}{L} \sum_ {n = 1} ^ {L} \widehat {\mathcal {L}} (\boldsymbol {\theta}, \boldsymbol {\phi}; \boldsymbol {y} _ {n}).
+$$
+
+Note that the variational model parameter  $\phi$  is also optimized for best ELBO approximation given the structure of  $q_{\phi}$ . Additionally, we should mention the estimation of the latent variables once  $(\theta, \phi)$  is obtained from the VI problem. Consider the minimum mean square error (MMSE) estimate  $\widehat{\pmb{z}}_n = \mathbb{E}_{p_\theta(\pmb{z}|\pmb{y}_n)}[\pmb{z}]$ . There is no known tractable equation for  $p_\theta(\pmb{z}|\pmb{y})$ . The variational distribution  $q_{\phi}(\pmb{z}|\pmb{y})$  can be seen as an approximation of  $p_{\theta}(\pmb{z}|\pmb{y})$  because the ELBO attains equality if and only if  $q_{\phi}(\pmb{z}|\pmb{y}) = p_{\theta}(\pmb{z}|\pmb{y})$ . This leads us to employ
+
+$$
+\widehat {z} _ {n} = \mathbb {E} _ {q _ {\phi} (\boldsymbol {z} | \boldsymbol {y} _ {n})} [ \boldsymbol {z} ]. \tag {6}
+$$
+
+# 3. Dimension-Reducing Diffusion VI for MMF
+
+Our endeavor is to take insight from variational diffusion models (Ho et al., 2020; Kingma et al., 2021) to develop an alternative VI scheme for MMF.
+
+# 3.1. Generative Model
+
+The generative model we consider is a modification of that in Section 2.1. Denote
+
+$$
+\boldsymbol {x} _ {0} = \boldsymbol {y}, \quad \boldsymbol {x} _ {T} = \boldsymbol {z}.
+$$
+
+The generation of the data point  $\boldsymbol{x}_0$  from the latent variable  $\boldsymbol{x}_T$  follows a Markov process
+
+$$
+\boldsymbol {x} _ {t - 1} = f _ {t, \boldsymbol {\theta}} (\boldsymbol {x} _ {t}) + \boldsymbol {v} _ {t}, \quad t = T, T - 1, \dots , 1,
+$$
+
+where  $\pmb{x}_t\in \mathbb{R}^{d_t}$  is the latent variable at layer  $t$
+
+$$
+f _ {t, \boldsymbol {\theta}} (\boldsymbol {x} _ {t}) = \left\{ \begin{array}{l l} \rho (\boldsymbol {A} _ {1} \boldsymbol {x} _ {1}), & t = 1 \\ \boldsymbol {B} _ {t} \boldsymbol {x} _ {t} + \boldsymbol {C} _ {t} \rho (\boldsymbol {A} _ {t} \boldsymbol {x} _ {t}), & 2 \leq t \leq T - 1; \\ \boldsymbol {A} _ {T} \boldsymbol {x} _ {T}, & t = T \end{array} \right. \tag {7a}
+$$
+
+$\pmb{v}_t \sim \mathcal{N}(\pmb{0}, \pmb{\Sigma}_t)$  represents the modeling error at layer  $t - 1$  for  $t \geq 2$  and noise for  $t = 1$ . The base latent variable  $\pmb{x}_T$
+
+is distributed according to a given latent prior  $p(\pmb{x}_T)$ , such as (3) and (4). The model parameter  $\pmb{\theta}$  contains all the  $A_t$ 's,  $B_t$ 's,  $C_t$ 's, and  $\Sigma_t$ 's.
+
+Some justification should be provided for the model of  $f_{t,\theta}$ . Eq. (7b) has a more general structure than its counterpart in (2), which basically considers  $f_{t,\theta} = \rho(\mathbf{A}_t\mathbf{x}_t)$ . The merit will be clear as we proceed to diffusion VI. Eq. (7c) has no activation function  $\rho$ . This is to facilitate our VI development which will be described later. Taking out  $\rho$  does not pose an issue: If  $\mathbf{A}_T = \mathbf{I}$  and  $\boldsymbol{\Sigma}_{T} \simeq 0$ , then  $\mathbf{x}_{T-1} \simeq \mathbf{x}_T$  and  $\mathbf{x}_{T-2} \simeq f_{T-1,\boldsymbol{\theta}}(\mathbf{x}_T) + \mathbf{v}_{T-1}$ . As a key assumption, we assume that the dimension of  $\mathbf{x}_t$  is gradually decreasing:
+
+$$
+M = d _ {0} \geq d _ {1} \geq d _ {2} \geq \dots \geq d _ {T} = N.
+$$
+
+# 3.2. Dimension-Reducing Diffusion Model
+
+Now we describe our proposed diffusion variational process. Consider the following process as our chosen variational process:
+
+$$
+\boldsymbol {x} _ {t} = \sqrt {a _ {t}} \boldsymbol {U} _ {t} ^ {\top} \boldsymbol {x} _ {t - 1} + \sqrt {1 - a _ {t}} \boldsymbol {e} _ {t}, t = 1, \dots , T - 1, (8 \mathrm {a})
+$$
+
+$$
+\boldsymbol {x} _ {T} \sim q _ {\gamma} (\boldsymbol {x} _ {T} | \boldsymbol {x} _ {T - 1}), \tag {8b}
+$$
+
+where  $0 < a_{t} < 1$ ;  $\pmb{e}_{t}\sim \mathcal{N}(\pmb {0},\pmb {I})$  and the  $\pmb{e}_{t}$  's are independent;  $\pmb {U}_t\in \mathbb{R}^{d_{t - 1}\times d_t}$  is semi-orthogonal;  $q_{\gamma}(\pmb{x}_T|\pmb{x}_{T - 1})$  is latent-prior-dependent. In SCA (cf. (3)), we may choose  $q_{\gamma}(\pmb{x}_T|\pmb{x}_{T - 1})$  as a Dirichlet distribution
+
+$$
+q _ {\gamma} \left(\boldsymbol {x} _ {T} \mid \boldsymbol {x} _ {T - 1}\right) = \operatorname {D i r} \left(\boldsymbol {x} _ {T}; \boldsymbol {\alpha} _ {\gamma} \left(\boldsymbol {x} _ {T - 1}\right)\right),
+$$
+
+where  $\mathrm{Dir}(\pmb{x};\pmb{\alpha})$  denotes a Dirichlet distribution with parameter  $\alpha$ ;  $\pmb{\alpha}_{\gamma}$  is a neural network with parameter  $\gamma$ . For the non-negative latent prior in (4), we may choose  $q_{\gamma}$  as a Beta distribution. For ICA, we may choose a Gaussian distribution in the same way as that of the VAE (cf. Section 2.2). The selection criteria of  $q_{\gamma}(\pmb{x}_T|\pmb{x}_{T-1})$  are that (i) the support of  $q_{\gamma}(\pmb{x}_T|\pmb{x}_{T-1})$  is the same as that of the latent prior  $p(\pmb{x}_T)$ ; and that (ii) it has analytical expressions with its mean, covariance, and entropy. Table 3 in Appendix A.5 shows some examples. The variational model parameter  $\phi$  contains all the  $\pmb{a}_t$ 's,  $\pmb{U}_t$ 's, and  $\gamma$ .
+
+It is important to note that if  $d_0 = \dots = d_T$  and  $U_{t} = I$ , (8a) is exactly the diffusion model in the context of generative models. To the best of our knowledge, the dimension-reducing diffusion model in (8a) has not been considered before. The objective is not only to gradually add noise to the data point  $x_0$  a key part with the previous diffusion models—but also to gradually reduce the dimension. The dimension reduction feature is particularly relevant to MMF or the low-dimensional representation of high-dimensional data.
+
+# 3.3. Dimension-Reducing Diffusion VI
+
+Let us examine the ELBO under the above model. We have:
+
+$$
+p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {0: T}\right) = p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {0} \mid \boldsymbol {x} _ {1}\right) \dots p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {T - 1} \mid \boldsymbol {x} _ {T}\right) p \left(\boldsymbol {x} _ {T}\right), \tag {9a}
+$$
+
+$$
+q _ {\phi} \left(\boldsymbol {x} _ {1: T} \mid \boldsymbol {x} _ {0}\right) = q _ {\phi} \left(\boldsymbol {x} _ {T} \mid \boldsymbol {x} _ {T - 1}\right) \dots q _ {\phi} \left(\boldsymbol {x} _ {1} \mid \boldsymbol {x} _ {0}\right), \tag {9b}
+$$
+
+where
+
+$$
+p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t}\right) = \mathcal {N} \left(\boldsymbol {x} _ {t - 1}; f _ {t, \boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}\right), \boldsymbol {\Sigma} _ {t}\right), \tag {10}
+$$
+
+$$
+q _ {\phi} \left(\boldsymbol {x} _ {t} \mid \boldsymbol {x} _ {t - 1}\right) = \mathcal {N} \left(\boldsymbol {x} _ {t}; \sqrt {a _ {t}} \boldsymbol {U} _ {t} ^ {\top} \boldsymbol {x} _ {t - 1}, (1 - a _ {t}) \boldsymbol {I}\right),
+$$
+
+for  $t \leq T - 1$ . When applying (9a) and (9b) to the ELBO
+
+$$
+\widehat {\mathcal {L}} (\boldsymbol {\theta}, \phi ; \boldsymbol {x} _ {0}) = \mathbb {E} _ {q _ {\phi} \left(\boldsymbol {x} _ {1: T} \mid \boldsymbol {x} _ {0}\right)} \left[ \log \frac {p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {0 : T}\right)}{q _ {\phi} \left(\boldsymbol {x} _ {1 : T} \mid \boldsymbol {x} _ {0}\right)} \right], \tag {11}
+$$
+
+it is natural to match  $p_{\theta}(\pmb{x}_{t-1}|\pmb{x}_t)$  and  $q_{\phi}(\pmb{x}_t|\pmb{x}_{t-1})$  and then to derive a multilayer ELBO expression; this is exactly what hierarchical VAEs do; see, e.g., Section 2.3 in (Luo, 2022). But this is not what diffusion VI does. It considers this alternative expression of  $q_{\phi}(\pmb{x}_{1:T}|\pmb{x}_0)$ :
+
+$$
+q _ {\phi} \left(\boldsymbol {x} _ {1: T} \mid \boldsymbol {x} _ {0}\right) = q _ {\phi} \left(\boldsymbol {x} _ {T} \mid \boldsymbol {x} _ {0}\right) \prod_ {t = 2} ^ {T} q _ {\phi} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}\right), \tag {12}
+$$
+
+which is obtained by applying  $q_{\phi}(\pmb{x}_t|\pmb{x}_{t-1}) = q_{\phi}(\pmb{x}_t|\pmb{x}_{t-1}, \pmb{x}_0)$  and  $q_{\phi}(\pmb{x}_t|\pmb{x}_{t-1}, \pmb{x}_0)q_{\phi}(\pmb{x}_{t-1}|\pmb{x}_0) = q_{\phi}(\pmb{x}_{t-1}|\pmb{x}_t, \pmb{x}_0)q_{\phi}(\pmb{x}_t|\pmb{x}_0)$  to (9b). With (12), we can express  $\widehat{\mathcal{L}}$  as
+
+$$
+\widehat {\mathcal {L}} (\boldsymbol {\theta}, \phi ; \boldsymbol {x} _ {0}) = \sum_ {t = 1} ^ {T} \widehat {\mathcal {L}} _ {t} (\boldsymbol {\theta}, \phi ; \boldsymbol {x} _ {0}), \tag {13}
+$$
+
+where
+
+$$
+\widehat {\mathcal {L}} _ {1} = \mathbb {E} _ {q _ {\phi} \left(\boldsymbol {x} _ {1} \mid \boldsymbol {x} _ {0}\right)} \left[ \log p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {0} \mid \boldsymbol {x} _ {1}\right) \right], \tag {14}
+$$
+
+$$
+\widehat {\mathcal {L}} _ {t} = \mathbb {E} _ {q _ {\phi} \left(\boldsymbol {x} _ {t - 1}, \boldsymbol {x} _ {t} \mid \boldsymbol {x} _ {0}\right)} \left[ \log \frac {p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t}\right)}{q _ {\phi} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t} , \boldsymbol {x} _ {0}\right)} \right] \tag {15}
+$$
+
+for  $2\leq t\leq T - 1$  , and
+
+$$
+\widehat {\mathcal {L}} _ {T} = \mathbb {E} _ {q _ {\phi} \left(\boldsymbol {x} _ {T - 1}, \boldsymbol {x} _ {T} \mid \boldsymbol {x} _ {0}\right)} \left[ \log \frac {p \left(\boldsymbol {x} _ {T}\right) p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {T - 1} \mid \boldsymbol {x} _ {T}\right)}{q _ {\phi} \left(\boldsymbol {x} _ {T - 1} , \boldsymbol {x} _ {T} \mid \boldsymbol {x} _ {0}\right)} \right]. \tag {16}
+$$
+
+In the following, we will deal with each  $\widehat{\mathcal{L}}_t$ .
+
+# 3.3.1. LAYER-1 TERM  $\widehat{\mathcal{L}}_1$
+
+First, consider (14). Denote  $\| \pmb{x}\|_{\pmb{\Sigma}}^{2} = \pmb{x}^{\top}\pmb{\Sigma}^{-1}\pmb{x}$ . It can be shown that
+
+$$
+- \widehat {\mathcal {L}} _ {1} \propto \underbrace {\frac {1}{2} \left(\mathbb {E} _ {q _ {\phi} \left(\boldsymbol {x} _ {1} \mid \boldsymbol {x} _ {0}\right)} \left[ \| \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {1} \boldsymbol {x} _ {1}) \| _ {\boldsymbol {\Sigma} _ {1}} ^ {2} \right] + \log | \boldsymbol {\Sigma} _ {1} |\right)} _ {r _ {1} (\boldsymbol {\theta}, \phi ; \boldsymbol {x} _ {0})} \tag {17}
+$$
+
+and that  $q_{\phi}(\pmb{x}_1|\pmb{x}_0) = \mathcal{N}(\pmb{x}_1; \sqrt{a_1}\pmb{U}_1^\top \pmb{x}_0, (1 - a_1)\pmb{I})$ . This term can be readily handled by stochastic approximation.
+
+# 3.3.2. LAYER-  $t$  TERM  $\widehat{\mathcal{L}}_t$ ,  $2 \leq t \leq T - 1$
+
+Second, consider (15). As a key result in diffusion models,  $q_{\phi}(\pmb{x}_{t-1}, \pmb{x}_t | \pmb{x}_0)$  has an analytical expression. It can be shown that, for  $2 \leq t \leq T - 1$ ,
+
+$$
+q \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}\right) = \mathcal {N} \left(\boldsymbol {x} _ {t - 1}; \boldsymbol {\mu} _ {t, \phi} \left(\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}\right), \boldsymbol {\Psi} _ {t, \phi}\right), \tag {18}
+$$
+
+where
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {t, \phi} = \frac {\sqrt {a _ {t}} (1 - \bar {a} _ {t - 1})}{1 - \bar {a} _ {t}} \boldsymbol {U} _ {t} \boldsymbol {x} _ {t} + \sqrt {\bar {a} _ {t - 1}} \bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} \tag {19} \\ + \frac {\sqrt {\bar {a} _ {t - 1}} (\bar {a} _ {t} - a _ {t})}{1 - \bar {a} _ {t}} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top} \bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0}; \\ \end{array}
+$$
+
+$$
+\Psi_ {t, \phi} = (1 - \bar {a} _ {t - 1}) \left(\boldsymbol {I} - \frac {a _ {t} - \bar {a} _ {t}}{1 - \bar {a} _ {t}} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top}\right); \tag {20}
+$$
+
+$\bar{U}_t = U_1U_2\dots U_t;\bar{a}_t = a_1a_2\dots a_t.$  This result is shown by using the fact that  $q(\pmb{x}_{t - 1},\pmb{x}_t|\pmb{x}_0)$  is Gaussian (for  $2\leq t\leq$ $T - 1$  ). The derivations of (18) are relegated to Appendix A.1. Eq. (18) leads to a simplified result for  $\widehat{\mathcal{L}}_t$  . From (15) we can write
+
+$$
+- \widehat {\mathcal {L}} _ {t} = \mathbb {E} _ {q _ {\phi} (\boldsymbol {x} _ {t} | \boldsymbol {x} _ {0})} \underbrace {\left[ D _ {\mathrm {K L}} \left(q _ {\phi} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t} , \boldsymbol {x} _ {0}\right) \right\lvert   \left| p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t}\right)\right) \right]} _ {:= D _ {t} \left(\boldsymbol {x} _ {t}; \boldsymbol {x} _ {0}\right)}, \tag {21}
+$$
+
+where  $D_{\mathrm{KL}}(q||p) = \int q(\boldsymbol{x})\log (q(\boldsymbol{x}) / p(\boldsymbol{x}))\mathrm{d}\boldsymbol{x}$  is the Kullback-Leibler (KL) divergence of two distributions  $p$  and  $q$ . Let  $\mathbb{S}_{+}^{d}$  and  $\mathbb{S}_{++}^{d}$  denote the sets of all symmetric positive semidefinite and positive definite matrices in  $\mathbb{R}^{d\times d}$ , respectively. Consider the following lemma.
+
+Lemma 3.1. Let  $\pmb{\mu}_1, \pmb{\mu}_2 \in \mathbb{R}^d$ ,  $\pmb{\Sigma}_1, \pmb{\Sigma}_2 \in \mathbb{S}_{++}^d$ ,  $\pmb{\Psi} \in \mathbb{S}_+^d$ . Let  $p(\pmb{x}) = \mathcal{N}(\pmb{x}; \pmb{\mu}_1, \pmb{\Sigma}_1)$  and  $q(\pmb{x}) = \mathcal{N}(\pmb{x}; \pmb{\mu}_2, \pmb{\Sigma}_2)$ .
+
+Consider
+
+$$
+f \left(\boldsymbol {\Sigma} _ {1}\right) = D _ {\mathrm {K L}} (q | | p) + \operatorname {t r} \left(\boldsymbol {\Sigma} _ {1} ^ {- 1} \boldsymbol {\Psi}\right).
+$$
+
+It holds that
+
+$$
+\min  _ {\boldsymbol {\Sigma} _ {1} \in \mathbb {S} _ {+ +} ^ {d}} f \left(\boldsymbol {\Sigma} _ {1}\right) = \frac {1}{2} \log \left| \boldsymbol {I} + \boldsymbol {\Sigma} _ {2} ^ {- 1 / 2} (\boldsymbol {W} + \boldsymbol {\Psi}) \boldsymbol {\Sigma} _ {2} ^ {- 1 / 2} \right|,
+$$
+
+where  $\pmb{W} = (\pmb{\mu}_2 - \pmb{\mu}_1)(\pmb{\mu}_2 - \pmb{\mu}_1)^\top$ , and the solution to the above problem is  $\pmb{\Sigma}_1^{\star} = \pmb{\Sigma}_2 + \pmb{W} + \pmb{\Psi}$ .
+
+The proof is provided in Appendix A.2. By applying Lemma 3.1 to  $D_{t}$ , and noting that  $p_{\theta}(\boldsymbol{x}_{t-1}|\boldsymbol{x})$  in (10) and  $q_{\phi}(\boldsymbol{x}_{t-1}|\boldsymbol{x}_t, \boldsymbol{x}_0)$  in (18) are Gaussian, we obtain
+
+$$
+\min  _ {\boldsymbol {\Sigma} _ {t} \in \mathbb {S} _ {+ +} ^ {d}} D _ {t} = \frac {1}{2} \log \left(1 + \left\| f _ {t, \boldsymbol {\theta}} (\boldsymbol {x} _ {t}) - \boldsymbol {\mu} _ {t, \boldsymbol {\phi}} (\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}) \right\| _ {\boldsymbol {\Psi} _ {t, \boldsymbol {\phi}}} ^ {2}\right);
+$$
+
+we have used  $|\pmb {I} + \pmb {A}\pmb {B}| = |\pmb {I} + \pmb {B}\pmb {A}|$  to get the above equation. To facilitate the VI, we apply an approximation
+
+$$
+\min  _ {\boldsymbol {\Sigma} _ {t} \in \mathbb {S} _ {+ +} ^ {d}} D _ {t} \leq \frac {1}{2} \left\| f _ {t, \boldsymbol {\theta}} (\boldsymbol {x} _ {t}) - \boldsymbol {\mu} _ {t, \boldsymbol {\phi}} (\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}) \right\| _ {\boldsymbol {\Psi} _ {t, \boldsymbol {\phi}}} ^ {2} := \tilde {r} _ {t},
+$$
+
+which is due to  $\log (x)\leq x - 1$  for  $x > 0$ ; this approximation is good if  $\tilde{r}_t$  is small.
+
+The above derivations show that VI intends to match  $f_{t,\theta}(\pmb{x}_t)$  and  $\pmb{\mu}_{t,\phi}(\pmb{x}_t,\pmb{x}_0)$ . This motivates us to fix  $\pmb{B}_t$  and  $\pmb{C}_t$  in (7b) such that the structure of  $f_{t,\theta}(\pmb{x}_t)$  matches that of  $\pmb{\mu}_{t,\phi}(\pmb{x}_t,\pmb{x}_0)$  in (19). Specifically,
+
+$$
+\boldsymbol {B} _ {t} = \frac {\sqrt {a _ {t}} (1 - \bar {a} _ {t - 1})}{1 - \bar {a} _ {t}} \boldsymbol {U} _ {t}, \tag {22}
+$$
+
+$$
+\boldsymbol {C} _ {t} = \frac {\sqrt {\bar {a} _ {t - 1}} (\bar {a} _ {t} - a _ {t})}{1 - \bar {a} _ {t}} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top} + \sqrt {\bar {a} _ {t - 1}} \boldsymbol {I}.
+$$
+
+With this choice,  $\tilde{r}_t$  is simplified to
+
+$$
+\tilde {r} _ {t} = \frac {1}{2} \left\| \boldsymbol {C} _ {t} \left(\bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x} _ {t})\right) \right\| _ {\boldsymbol {\Psi} _ {t, \phi}} ^ {2}. \tag {23}
+$$
+
+In fact,  $\tilde{r}_t$  can be further simplified to
+
+$$
+\tilde {r} _ {t} = \frac {\bar {a} _ {t - 1}}{2 \left(1 - \bar {a} _ {t - 1}\right)} \left\| \bar {U} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x} _ {t}) \right\| _ {2} ^ {2} \tag {24}
+$$
+
+$$
++ \frac {\bar {a} _ {t}}{2 (\bar {a} _ {t} - 1)} \left\| \boldsymbol {U} _ {t} ^ {\top} \left(\bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x} _ {t})\right) \right\| _ {2} ^ {2}.
+$$
+
+We relegate the derivation to Appendix A.3. This gives the following result
+
+$$
+\max  _ {\boldsymbol {B} _ {t}, \boldsymbol {C} _ {t}, \boldsymbol {\Sigma} _ {t}} \widehat {\mathcal {L}} _ {t} \geq - \underbrace {\mathbb {E} _ {q _ {\phi} (\boldsymbol {x} _ {t} \mid \boldsymbol {x} _ {0})} [ \tilde {r} _ {t} (\boldsymbol {x} _ {t} ; \boldsymbol {x} _ {0}) ]} _ {: = r _ {t} (\boldsymbol {\theta}, \phi ; \boldsymbol {x} _ {0})}. \tag {25}
+$$
+
+Note that  $q_{\phi}(\pmb{x}_t|\pmb{x}_0) = \mathcal{N}(\pmb{x}_t;\sqrt{\bar{a}_t}\bar{\pmb{U}}_t^\top \pmb{x}_0,(1 - \bar{a}_t)\pmb {I})$  ; this can be derived from  $q_{\phi}(\pmb {x}_t|\pmb{x}_{t - 1})$  . The function  $r_t$  can be readily handled by stochastic approximation.
+
+# 3.3.3. SOME INSIGHT
+
+Let us pause a moment and try to get some intuitive insight. Eq. (25) suggests that VI intends to approximate
+
+$$
+\bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} \approx \rho (\boldsymbol {A} _ {t} \boldsymbol {x} _ {t}).
+$$
+
+In particular, the left-hand side is a dimension-reduced  $x_0$  ( $\bar{U}_{t-1}$  is semi-orthogonal) while the right-hand side is a nonlinear low-dimension representation. Take the case of  $\rho$  being a ReLU function as an example. We may want the dimension-reduced data point to be non-negative and possess a latent lower-dimensional structure. And this gradually happens from layer 1 to layer  $T$ .
+
+# 3.3.4. LAYER-T TERM  $\hat{\mathcal{L}}_T$
+
+Third, consider (16). If  $q_{\gamma}(\boldsymbol{x}_T|\boldsymbol{x}_{T-1})$  takes a Gaussian form, then (16) may be handled in a similar way as in Section 3.3.2. If not, more work needs to be done; particularly, the key result in (18) no longer applies. Our derivations are as follows. We can decompose (16) as
+
+$$
+- \hat {\mathcal {L}} _ {T} \propto \tilde {r} _ {T} (\boldsymbol {\theta}, \phi ; \boldsymbol {x} _ {0}) + r _ {T + 1} (\boldsymbol {\theta}, \phi ; \boldsymbol {x} _ {0}), \tag {26}
+$$
+
+where
+
+$$
+\begin{array}{l} \tilde {r} _ {T} = \mathbb {E} _ {q _ {\phi} (\boldsymbol {x} _ {T - 1} | \boldsymbol {x} _ {0})} [ \check {r} _ {T} (\boldsymbol {\theta}, \phi , \boldsymbol {x} _ {T - 1}; \boldsymbol {x} _ {0}) \tag {27} \\ - \log q _ {\phi} (\boldsymbol {x} _ {T - 1} | \boldsymbol {x} _ {0}) ]; \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \tilde {r} _ {T} = \mathbb {E} _ {q _ {\gamma} (\boldsymbol {x} _ {T} | \boldsymbol {x} _ {T - 1})} [ \log p _ {\boldsymbol {\theta}} (\boldsymbol {x} _ {T - 1} | \boldsymbol {x} _ {T}) ]; \\ r _ {T + 1} = \mathbb {E} _ {q _ {\phi} \left(\boldsymbol {x} _ {T - 1} \mid \boldsymbol {x} _ {0}\right)} [ H \left(q _ {\gamma} \left(\boldsymbol {x} _ {T} \mid \boldsymbol {x} _ {T - 1}\right)\right) ]; \tag {28} \\ \end{array}
+$$
+
+$H(p(\pmb{x})) = \int p(\pmb{x})\log p(\pmb{x})\mathrm{d}\pmb{x}$  denotes the negative entropy of  $p(\pmb{x})$ . Also note that
+
+$$
+q _ {\phi} (\boldsymbol {x} _ {T - 1} | \boldsymbol {x} _ {0}) = \mathcal {N} (\boldsymbol {x} _ {T - 1}; \sqrt {\bar {a} _ {T - 1}} \bar {\boldsymbol {U}} _ {T - 1} ^ {\top} \boldsymbol {x} _ {0}, (1 - \bar {a} _ {T - 1}) \boldsymbol {I}).
+$$
+
+It can be shown that
+
+$$
+\begin{array}{l} \check {r} _ {T} = \log \mathcal {N} \left(\boldsymbol {x} _ {T - 1}; \boldsymbol {A} _ {T} \boldsymbol {\mu} _ {T, \gamma} (\boldsymbol {x} _ {T - 1}), \boldsymbol {\Sigma} _ {T}\right) \\ - \frac {1}{2} \operatorname {t r} \left(\boldsymbol {\Sigma} _ {T} ^ {- 1} \boldsymbol {A} _ {T} \boldsymbol {\Psi} _ {T, \gamma} (\boldsymbol {x} _ {T - 1}) \boldsymbol {A} _ {T} ^ {\top}\right), \end{array} \tag {29}
+$$
+
+$$
+\boldsymbol {\mu} _ {T, \gamma} (\boldsymbol {x} _ {T - 1}) = \mathbb {E} _ {q _ {\gamma} (\boldsymbol {x} _ {T} | \boldsymbol {x} _ {T - 1})} [ \boldsymbol {x} _ {T} ], \tag {30}
+$$
+
+$$
+\Psi_ {T, \gamma} (\boldsymbol {x} _ {T - 1}) = \operatorname {C o v} _ {q _ {\gamma} \left(\boldsymbol {x} _ {T} \mid \boldsymbol {x} _ {T - 1}\right)} (\boldsymbol {x} _ {T}); \tag {31}
+$$
+
+see Appendix A.4. Consider the following lemma.
+
+Lemma 3.2. Consider the same settings in Lemma 3.1, except that  $\pmb{\mu}_1$  is changed to  $\pmb{\mu}_1(\pmb{x})$ , which is a function of  $\pmb{x}$ . Let  $\pmb{R} = \mathbb{E}_{q(\pmb{x})}\left[(\pmb{x} - \pmb{\mu}_1(\pmb{x}))(\pmb{x} - \pmb{\mu}_1(\pmb{x}))^\top\right]$ . Suppose  $\pmb{R} + \pmb{\Psi}$  is positive definite. Then
+
+$$
+\min_{\boldsymbol{\Sigma}_{1}\in \mathbb{S}^{d}_{+ + }}f(\boldsymbol{\Sigma}_{1}) = \frac{1}{2}\log |\boldsymbol{\Sigma}_{2}^{-1 / 2}(\boldsymbol {R} + \boldsymbol {\Psi})\boldsymbol{\Sigma}_{2}^{-1 / 2}|,
+$$
+
+and the solution to the above problem is  $\pmb{\Sigma}_{1}^{\star} = \pmb{R} + \pmb{\Psi}$
+
+The proof is provided in Appendix A.2. By applying Lemma 3.2 to (27) and (29), we obtain
+
+$$
+\min  _ {\boldsymbol {\Sigma} _ {T} \in \mathbb {S} _ {+ +} ^ {d _ {T}}} \tilde {r} _ {T} = \frac {1}{2} \log \left| \frac {1}{1 - \bar {a} _ {T - 1}} (\boldsymbol {R} + \boldsymbol {G}) \right|, \tag {32}
+$$
+
+where
+
+$$
+\begin{array}{l} \boldsymbol {R} = \mathbb {E} _ {q \left(\boldsymbol {x} _ {T - 1} \mid \boldsymbol {x} _ {0}\right)} \left[ \left(\boldsymbol {x} _ {T - 1} - \boldsymbol {\mu} _ {T, \gamma} (\boldsymbol {x} _ {T - 1})\right) \right. \\ \left(\pmb {x} _ {T - 1} - \pmb {\mu} _ {T, \pmb {\gamma}} (\pmb {x} _ {T - 1})\right) ^ {\top} ], \\ \end{array}
+$$
+
+$$
+\boldsymbol {G} = \mathbb {E} _ {q (\boldsymbol {x} _ {T - 1} | \boldsymbol {x} _ {0})} \left[ \boldsymbol {A} _ {T} \boldsymbol {\Psi} _ {T, \gamma} (\boldsymbol {x} _ {T - 1}) \boldsymbol {A} _ {T} ^ {\top} \right].
+$$
+
+We assume that  $\mathbf{R} + \mathbf{G}$  is positive definite, which is a fairly mild assumption. Eq.(32) looks complicated. To facilitate VI, we consider
+
+$$
+\min  _ {\boldsymbol {\Sigma} _ {T} \in \mathbb {S} _ {+ +} ^ {d _ {T}}} \tilde {r} _ {T} \leq \frac {1}{2} \operatorname {t r} \left(\frac {1}{1 - \bar {a} _ {T - 1}} (\boldsymbol {R} + \boldsymbol {G})\right) - \frac {d _ {T}}{2}
+$$
+
+$$
+\begin{array}{l} \propto \frac {1}{2 (1 - \bar {a} _ {T - 1})} \left(\mathbb {E} _ {q _ {\phi} (\boldsymbol {x} _ {T - 1} | \boldsymbol {x} _ {0})} \left[ \| \boldsymbol {x} _ {T - 1} - \boldsymbol {\mu} _ {T, \boldsymbol {\gamma}} (\boldsymbol {x} _ {T - 1}) \| _ {2} ^ {2} \right. \right. \\ \left. + \operatorname {t r} \left(\boldsymbol {A} _ {T} \boldsymbol {\Psi} _ {T, \gamma} \left(\boldsymbol {x} _ {T - 1}\right) \boldsymbol {A} _ {T} ^ {\top}\right)\right]\left. \right) := r _ {T} (\boldsymbol {\theta}, \phi ; \boldsymbol {x} _ {0}), \tag {33} \\ \end{array}
+$$
+
+where we have used  $\log (|\pmb {A}|)\leq \mathrm{tr}(\pmb {A}) - d$  for  $\pmb {A}\in \mathbb{S}_{++}^{d}$  As described previously,  $q_{\gamma}(\pmb{x}_T|\pmb{x}_{T-1})$  is chosen such that  $\pmb{\mu}_{T,\gamma}(\pmb{x}_{T-1}),\pmb{\Psi}_{T,\gamma}(\pmb{x}_{T-1})$  , and  $H(q_{\gamma}(\pmb{x}_T|\pmb{x}_{T-1}))$  have analytical expressions. Table 3 in Appendix A.5 gives some examples. The terms  $r_T$  and  $r_{T + 1}$  can hence be handled by stochastic approximation.
+
+# 3.3.5. REMAINING ASPECTS
+
+Let us assemble the components together. The VI problem is
+
+$$
+\min  _ {\boldsymbol {\phi}, \boldsymbol {\theta}} \frac {1}{L} \sum_ {n = 1} ^ {L} \sum_ {t = 1} ^ {T + 1} r _ {t} (\boldsymbol {\theta}, \boldsymbol {\phi}; \boldsymbol {y} _ {n}) + \lambda \sum_ {t = 1} ^ {T - 1} \left\| \boldsymbol {U} _ {t} ^ {\top} \boldsymbol {U} _ {t} - \boldsymbol {I} \right\| _ {\mathrm {F}} ^ {2}, \tag {34}
+$$
+
+where the  $r_t$ 's are given in (17), (25), (33), and (28); a regularization term is added to enforce the semi-orthogonality of  $\pmb{U}_t$ 's;  $\lambda \geq 0$  is given. Also,  $\pmb{\theta}$  is modified as  $\pmb{\theta} = \{\pmb{\Sigma}_1, \pmb{A}_1, \dots, \pmb{A}_T\}$ . The latent variable estimate in (6) is given by
+
+$$
+\widehat {\boldsymbol {z}} _ {n} = \mathbb {E} _ {q _ {\phi} (\boldsymbol {x} _ {T} | \boldsymbol {y} _ {n})} [ \boldsymbol {x} _ {T} ] = \mathbb {E} _ {q _ {\phi} (\boldsymbol {x} _ {T - 1} | \boldsymbol {y} _ {n})} [ \boldsymbol {\mu} _ {T, \gamma} (\boldsymbol {x} _ {T - 1}) ], \tag {35}
+$$
+
+where  $\pmb{\mu}_{T,\gamma}(\pmb{x}_{T - 1})$  is given in (30) and is assumed to have an analytical expression; Monte Carlo sampling may be used to compute  $\widehat{z}_n$ . Alternatively, we can consider
+
+$$
+\begin{array}{l} \widehat {\mathbf {z}} _ {n} \approx \boldsymbol {\mu} _ {T, \gamma} \left(\mathbb {E} _ {q _ {\phi} (\boldsymbol {x} _ {T - 1} | \boldsymbol {y} _ {n})} [ \boldsymbol {x} _ {T - 1} ]\right) \\ = \boldsymbol {\mu} _ {T, \boldsymbol {\gamma}} \left(\sqrt {\bar {a} _ {T - 1}} \bar {\boldsymbol {U}} _ {T - 1} ^ {\top} \boldsymbol {y} _ {n}\right), \\ \end{array}
+$$
+
+which does not require Monte Carlo sampling.
+
+# 4. Numerical Results
+
+Table 1. Hyperspectral images for experiments.  
+
+<table><tr><td>DATASET</td><td>L</td><td>dT</td><td>d0</td></tr><tr><td>SAMSON</td><td>95 × 95</td><td>3</td><td>156</td></tr><tr><td>JASPER</td><td>100 × 100</td><td>4</td><td>198</td></tr><tr><td>APEX</td><td>111 × 122</td><td>4</td><td>285</td></tr><tr><td>URBAN</td><td>307 × 307</td><td>6</td><td>162</td></tr></table>
+
+In this section, we test the proposed DRD-VI for MMF with the latent priors in (3) and (4). For the uniform simplex prior in (3), the variational distribution  $q_{\gamma}(\boldsymbol{x}_T|\boldsymbol{x}_{T-1})$  is chosen as a Dirichlet distribution
+
+$$
+q _ {\gamma} \left(\boldsymbol {x} _ {T} \mid \boldsymbol {x} _ {T - 1}\right) = \operatorname {D i r} \left(\boldsymbol {x} _ {T}; \boldsymbol {\alpha} _ {\gamma} \left(\boldsymbol {x} _ {T - 1}\right)\right),
+$$
+
+where
+
+$$
+\boldsymbol {\alpha} _ {\gamma} \left(\boldsymbol {x} _ {T - 1}\right) = \exp \left(\boldsymbol {W} \boldsymbol {x} _ {T - 1}\right)
+$$
+
+is a one-layer network, and with  $\gamma = W$ . For the nonnegative bounded uniform prior in (4), the variational distribution is chosen as a Beta distribution
+
+$$
+q _ {\gamma} \left(\boldsymbol {x} _ {T} \mid \boldsymbol {x} _ {T - 1}\right) = \mathcal {B} \left(\boldsymbol {x} _ {T}; \boldsymbol {\alpha} _ {\gamma} \left(\boldsymbol {x} _ {T - 1}\right), \boldsymbol {\beta} _ {\gamma} \left(\boldsymbol {x} _ {T - 1}\right)\right),
+$$
+
+Table 2. MSE averaged over all EMs (the best MSE among 10 independent trails/standard deviation).  
+
+<table><tr><td>DATASET</td><td>SISAL</td><td>PRISM</td><td>CNNAEU</td><td>MISICNET</td><td>VASCA</td><td>DRD-VI</td></tr><tr><td>SAMSON</td><td>0.555/0.00</td><td>0.646/0.13</td><td>0.453/0.08</td><td>0.461/0.00</td><td>0.401/0.14</td><td>0.328/0.00</td></tr><tr><td>JASPER</td><td>0.516/0.00</td><td>0.452/0.15</td><td>0.667/0.08</td><td>0.518/0.00</td><td>0.634/0.05</td><td>0.305/0.09</td></tr><tr><td>APEX</td><td>0.743/0.00</td><td>0.645/0.15</td><td>0.812/0.06</td><td>0.413/0.00</td><td>0.633/0.05</td><td>0.609/0.02</td></tr><tr><td>URBAN</td><td>0.796/0.02</td><td>0.824/0.12</td><td>0.700/0.12</td><td>0.955/0.00</td><td>0.785/0.03</td><td>0.677/0.04</td></tr></table>
+
+where
+
+$$
+\boldsymbol {\alpha} _ {\gamma} = \exp \left(\boldsymbol {W} _ {\alpha} \boldsymbol {x} _ {T - 1}\right), \quad \boldsymbol {\beta} _ {\gamma} = \exp \left(\boldsymbol {W} _ {\beta} \boldsymbol {x} _ {T - 1}\right), \tag {36}
+$$
+
+are one-layer networks, with  $\boldsymbol{\gamma} = (\boldsymbol{W}_{\alpha}, \boldsymbol{W}_{\beta})$ . The activation function  $\rho$  is set as the ReLU function. In the experiments, we constrain  $\Sigma_{1} = \sigma^{2}\boldsymbol{I}$ . We adopt the Adam algorithm (Kingma & Ba, 2015) for optimization.
+
+# 4.1. Abundance Estimation in Hyperspectral Images
+
+We first apply the proposed DRD-VI, with the uniform simplex prior (3), to the problem of estimating material abundance in hyperspectral images. This is a representative blind inverse problem in geoscience and remote sensing.
+
+We briefly provide the background. In hyperspectral imaging, each image pixel is a  $d_0$ -dimensional vector capturing the electromagnetic reflectances of materials across  $d_0$  spectral bands, known as spectral signatures or endmembers (EMs). Due to limited spatial resolution, a single pixel may contain mixed reflectances from multiple materials. The proportions of these EMs are modeled by a unit simplex variable  $\pmb{x}_T \in \mathbb{R}^{d_T}$  with  $d_T \ll d_0$ . Without the precise knowledge of the mixing process, abundance estimation aims to recover the abundance map  $\pmb{X}_T \in \mathbb{R}^{d_T \times L}$ , where  $L$  is the number of pixels. Here, each column of  $\pmb{X}_T$  represents the EMs' abundance in a single pixel, while each row shows the spatial distribution of an EM across the image. The task is to retrieve the low-dimensional simplex structures from the high-dimensional hyperspectral image.
+
+We conduct experiments on four hyperspectral image datasets as listed in Table 1. We evaluate the mean squared error (MSE) defined as
+
+$$
+\operatorname {M S E} \left(\boldsymbol {X} _ {T}, \boldsymbol {X} ^ {\star}\right) = \frac {1}{d _ {T}} \sum_ {i = 1} ^ {d _ {T}} \| \tilde {\boldsymbol {x}} _ {i} - \tilde {\boldsymbol {x}} _ {i} ^ {\star} \| _ {2} / \| \tilde {\boldsymbol {x}} _ {i} ^ {\star} \| _ {2}
+$$
+
+where  $X^{\star}$  is the reference ground truth provided by each dataset;  $\tilde{x}_i$  and  $\tilde{x}_i^{\star}$  are the  $i$ -th row of  $X_T$  and  $X^{\star}$ , respectively. We use the Hungarian algorithm (Kuhn, 1955) to align the rows of  $X_T$  returned by algorithms with the rows of  $X^{\star}$ .
+
+The benchmark algorithms are as follows: Simplex identification via split augmented Lagrangian (SISAL) (Bioucas-Dias, 2009); PRobabIistic SiMplex (PRISM) component
+
+analysis method (Wu et al., 2021); and deep structures, CN-NAEU $^{1}$  (Palsson et al., 2020) and  $\mathrm{MiSiCNet}^2$  (Rasti et al., 2022). We also consider the VAE method with the log-norm variational distribution proposed in (Li et al., 2024), termed VASCA. VASCA employs a linear decoder. To make the comparison fair, we extend the linear decoder to the nonlinear generative model (2). The dimensions of the nonlinear decoder are the same as those of DRD-VI.
+
+The experimental settings of DRD-VI, detailed in Appendix B.1, are consistent for all the tested hyperspectral images. Each algorithm is executed with 10 random initializations. Table 2 reports the overall MSE results, while the MSE contributions from each EM are provided in Appendix B.1.
+
+Fig. 1 presents the estimated abundance map corresponding to the hyperspectral image Jasper. The abundance map results for other images are provided in Appendix B.1.
+
+The results demonstrate that DRD-VI performs competitively, surpassing the state-of-the-art deep structures on some datasets and consistently outperforming VASCA.
+
+# 4.2. Low-Dimensional Representation Learning
+
+In this subsection, we consider DRD-VI with the non-negative bounded uniform prior in (4). We compare DRD-VI with other state-of-the-art MMF methods following prior work on MMF (e.g., (Trigeorgis et al., 2016)) that evaluates MMF methods by analyzing the learned low-dimensional representations. Specifically, given a data matrix  $\mathbf{X}_0$  with columns as i.i.d. samples, we apply clustering algorithms such as K-means to the low-dimensional representation matrix  $\mathbf{X}_T$  produced by MF and MMF methods. The clustering results are evaluated using three standard metrics: adjusted rand index (ARI) (Hubert & Arabie, 1985), accuracy (Acc), and normalized mutual information (NMI) (Cai et al., 2005). Higher values of the three metrics indicate better performance, with a maximum of 1. It is believed that higher clustering performance indicates better-learned low-dimensional representations.
+
+The benchmark algorithms are as follows: the one-layer semi-nonnegative matrix factorization (SNMF)3(Ding et al.,
+
+<sup>1</sup>Codes for CNNAEU.  
+$^{2}$ Codes for MiSiCNet.  
+3Codes for SNMF.
+
+![](images/51af970a7c4fa1a777a8c1c308154df64dac9fd484d2a114ec03ba3110ed4eba.jpg)  
+Figure 1. Estimated abundances for the hyperspectral image Jasper.
+
+![](images/bb10c2aebc73cf76d52d26871398ab95e35cbb4a82433e1bfd3498380c30110d.jpg)  
+Figure 2. Performance comparison of the MF and MMF methods across the six datasets. The latent space dimensions equal 16 for gray image datasets and  $16 \times 3$  for color ones.
+
+2008), layer-centric deep matrix factorization (LC-DMF) (De Handschutter & Gillis, 2023), deep matrix factorization  $(\mathrm{DMF})^4$  (Fan & Cheng, 2018), deep semi-nonnegative matrix factorization (Deep Semi-NMF) $^5$  (Trigeorgis et al., 2016), and deep autoencoder-like nonnegative matrix factorization (DANMF) $^6$  (Ye et al., 2018). In the experiments, the model dimensions for the MMF methods and the dimension of the base latent variable for all methods are identical. We test the methods on six datasets: a freely available version
+
+of CMU PIE (Sim et al., 2002), Caltech 101 Silhouettes $^{7}$ , Fashion MNIST(Xiao, 2017), GTSRB(Houben et al., 2013), DTD(Cimpoi et al., 2014), and Oxford-IIIT Pet (Parkhi et al., 2012). Descriptions of the datasets and details of the experimental setups are provided in Appendix B.2.
+
+The MF and MMF methods are applied to each dataset with 10 independent random initializations. For each trial, K-means clustering is performed on the learned representations with 50 independent random initializations, and the best clustering is recorded. The best results among the 10 trials are shown in Fig. 2. DRD-VI generally performs well and is
+
+comparable to, or in some cases outperforms, other state-of-the-art MMF methods. Due to space limitations, additional experimental results, including specific metric values and the effects of varying latent space dimensions, are provided in Appendix B.2.
+
+# 5. Conclusion
+
+This paper considered the application of diffusion model (DM)-based VI for MMF. We expanded on the idea of the existing variational DM, which assumes equal dimension with the latent variables, to propose a dimension-reducing variational DM for MMF. Each layer of the DM is associated with a layer of the MMF model, the latter of which can be seen as a shallow one-layer network (rather than a deep network in DMs for generative models). DMs are known to have the benefit of simple VI, and we turned that benefit to build a per-layer light-weight scheme for the VI of MMF. Experimental results showed that our proposed dimension-reducing DM-based VI scheme yields promising performance, suggesting the potential of variational DMs for MMF.
+
+# Acknowledgements
+
+This work was supported by a General Research Fund (GRF) of Hong Kong Research Grant Council (RGC) under Project ID CUHK 14203721.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
+# References
+
+Arora, S., Ge, R., and Moitra, A. Learning topic models—going beyond SVD. In Proc. Annu. IEEE Symp. Found. Comput. Sci., pp. 1-10. IEEE, 2012.  
+Bioucas-Dias, J. M. A variable splitting augmented Lagrangian approach to linear spectral unmixing. In Proc. Workshop Hyperspectral Image Signal Process. Evol. Remote Sens., pp. 1-4. IEEE, 2009.  
+Boyd, S. Convex optimization. Cambridge UP, 2004.  
+Cai, D., He, X., and Han, J. Document clustering using locality preserving indexing. IEEE Trans. Knowl. Data Eng., 17(12):1624-1637, 2005.  
+Cimpoi, M., Maji, S., Kokkinos, I., Mohamed, S., and Vedaldi, A. Describing textures in the wild. In Proc. IEEE Conf. Comput. Vis. Pattern Recognit., 2014.
+
+De Handschutter, P. and Gillis, N. A consistent and flexible framework for deep matrix factorizations. Pattern Recognit., 134:109102, 2023.  
+De Handschutter, P., Gillis, N., and Siebert, X. A survey on deep matrix factorizations. Comput. Sci. Rev., 42:100423, 2021.  
+Ding, C. H., Li, T., and Jordan, M. I. Convex and semi-nonnegative matrix factorizations. IEEE Trans. Pattern Anal. Mach. Intell., 32(1):45-55, 2008.  
+Fan, J. Multi-mode deep matrix and tensor factorization. In Proc. Int. Conf. Learn. Represent., 2021.  
+Fan, J. and Cheng, J. Matrix completion by deep matrix factorization. Neural Netw., 98:34-41, 2018.  
+Gillis, N. Nonnegative matrix factorization. SIAM, 2020.  
+Ho, J., Jain, A., and Abbeel, P. Denoising diffusion probabilistic models. Proc. Adv. Neural Inf. Process. Syst., 33: 6840-6851, 2020.  
+Houben, S., Stallkamp, J., Salmen, J., Schlipsing, M., and Igel, C. Detection of traffic signs in real-world images: The German Traffic Sign Detection Benchmark. In Proc. Int. Joint Conf. Neural Netw., pp. 1-8. IEEE, 2013.  
+Hubert, L. and Arabie, P. Comparing partitions. J. Classif., 2:193-218, 1985.  
+Hyvärinen, A., Khemakhem, I., and Morioka, H. Nonlinear independent component analysis for principled disentanglement in unsupervised deep learning. *Patterns*, 4(10), 2023.  
+Jing, B., Corso, G., Berlinghieri, R., and Jaakkola, T. Subspace diffusion generative models. In *Eur. Conf. Comput. Vis.*, pp. 274-289. Springer, 2022.  
+Khemakhem, I., Kingma, D., Monti, R., and Hyvarinen, A. Variational autoencoders and nonlinear ICA: A unifying framework. In Proc. Int. Conf. Artif. Intell. Stat., volume 108, pp. 2207-2217. PMLR, 26-28 Aug 2020.  
+Kingma, D., Salimans, T., Poole, B., and Ho, J. Variational diffusion models. Proc. Adv. Neural Inf. Process. Syst., 34:21696-21707, 2021.  
+Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. In Proc. Int. Conf. Learn. Represent. San Diego, CA, USA, 2015.  
+Kingma, D. P. and Welling, M. Auto-encoding variational Bayes. In Proc. Int. Conf. Learn. Represent., 2013.  
+Kuhn, H. W. The Hungarian method for the assignment problem. Naval research logistics quarterly, 2(1-2):83-97, 1955.
+
+Li, Y., Fu, X., and Ma, W.-K. Probabilistic simplex component analysis via variational auto-encoding. In Proc. IEEE Int. Conf. Acoust. Speech Signal Process., pp. 9671-9675. IEEE, 2024.  
+Luo, C. Understanding diffusion models: A unified perspective. arXiv preprint arXiv:2208.11970, 2022.  
+Ma, W.-K., Bioucas-Dias, J. M., Chan, T.-H., Gillis, N., Gader, P., Plaza, A. J., Ambikapathi, A., and Chi, C.-Y. A signal processing perspective on hyperspectral unmixing: Insights from remote sensing. IEEE Signal Process. Mag., 31(1):67-81, 2013.  
+Meyer, G. P. An alternative probabilistic interpretation of the Huber loss. In Proc. Conf. Comput. Vis. Pattern Recognit., pp. 5261-5269, 2021.  
+Palsson, B., Ulfarsson, M. O., and Sveinsson, J. R. Convolutional autoencoder for spectral-spatial hyperspectral unmixing. IEEE Trans. Geosci. Remote Sens., 59(1): 535-549, 2020.  
+Parkhi, O. M., Vedaldi, A., Zisserman, A., and Jawahar, C. Cats and dogs. In Proc. Conf. Comput. Vis. Pattern Recognit., pp. 3498-3505. IEEE, 2012.  
+Ranganath, R., Tang, L., Charlin, L., and Blei, D. Deep exponential families. In Artif. Intell. Stat., pp. 762-771. PMLR, 2015.  
+Ranganath, R., Tran, D., and Blei, D. Hierarchical variational models. In Proc. Int. Conf. Mach. Learn., pp. 324-333. PMLR, 2016.  
+Rasti, B., Koirala, B., Scheunders, P., and Chanussot, J. MiSiCNet: Minimum simplex convolutional network for deep hyperspectral unmixing. IEEE Trans. Geosci. Remote Sens., 60:1-15, 2022.  
+Rezende, D. J., Mohamed, S., and Wierstra, D. Stochastic backpropagation and approximate inference in deep generative models. In Proc. Int. Conf. Mach. Learn., volume 32, pp. 1278-1286. PMLR, 2014.  
+Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. High-resolution image synthesis with latent diffusion models. In Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., pp. 10684-10695, 2022.  
+Saul, L. K. A nonlinear matrix decomposition for mining the zeros of sparse data. J. Math. Data Sci., 4(2):431-463, 2022.  
+Sim, T., Baker, S., and Bsat, M. The CMU pose, illumination, and expression (PIE) database. In Proc. Int. Conf. Autom. Face Gesture Recognit., pp. 53-58. IEEE, 2002.
+
+Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N., and Ganguli, S. Deep unsupervised learning using nonequilibrium thermodynamics. In Proc. Int. Conf. Mach. Learn., pp. 2256-2265. PMLR, 2015.  
+Sonderby, C. K., Raiko, T., Maaløe, L., Sonderby, S. K., and Winther, O. Ladder variational autoencoders. Proc. Adv. Neural Inf. Process. Syst., 29, 2016.  
+Song, Y. and Ermon, S. Generative modeling by estimating gradients of the data distribution. In Proc. Adv. Neural Inf. Process. Syst., volume 32, 2019.  
+Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., and Poole, B. Score-based generative modeling through stochastic differential equations. In Proc. Int. Conf. Learn. Represent., 2021.  
+Trigeorgis, G., Bousmalis, K., Zafeiriou, S., and Schuller, B. W. A deep matrix factorization method for learning attribute representations. IEEE Trans. Pattern Anal. Mach. Intell., 39(3):417-429, 2016.  
+Vahdat, A. and Kautz, J. Nvae: A deep hierarchical variational autoencoder. Proc. Adv. Neural Inf. Process. Syst., 33:19667-19679, 2020.  
+Wang, W., Xu, Y., Feng, F., Lin, X., He, X., and Chua, T.-S. Diffusion recommender model. In Proc. Int. ACM SIGIR Conf. Res. Dev. Inf. Retr., pp. 832-841, 2023.  
+Wu, R., Ma, W.-K., Li, Y., So, A. M.-C., and Sidiropoulos, N. D. Probabilistic simplex component analysis. IEEE Trans. Signal Process., 70:582-599, 2021.  
+Xiao, H. Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017.  
+Xue, H.-J., Dai, X., Zhang, J., Huang, S., and Chen, J. Deep matrix factorization models for recommender systems. In Proc. Int. Joint Conf. Artif. Intell., volume 17, pp. 3203-3209. Melbourne, Australia, 2017.  
+Yang, J. and Leskovec, J. Overlapping community detection at scale: A nonnegative matrix factorization approach. In Proc. ACM Int. Conf. Web Search Data Min., pp. 587-596, 2013.  
+Ye, F., Chen, C., and Zheng, Z. Deep autoencoder-like nonnegative matrix factorization for community detection. In Proc. ACM Int. Conf. Inf. Knowl. Manag., pp. 1393-1402, 2018.  
+Zhang, H., Feng, R., Yang, Z., Huang, L., Liu, Y., Zhang, Y., Shen, Y., Zhao, D., Zhou, J., and Cheng, F. Dimensionality-varying diffusion process. In Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., pp. 14307-14316, 2023.
+
+Zhao, H., Ding, Z., and Fu, Y. Multi-view clustering via deep matrix factorization. In Proc. AAAI Conf. Artif. Intell., volume 31, 2017.
+
+# A. Derivations and Proofs
+
+# A.1. Derivation of (18)
+
+Using Bayes' rule, we have
+
+$$
+\begin{array}{l} \log q (\boldsymbol {x} _ {t - 1} | \boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}) = \log \frac {q (\boldsymbol {x} _ {t} | \boldsymbol {x} _ {t - 1} , \boldsymbol {x} _ {0}) q (\boldsymbol {x} _ {t - 1} | \boldsymbol {x} _ {0})}{q (\boldsymbol {x} _ {t} | \boldsymbol {x} _ {0})} \\ = \log \frac {\mathcal {N} \left(\boldsymbol {x} _ {t} ; \sqrt {a _ {t}} \boldsymbol {U} _ {t} ^ {\top} \boldsymbol {x} _ {t - 1} , (1 - a _ {t}) \boldsymbol {I}\right) \mathcal {N} \left(\boldsymbol {x} _ {t - 1} ; \sqrt {\bar {a} _ {t - 1}} \bar {U} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} , (1 - \bar {a} _ {t - 1}) \boldsymbol {I}\right)}{\mathcal {N} \left(\boldsymbol {x} _ {t} ; \sqrt {\bar {a} _ {t}} \bar {U} _ {t} ^ {\top} \boldsymbol {x} _ {0} , (1 - \bar {a} _ {t}) \boldsymbol {I}\right)} \tag {37} \\ = - \frac {1}{2} \left(\frac {a _ {t} \boldsymbol {x} _ {t - 1} ^ {\top} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top} \boldsymbol {x} _ {t - 1} - 2 \sqrt {a _ {t}} \boldsymbol {x} _ {t - 1} ^ {\top} \boldsymbol {U} _ {t} \boldsymbol {x} _ {t}}{1 - a _ {t}} + \frac {\boldsymbol {x} _ {t - 1} ^ {\top} \boldsymbol {x} _ {t - 1} - 2 \sqrt {\bar {a} _ {t - 1}} \boldsymbol {x} _ {t - 1} ^ {\top} \bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0}}{1 - \bar {a} _ {t - 1}}\right) + \text {c o n s t a n t} \\ = - \frac {1}{2} \boldsymbol {x} _ {t - 1} ^ {\top} \left(\frac {a _ {t}}{1 - a _ {t}} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top} + \frac {1}{1 - \bar {a} _ {t}} \boldsymbol {I}\right) \boldsymbol {x} _ {t - 1} + \boldsymbol {x} _ {t - 1} ^ {\top} \left(\frac {\sqrt {a _ {t}}}{1 - a _ {t}} \boldsymbol {U} _ {t} \boldsymbol {x} _ {t} + \frac {\sqrt {\bar {a} _ {t - 1}}}{1 - \bar {a} _ {t - 1}} \bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0}\right) + \text {c o n s t a n t}. \\ \end{array}
+$$
+
+We see that the above takes a quadratic form and it can be shown that  $q(\pmb{x}_{t-1}|\pmb{x}_t, \pmb{x}_0)$  is still a Gaussian distribution. The covariance matrix is given by
+
+$$
+\begin{array}{l} \boldsymbol {\Psi} _ {t, \phi} = \left(\frac {a _ {t}}{1 - a _ {t}} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top} + \frac {1}{1 - \bar {a} _ {t - 1}} \boldsymbol {I}\right) ^ {- 1} \\ = \left(1 - \bar {a} _ {t - 1}\right) \left(\boldsymbol {I} - \frac {a _ {t} \left(1 - \bar {a} _ {t - 1}\right)}{1 - a _ {t}} \boldsymbol {U} _ {t} \left(\boldsymbol {I} + \frac {a _ {t} \left(1 - \bar {a} _ {t - 1}\right)}{1 - a _ {t}} \boldsymbol {U} _ {t} ^ {\top} \boldsymbol {U} _ {t}\right) ^ {- 1} \boldsymbol {U} _ {t} ^ {\top}\right) \tag {38} \\ = (1 - \bar {a} _ {t - 1}) \left(\boldsymbol {I} - \frac {a _ {t} (1 - \bar {a} _ {t - 1})}{1 - a _ {t}} \left(1 + \frac {a _ {t} (1 - \bar {a} _ {t - 1})}{1 - a _ {t}}\right) ^ {- 1} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top}\right) \\ = \left(1 - \bar {a} _ {t - 1}\right) \left(\boldsymbol {I} - \frac {a _ {t} - \bar {a} _ {t}}{1 - \bar {a} _ {t}} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top}\right) \\ \end{array}
+$$
+
+where we have used the matrix inverse formula
+
+$$
+\left(\boldsymbol {I} + \boldsymbol {X} \boldsymbol {Y}\right) ^ {- 1} = \boldsymbol {I} - \boldsymbol {X} \left(\boldsymbol {I} + \boldsymbol {Y} \boldsymbol {X}\right) ^ {- 1} \boldsymbol {Y} \tag {39}
+$$
+
+for matrices  $\mathbf{X}$  and  $\mathbf{Y}$  with proper sizes in the second line, and the semi-orthogonality of  $U_{t}$  in the third line. The mean is given by
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {t, \phi} (\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}) = \boldsymbol {\Psi} _ {t, \phi} \frac {\sqrt {a _ {t}}}{1 - a _ {t}} \boldsymbol {U} _ {t} \boldsymbol {x} _ {t} + \boldsymbol {\Psi} _ {t, \phi} \frac {\sqrt {\bar {a} _ {t - 1}}}{1 - \bar {a} _ {t - 1}} \bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} \\ = \left(\frac {\sqrt {a _ {t}} (1 - \bar {a} _ {t - 1})}{1 - a _ {t}} - \frac {a _ {t} \sqrt {a _ {t}} (1 - \bar {a} _ {t - 1}) ^ {2}}{(1 - a _ {t}) (1 - \bar {a} _ {t})}\right) \boldsymbol {U} _ {t} \boldsymbol {x} _ {t} \tag {40} \\ + \sqrt {\bar {a} _ {t - 1}} \bar {U} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \frac {\sqrt {\bar {a} _ {t - 1}} a _ {t} (1 - \bar {a} _ {t - 1})}{1 - \bar {a} _ {t}} U _ {t} U _ {t} ^ {\top} \bar {U} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} \\ = \frac {\sqrt {a _ {t}} (1 - \bar {a} _ {t - 1})}{1 - \bar {a} _ {t}} \boldsymbol {U} _ {t} \boldsymbol {x} _ {t} + \sqrt {\bar {a} _ {t - 1}} \bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} + \frac {\sqrt {\bar {a} _ {t - 1}} (\bar {a} _ {t} - a _ {t})}{1 - \bar {a} _ {t}} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top} \bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0}. \\ \end{array}
+$$
+
+# A.2. Proof of Lemma 3.1 and Lemma 3.2
+
+First, we consider Lemma 3.1. It can be verified that
+
+$$
+\mathbb {E} _ {q (\boldsymbol {x})} [ \log p (\boldsymbol {x}) ] = - \frac {1}{2} \log | \boldsymbol {\Sigma} _ {1} | - \frac {1}{2} \operatorname {t r} \left(\boldsymbol {\Sigma} _ {1} ^ {- 1} \underbrace {\mathbb {E} _ {q (\boldsymbol {x})} \left[ (\boldsymbol {x} - \boldsymbol {\mu} _ {1}) (\boldsymbol {x} - \boldsymbol {\mu} _ {1}) ^ {\top} \right]}\right) - \frac {d}{2} \log (2 \pi), \tag {41}
+$$
+
+$$
+\mathbb {E} _ {q (\boldsymbol {x})} [ \log q (\boldsymbol {x}) ] = - \frac {1}{2} \log | \boldsymbol {\Sigma} _ {2} | - \frac {d}{2} (1 + \log (2 \pi)). \tag {42}
+$$
+
+The function  $f$  can be written as
+
+$$
+f \left(\boldsymbol {\Sigma} _ {1}\right) = \frac {1}{2} \left(\log \left| \boldsymbol {\Sigma} _ {1} \right| + \operatorname {t r} \left(\boldsymbol {\Sigma} _ {1} ^ {- 1} (\boldsymbol {R} + \boldsymbol {\Psi})\right)\right) + c \tag {43}
+$$
+
+where  $c = -\frac{1}{2} (\log |\Sigma_2| + d)$ . It is known that the solution to  $\min_{\Sigma_1 \in \mathbb{S}_{++}^d} f(\Sigma_1)$  is uniquely given by  $\boldsymbol{\Sigma}_1^\star = \boldsymbol{R} + \boldsymbol{\Psi}$  if  $\boldsymbol{R} + \boldsymbol{\Psi}$  is positive definite (PD). To put this into context, consider the change of variable  $Y = \Sigma_1^{-1}$ . The corresponding objective function
+
+$$
+f (\boldsymbol {Y}) = \frac {1}{2} \left(- \log | \boldsymbol {Y} | + \operatorname {t r} \left(\boldsymbol {Y} (\boldsymbol {R} + \boldsymbol {\Psi})\right)\right) + c
+$$
+
+is convex, and its gradient equals
+
+$$
+\nabla f (\boldsymbol {Y}) = \frac {1}{2} \left(- \boldsymbol {Y} ^ {- 1} + \boldsymbol {R} + \boldsymbol {\Psi}\right);
+$$
+
+see, e.g., (Boyd, 2004). It is easy to verify that
+
+$$
+\boldsymbol {R} = \boldsymbol {\Sigma} _ {2} + \underbrace {\left(\boldsymbol {\mu} _ {2} - \boldsymbol {\mu} _ {1}\right) \left(\boldsymbol {\mu} _ {2} - \boldsymbol {\mu} _ {1}\right) ^ {\top}} _ {= W}. \tag {44}
+$$
+
+Also, since  $\Sigma_{2}$  is PD,  $R$  is also PD. Putting the optimal solution  $\Sigma_{1}^{\star}$  into  $f$  gives
+
+$$
+f \left(\boldsymbol {\Sigma} ^ {\star}\right) = \frac {1}{2} \left(\log | \boldsymbol {R} + \boldsymbol {\Psi} | - \log | \boldsymbol {\Sigma} _ {2} |\right) = \frac {1}{2} \log | \boldsymbol {\Sigma} _ {2} ^ {- 1 / 2} (\boldsymbol {R} + \boldsymbol {\Psi}) \boldsymbol {\Sigma} _ {2} ^ {- 1 / 2} |, \tag {45}
+$$
+
+and applying (44) to (45) gives the desired result.
+
+Next, we consider Lemma 3.2. The proof is identical to the above, with the previous  $\pmb{R}$  being replaced by
+
+$$
+\boldsymbol {R} = \mathbb {E} _ {q (\boldsymbol {x})} \left[ (\boldsymbol {x} - \boldsymbol {\mu} (\boldsymbol {x})) (\boldsymbol {x} - \boldsymbol {\mu} (\boldsymbol {x})) ^ {\top} \right].
+$$
+
+# A.3. Derivation of (24)
+
+Recall that the covariance matrices of  $q_{\phi}(\pmb{x}_{t-1}|\pmb{x}_t, \pmb{x}_0)$  and  $p_{\theta}(\pmb{x}_{t-1}|\pmb{x}_t)$  are the same. The KL divergence can be written as
+
+$$
+\begin{array}{l} D _ {\mathrm {K L}} \left(q _ {\phi} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}\right) | | p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t}\right)\right) \\ = \frac {1}{2} \left\| f _ {t, \boldsymbol {\theta}} (\boldsymbol {x} _ {t}) - \boldsymbol {\mu} _ {t, \phi} (\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}) \right\| _ {\boldsymbol {\Psi} _ {t, \phi}} ^ {2} + \text {c o n s t a n t} \\ = \frac {1}{2} \left(f _ {t, \boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}\right) - \boldsymbol {\mu} _ {t, \phi} \left(\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}\right)\right) ^ {\top} \left(\frac {a _ {t}}{1 - a _ {t}} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top} + \frac {1}{1 - \bar {a} _ {t - 1}} \boldsymbol {I}\right) \left(f _ {t, \boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}\right) - \boldsymbol {\mu} _ {t, \phi} \left(\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}\right)\right) + \text {c o n s t a n t} \tag {46} \\ = \frac {1}{2} \left(\frac {a _ {t}}{1 - a _ {t}} \left\| \boldsymbol {U} _ {t} ^ {\top} \left(f _ {t, \boldsymbol {\theta}} (\boldsymbol {x} _ {t}) - \boldsymbol {\mu} _ {t, \phi} (\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0})\right) \right\| _ {2} ^ {2} + \frac {1}{1 - \bar {a} _ {t - 1}} \left\| f _ {t, \boldsymbol {\theta}} (\boldsymbol {x} _ {t}) - \boldsymbol {\mu} _ {t, \phi} (\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}) \right\| _ {2} ^ {2}\right) + \text {c o n s t a n t}. \\ \end{array}
+$$
+
+Based on (22), we can further write
+
+$$
+\begin{array}{l} \frac {a _ {t}}{1 - a _ {t}} \left\| \boldsymbol {U} _ {t} ^ {\top} \left(f _ {t, \boldsymbol {\theta}} (\boldsymbol {x} _ {t}) - \boldsymbol {\mu} _ {t, \phi} (\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0})\right) \right\| _ {2} ^ {2} = \frac {a _ {t} \bar {a} _ {t - 1}}{1 - a _ {t}} \left\| \left(\boldsymbol {U} _ {t} ^ {\top} - \frac {a _ {t} (1 - \bar {a} _ {t - 1})}{1 - \bar {a} _ {t}} \boldsymbol {U} _ {t} ^ {\top}\right) \left(\bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x})\right) \right\| _ {2} ^ {2} \tag {47} \\ = \frac {\bar {a} _ {t} (1 - a _ {t})}{(1 - \bar {a} _ {t}) ^ {2}} \left\| \boldsymbol {U} _ {t} ^ {\top} \left(\bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x})\right) \right\| _ {2} ^ {2}; \\ \end{array}
+$$
+
+and
+
+$$
+\begin{array}{l} \frac {1}{1 - \bar {a} _ {t - 1}} \| f _ {t, \boldsymbol {\theta}} (\boldsymbol {x} _ {t}) - \boldsymbol {\mu} _ {t, \phi} (\boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}) \| _ {2} ^ {2} \\ = \frac {\bar {a} _ {t - 1}}{1 - \bar {a} _ {t - 1}} \left\| \left(\boldsymbol {I} - \frac {a _ {t} (1 - \bar {a} _ {t - 1})}{1 - \bar {a} _ {t}} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top}\right) \left(\bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x})\right) \right\| _ {2} ^ {2} \tag {48} \\ = \frac {\bar {a} _ {t - 1}}{1 - \bar {a} _ {t - 1}} \left(\bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x})\right) ^ {\top} \left(\boldsymbol {I} - \frac {(a _ {t} - \bar {a} _ {t}) (2 - a _ {t} - \bar {a} _ {t})}{(1 - \bar {a} _ {t}) ^ {2}} \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top}\right) \left(\bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x})\right) \\ = \frac {\bar {a} _ {t - 1}}{1 - \bar {a} _ {t - 1}} \left\| \bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x}) \right\| _ {2} ^ {2} - \frac {\bar {a} _ {t} (2 - a _ {t} - \bar {a} _ {t})}{(1 - \bar {a} _ {t}) ^ {2}} \left\| \boldsymbol {U} _ {t} ^ {\top} \left(\bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x})\right) \right\| _ {2} ^ {2}. \\ \end{array}
+$$
+
+Adding (47) and (48) up gives
+
+$$
+\begin{array}{l} D _ {\mathrm {K L}} \left(q _ {\phi} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t}, \boldsymbol {x} _ {0}\right) \| p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t}\right)\right) \\ = \frac {\bar {a} _ {t - 1}}{2 \left(1 - \bar {a} _ {t - 1}\right)} \left\| \bar {U} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x}) \right\| _ {2} ^ {2} + \frac {1}{2} \left(\frac {\bar {a} _ {t} (1 - a _ {t})}{(1 - \bar {a} _ {t}) ^ {2}} - \frac {\bar {a} _ {t} (2 - a _ {t} - \bar {a} _ {t})}{(1 - \bar {a} _ {t}) ^ {2}}\right) \left\| \boldsymbol {U} _ {t} ^ {\top} \left(\bar {U} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x})\right) \right\| _ {2} ^ {2} \tag {49} \\ = \frac {\bar {a} _ {t - 1}}{2 (1 - \bar {a} _ {t - 1})} \left\| \bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x}) \right\| _ {2} ^ {2} + \frac {\bar {a} _ {t}}{2 (\bar {a} _ {t} - 1)} \left\| \boldsymbol {U} _ {t} ^ {\top} \left(\bar {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {x} _ {0} - \rho (\boldsymbol {A} _ {t} \boldsymbol {x})\right) \right\| _ {2} ^ {2}, \\ \end{array}
+$$
+
+which leads to the  $\tilde{r}_t$  in (24).
+
+# A.4. Derivation of (29)
+
+We can write  $\tilde{r}_T$  as
+
+$$
+\begin{array}{l} \check {r} _ {T} = \mathbb {E} _ {q _ {\gamma} \left(\boldsymbol {x} _ {T} \mid \boldsymbol {x} _ {T - 1}\right)} \left[ \log p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {T - 1} \mid \boldsymbol {x} _ {T}\right) \right] \\ = \mathbb {E} _ {q _ {\gamma} (\boldsymbol {x} _ {T} | \boldsymbol {x} _ {T - 1})} \left[ \log \mathcal {N} (\boldsymbol {x} _ {T - 1}; \boldsymbol {A} _ {T} \boldsymbol {x} _ {T}, \boldsymbol {\Sigma} _ {T}) \right] \\ = \mathbb {E} _ {q _ {\gamma} (\boldsymbol {x} _ {T} | \boldsymbol {x} _ {T - 1})} \left[ - \frac {1}{2} \| \boldsymbol {x} _ {T - 1} - \boldsymbol {A} _ {T} \boldsymbol {x} _ {T} \| _ {\boldsymbol {\Sigma} _ {T}} ^ {2} \right] - \frac {1}{2} \log (2 \pi | \boldsymbol {\Sigma} _ {T} |) \\ = \mathbb {E} _ {q _ {\gamma} (\boldsymbol {x} _ {T} | \boldsymbol {x} _ {T - 1})} \left[ - \frac {1}{2} \left(\| \boldsymbol {x} _ {T - 1} \| _ {\boldsymbol {\Sigma} _ {T}} ^ {2} + \| \boldsymbol {A} _ {T} \boldsymbol {x} _ {T} \| _ {\boldsymbol {\Sigma} _ {T}} ^ {2} - 2 \boldsymbol {x} _ {T - 1} ^ {\top} \boldsymbol {\Sigma} _ {T} ^ {- 1} \boldsymbol {A} _ {T} \boldsymbol {x} _ {T}\right) \right] - \frac {1}{2} \log (2 \pi | \boldsymbol {\Sigma} _ {T} |) \\ = - \frac {1}{2} \left(\| \boldsymbol {x} _ {T - 1} \| _ {\boldsymbol {\Sigma} _ {T}} ^ {2} + \operatorname {t r} \left(\boldsymbol {A} ^ {\top} \boldsymbol {\Sigma} _ {T} ^ {- 1} \boldsymbol {A} _ {T} \left(\boldsymbol {\mu} _ {T, \boldsymbol {\gamma}} (\boldsymbol {x} _ {T - 1}) \boldsymbol {\mu} _ {T, \boldsymbol {\gamma}} (\boldsymbol {x} _ {T - 1}) ^ {\top} + \boldsymbol {\Psi} _ {T, \boldsymbol {\gamma}} (\boldsymbol {x} _ {T - 1})\right)\right) - 2 \boldsymbol {x} _ {T - 1} ^ {\top} \boldsymbol {\Sigma} _ {T} ^ {- 1} \boldsymbol {A} _ {T} \boldsymbol {\mu} _ {T, \boldsymbol {\gamma}} (\boldsymbol {x} _ {T - 1})\right) \\ - \frac {1}{2} \log (2 \pi | \boldsymbol {\Sigma} _ {T} |) \\ = - \frac {1}{2} \left(\| \boldsymbol {x} _ {T - 1} \| _ {\boldsymbol {\Sigma} _ {T}} ^ {2} + \| \boldsymbol {A} _ {T} \boldsymbol {\mu} _ {T, \gamma} (\boldsymbol {x} _ {T - 1}) \| _ {\boldsymbol {\Sigma} _ {T}} ^ {2} - 2 \boldsymbol {x} _ {T - 1} ^ {\top} \boldsymbol {\Sigma} _ {T} ^ {- 1} \boldsymbol {A} _ {T} \boldsymbol {x} _ {T}\right) - \frac {1}{2} \log (2 \pi | \boldsymbol {\Sigma} _ {T} |) \\ - \frac {1}{2} \operatorname {t r} \left(\boldsymbol {\Sigma} _ {T} ^ {- 1} \boldsymbol {A} _ {T} \boldsymbol {\Psi} _ {T, \gamma} (\boldsymbol {x} _ {T - 1}) \boldsymbol {A} ^ {\top}\right) \\ = \log \mathcal {N} \left(\boldsymbol {x} _ {T - 1}; \boldsymbol {A} _ {T} \boldsymbol {\mu} _ {T, \gamma} \left(\boldsymbol {x} _ {T - 1}\right), \boldsymbol {\Sigma} _ {T}\right) - \frac {1}{2} \operatorname {t r} \left(\boldsymbol {\Sigma} _ {T} ^ {- 1} \boldsymbol {A} _ {T} \boldsymbol {\Psi} _ {T, \gamma} \left(\boldsymbol {x} _ {T - 1}\right) \boldsymbol {A} ^ {\top}\right), \tag {50} \\ \end{array}
+$$
+
+which gives the result in (29).
+
+# A.5. Examples of applicable prior distributions
+
+Table 3 presents examples of applicable distribution pairs. We clarify some notation. Given two vectors  $\mathbf{x}$  and  $\mathbf{y}$  of the same dimension,  $\mathbf{x} \odot \mathbf{y}$  and  $\mathbf{x}/\mathbf{y}$  denote the element-wise product and division, respectively. The symbols  $\Gamma(\cdot)$  and  $\psi(\cdot)$  denote the Gamma and Digamma functions, respectively. Given a vector  $\mathbf{x}$ ,  $|\mathbf{x}|$ ,  $\Gamma(\mathbf{x})$ ,  $\psi(\mathbf{x})$ ,  $\log(\mathbf{x})$ ,  $\exp(\mathbf{x})$ , and  $\mathbf{x}^2$  denote the element-wise operations of their scalar counterparts. The multivariate Beta distribution  $\mathcal{B}(\mathbf{x};\alpha,\beta)$  is defined as  $\prod_{i}\mathcal{B}(x_i;\alpha_i,\beta_i)$  where  $\mathcal{B}(x;\alpha,\beta)$  is the probability density function of the one-dimensional Beta distribution; the same applies to the Laplace distribution. The symbol  $c$  collects irrelevant constants.
+
+# B. Experimental Setups and Additional Results
+
+# B.1. Abundance Estimation in Hyperspectral Images
+
+The settings of DRD-VI are listed in Table 4. For VASCA, the model dimensions, learning rate, batch size, and the number of epochs are set the same. Table 5 provides a more comprehensive MSE results for each hyperspectral image including contributions from each EM. Figs. 3-5 show the estimated abundance maps. The results show that the deep structure MiSiCNet performs well on the Apex dataset. DRD-VI consistently outperforms VASCA and is very competitive in general.
+
+# B.2. Low-Dimensional Representation Learning
+
+The datasets used are summarized in Table 6. Table 7 presents the experiment settings of the DRD-VI methods which are the same for all the datasets. The model dimensions of all other MMF methods are set the same as those of DRD-VI. Tables 8,
+
+Table 3. Examples of distribution pairs.  
+
+<table><tr><td>DISTRIBUTION</td><td>SETTING</td><td>EXPRESSION</td></tr><tr><td>LAPLACE (MEYER, 2021)</td><td>p(xT)qγ(xT|xT-1)μT,γ(xT-1)ΨT,γ(xT)H(qγ(xT|xT-1))</td><td>L(xT,1,1)L(xT,μγ(xT-1),bγ(xT-1))μγDIAG(2bγ2)1T(b ⊙ exp (−|μγ|/b) + |μγ|−log b)+c</td></tr><tr><td>DIRICHLET</td><td>p(xT)qγ(xT|xT-1)μT,γ(xT-1)ΨT,γ(xT)H(qγ(xT|xT-1))</td><td>DIR(xT,1)DIR(xT,αγ(xT-1))αγ/(1Tαγ)(DIAG(μT,γ)−μT,γμT,γ)/(1+1Tαγ)(αγ−1T(ψ(αγ)−ψ(1Tαγ))−log1TΓ(αγ)/Γ(1Tαγ)+c</td></tr><tr><td rowspan="2">BETA</td><td rowspan="2">p(xT)qγ(xT|xT-1)μT,γ(xT-1)ΨT,γ(xT)H(qγ(xT|xT-1))</td><td>B(xT;1,1)B(xT;αγ(xT-1),βγ(xT-1))αγ/(αγ+βγ)DIAG((αγ ⊙ βγ)/(αγ+βγ)2/(αγ+βγ+1))</td></tr><tr><td>(αγ−1)Tψ(αγ)+(βγ−1)Tψ(βγ)-(αγ+βγ−21)Tψ(αγ+βγ)-1Tlog Γ(αγ)T(βγ)/Γ(αγ+βγ)+c</td></tr></table>
+
+Table 4. Experimental settings of DRD-VI in abundance estimation.  
+
+<table><tr><td>[d1,d2,...,dT]</td><td>λ</td><td>BATCH SIZE</td><td>EPOCH</td><td>LEARNING RATE</td></tr><tr><td>[64,32,16,8,dT]</td><td>10^5</td><td>ROUND(L/100)</td><td>500</td><td>0.001</td></tr></table>
+
+![](images/73c48aaf35569df8e107e424abfcad1261904b5e985a321bb8f14b10c0333277.jpg)  
+Figure 3. Estimated abundances for the hyperspectral image Samson.
+
+9, and 10 present the results of applying the MF and MMF methods to the datasets, with the latent space dimension being 16 for gray image datasets and  $16 \times 3$  for color ones. Note that the Deep Semi-NMF method uses a deterministic initialization.
+
+We also present the results of varying the latent space dimension, with the other settings kept the same as before. Fig. 6,
+
+Table 5. MSE results of abundance estimation (the best MSE among 10 independent trials/standard deviation).  
+
+<table><tr><td>DATASET</td><td>ENDMEMBER</td><td>SISAL</td><td>PRISM</td><td>CNNAEU</td><td>MiSICNET</td><td>VASCA</td><td>DRD-VI</td></tr><tr><td rowspan="4">SAMSON</td><td>SOIL</td><td>0.387/0.00</td><td>0.426/0.21</td><td>0.410/0.11</td><td>0.351/0.00</td><td>0.408/0.25</td><td>0.269/0.01</td></tr><tr><td>TREE</td><td>0.494/0.00</td><td>0.630/0.22</td><td>0.469/0.07</td><td>0.335/0.00</td><td>0.380/0.09</td><td>0.314/0.01</td></tr><tr><td>WATER</td><td>0.785/0.00</td><td>0.882/0.12</td><td>0.480/0.14</td><td>0.697/0.00</td><td>0.416/0.10</td><td>0.401/0.01</td></tr><tr><td>Avg. MSE</td><td>0.555/0.00</td><td>0.646/0.13</td><td>0.453/0.08</td><td>0.461/0.00</td><td>0.401/0.14</td><td>0.328/0.00</td></tr><tr><td rowspan="5">JASPER</td><td>TREE</td><td>0.432/0.00</td><td>0.498/0.14</td><td>0.450/0.13</td><td>0.190/0.00</td><td>0.469/0.05</td><td>0.275/0.01</td></tr><tr><td>WATER</td><td>0.457/0.00</td><td>0.142/0.28</td><td>0.266/0.07</td><td>0.214/0.00</td><td>0.575/0.02</td><td>0.220/0.08</td></tr><tr><td>SOIL</td><td>0.605/0.01</td><td>0.597/0.25</td><td>0.872/0.14</td><td>0.578/0.00</td><td>0.696/0.10</td><td>0.297/0.05</td></tr><tr><td>ROAD</td><td>0.569/0.00</td><td>0.570/0.15</td><td>1.079/0.16</td><td>1.093/0.00</td><td>0.794/0.22</td><td>0.428/0.23</td></tr><tr><td>Avg. MSE</td><td>0.516/0.00</td><td>0.452/0.15</td><td>0.667/0.08</td><td>0.518/0.00</td><td>0.634/0.05</td><td>0.305/0.09</td></tr><tr><td rowspan="5">APEX</td><td>ROAD</td><td>0.911/0.00</td><td>1.173/0.29</td><td>1.596/0.29</td><td>0.648/0.00</td><td>1.022/0.08</td><td>0.978/0.04</td></tr><tr><td>TREE</td><td>0.543/0.00</td><td>0.488/0.13</td><td>0.567/0.09</td><td>0.255/0.00</td><td>0.387/0.08</td><td>0.369/0.02</td></tr><tr><td>ROOF</td><td>0.664/0.00</td><td>0.427/0.15</td><td>0.615/0.09</td><td>0.299/0.00</td><td>0.619/0.05</td><td>0.389/0.02</td></tr><tr><td>WATER</td><td>0.853/0.01</td><td>0.491/0.22</td><td>0.471/0.15</td><td>0.452/0.00</td><td>0.503/0.09</td><td>0.698/0.01</td></tr><tr><td>Avg. MSE</td><td>0.743/0.00</td><td>0.645/0.15</td><td>0.812/0.06</td><td>0.413/0.00</td><td>0.633/0.05</td><td>0.609/0.02</td></tr><tr><td rowspan="7">URBAN</td><td>ASPHALT</td><td>0.816/0.02</td><td>0.923/0.17</td><td>0.626/0.10</td><td>0.539/0.00</td><td>0.720/0.07</td><td>0.617/0.05</td></tr><tr><td>GRASS</td><td>0.659/0.04</td><td>0.549/0.17</td><td>0.590/0.07</td><td>0.515/0.00</td><td>0.538/0.06</td><td>0.480/0.04</td></tr><tr><td>TREE</td><td>0.668/0.03</td><td>0.468/0.21</td><td>0.496/0.07</td><td>0.587/0.00</td><td>0.695/0.08</td><td>0.542/0.01</td></tr><tr><td>ROOF</td><td>1.004/0.11</td><td>1.024/0.45</td><td>0.581/0.09</td><td>0.638/0.00</td><td>0.895/0.09</td><td>0.566/0.01</td></tr><tr><td>METAL</td><td>0.907/0.05</td><td>1.016/0.21</td><td>0.977/0.71</td><td>2.823/0.00</td><td>0.979/0.10</td><td>1.337/0.14</td></tr><tr><td>DIRT</td><td>0.722/0.06</td><td>0.963/0.11</td><td>0.927/0.24</td><td>0.626/0.00</td><td>0.883/0.07</td><td>0.517/0.04</td></tr><tr><td>Avg. MSE</td><td>0.796/0.02</td><td>0.824/0.12</td><td>0.700/0.12</td><td>0.955/0.00</td><td>0.785/0.03</td><td>0.677/0.04</td></tr></table>
+
+![](images/301a7845216c516b73522b1672aef73f5a26204bca5915e4a53b61c6dcf6bed9.jpg)  
+Figure 4. Estimated abundances for the hyperspectral image Apex.
+
+7, and 8 present the results. DRD-VI is generally highly competitive compared to the other state-of-the-art methods and demonstrates a clearly better performance on the CMUPIE dataset.
+
+![](images/32fae920332a8637cca87997679a53750a1fc19c0bb894b5ce5d51f7a49b42a6.jpg)  
+Figure 5. Estimated abundances for the hyperspectral image Urban.
+
+Table 6. Image datasets used for low-dimensional representation learning.  
+
+<table><tr><td>DATASET</td><td>DIMENSION</td><td>SAMPLES</td><td>CLUSTERS</td><td>DATA TYPE</td></tr><tr><td>CMU PIE (FREE VERSION)</td><td>32 × 32</td><td>2856</td><td>68</td><td>FACES</td></tr><tr><td>FASHION-MNIST (TESTING SET)</td><td>28 × 28</td><td>10000</td><td>10</td><td>CLOTHES</td></tr><tr><td>CALTECH 101 SILHOUETTES</td><td>28 × 28</td><td>6407</td><td>101</td><td>OBJECT SILHOUETTES</td></tr><tr><td>GTSRB (TESTING SET)</td><td>32 × 32 × 3</td><td>12630</td><td>43</td><td>TRAFFIC SIGNS</td></tr><tr><td>OXFORD-IIIT PET (TESTING SET; RESIZED)</td><td>40 × 40 × 3</td><td>3680</td><td>37</td><td>PETS</td></tr><tr><td>DTD (TESTING SET; RESIZED)</td><td>50 × 50 × 3</td><td>1880</td><td>47</td><td>DESCRIBABLE TEXTURES</td></tr></table>
+
+Table 7. Low-dimensional representation learning experiment settings of DRD-VI.  
+
+<table><tr><td>Data Type</td><td>[d1,d2,···,dT]</td><td>λ</td><td>Batch Size</td><td>Epoch</td><td>Learning Rate</td></tr><tr><td>Gray Image</td><td>[256,128,64,32,16]</td><td>10^6</td><td rowspan="2">ROUND(L/100)</td><td rowspan="2">500</td><td rowspan="2">0.001</td></tr><tr><td>Color Image</td><td>[256,128,64,32,16] × 3</td><td>10^6</td></tr></table>
+
+Table 8. Accuracy (the best result among 10 independent trials/standard deviation).  
+
+<table><tr><td>METHODS</td><td>CMU PIE</td><td>GTSRB</td><td>Fashion-MNIST</td><td>OXFORD-IIIT PET</td><td>SILHOUETTES</td><td>DTD</td></tr><tr><td>SNMF</td><td>0.237/0.01</td><td>0.199/0.01</td><td>0.513/0.01</td><td>0.081/0.00</td><td>0.115/0.00</td><td>0.094/0.00</td></tr><tr><td>DANMF</td><td>0.223/0.01</td><td>0.186/0.00</td><td>0.497/0.00</td><td>0.079/0.00</td><td>0.118/0.00</td><td>0.098/0.00</td></tr><tr><td>LC-DMF</td><td>0.192/0.02</td><td>0.179/0.02</td><td>0.501/0.06</td><td>0.090/0.01</td><td>0.217/0.01</td><td>0.114/0.00</td></tr><tr><td>DMF</td><td>0.179/0.01</td><td>0.211/0.00</td><td>0.620/0.04</td><td>0.092/0.00</td><td>0.299/0.01</td><td>0.109/0.00</td></tr><tr><td>DEEP SEMI-NMF</td><td>0.395</td><td>0.279</td><td>0.452</td><td>0.081</td><td>0.304</td><td>0.085</td></tr><tr><td>DRD-VI</td><td>0.615/0.01</td><td>0.290/0.01</td><td>0.588/0.00</td><td>0.092/0.00</td><td>0.300/0.01</td><td>0.100/0.00</td></tr></table>
+
+Table 9. Normalized mutual information (the best result among 10 independent trials/standard deviation).  
+
+<table><tr><td>METHODS</td><td>CMU PIE</td><td>GTSRB</td><td>FASHION-MNIST</td><td>OXFORD-IIIT PET</td><td>SILHOUETTES</td><td>DTD</td></tr><tr><td>SNMF</td><td>0.428/0.01</td><td>0.288/0.01</td><td>0.457/0.01</td><td>0.076/0.00</td><td>0.320/0.00</td><td>0.157/0.00</td></tr><tr><td>DANMF</td><td>0.407/0.01</td><td>0.286/0.00</td><td>0.449/0.00</td><td>0.076/0.00</td><td>0.318/0.00</td><td>0.162/0.01</td></tr><tr><td>LC-DMF</td><td>0.421/0.03</td><td>0.281/0.03</td><td>0.414/0.06</td><td>0.109/0.01</td><td>0.441/0.02</td><td>0.201/0.00</td></tr><tr><td>DMF</td><td>0.417/0.02</td><td>0.332/0.01</td><td>0.549/0.02</td><td>0.108/0.00</td><td>0.535/0.00</td><td>0.163/0.01</td></tr><tr><td>DEEP SEMI-NMF</td><td>0.677</td><td>0.416</td><td>0.467</td><td>0.100</td><td>0.530</td><td>0.132</td></tr><tr><td>DRD-VI</td><td>0.792/0.01</td><td>0.412/0.01</td><td>0.601/0.00</td><td>0.103/0.00</td><td>0.532/0.00</td><td>0.160/0.00</td></tr></table>
+
+Table 10. Adjusted rand index (the best result among 10 independent trials/standard deviation).  
+
+<table><tr><td>METHODS</td><td>CMU PIE</td><td>GTSRB</td><td>FASHION-MNIST</td><td>OXFORD-IIIT PET</td><td>SILHOUETTES</td><td>DTD</td></tr><tr><td>SNMF</td><td>0.103/0.01</td><td>0.061/0.00</td><td>0.299/0.01</td><td>0.007/0.00</td><td>0.057/0.00</td><td>0.012/0.00</td></tr><tr><td>DANMF</td><td>0.092/0.00</td><td>0.058/0.00</td><td>0.272/0.00</td><td>0.007/0.00</td><td>0.059/0.00</td><td>0.014/0.00</td></tr><tr><td>LC-DMF</td><td>0.080/0.01</td><td>0.062/0.01</td><td>0.287/0.05</td><td>0.013/0.00</td><td>0.213/0.03</td><td>0.015/0.00</td></tr><tr><td>DMF</td><td>0.075/0.01</td><td>0.069/0.01</td><td>0.414/0.03</td><td>0.013/0.00</td><td>0.280/0.01</td><td>0.020/0.00</td></tr><tr><td>DEEP SEMI-NMF</td><td>0.304</td><td>0.151</td><td>0.296</td><td>0.011</td><td>0.263</td><td>0.008</td></tr><tr><td>DRD-VI</td><td>0.479/0.01</td><td>0.108/0.01</td><td>0.445/0.00</td><td>0.013/0.00</td><td>0.261/0.01</td><td>0.016/0.00</td></tr></table>
+
+# Acc performance across datasets
+
+![](images/303646111182d8cc91e72ebfd48ef25b3f671460a88a91deb9059fe0a3d7f762.jpg)  
+Figure 6. Accuracy vs. latent space dimensions.
+
+![](images/a5c26e884a8d5a60eb05048bf832d5cebb9e1e14f9a5e9af2b82f7d652429d5c.jpg)  
+NMI performance across datasets
+
+![](images/e38b94f43fae5452c45fd53256ad1ca83e9e555e24df86a872bd588ebf3e7472.jpg)
+
+![](images/d4e5cfbbdfb5522a12d47b7f1fcdeb6bf9563e2ee614f48ea8f7c25b95304cef.jpg)
+
+![](images/96b89295ca04cc2349d4d89775deefa4fd53cf6b810991a2c7e632a0bec91a86.jpg)  
+Figure 7. Normalized mutual information vs. latent space dimensions.
+
+![](images/300dd83f9e7c94f1fb8b6ebb2e3f40c70b37d3e9fab13ea42a562a48935635fc.jpg)
+
+![](images/ad7e86553def82c615cfc059c98c204b689ec13121a1f5917c7145a0f462d01d.jpg)
+
+![](images/d61416c7640e592e21d61b3dfe1d04c96cbecec0ae42a0d30fc3212a35f6a28f.jpg)  
+ARI performance across datasets
+
+![](images/7efde1d3a45716f9d61c69f06d831e33bc0370656dae8885089db8046fef3e71.jpg)
+
+![](images/4be57425390e2a025a88663ba483df50f86a8594b03054a6e88f059d698b5673.jpg)
+
+![](images/f3da8c0d49b24328c858e54f3c1886628f3353567c851391fe820a3dcf41df17.jpg)
+
+![](images/a7fa8d8f57b54138fc8d089645feef1d42598008743b5b40a09d2e7357a8f907.jpg)  
+Latent space dimension
+
+![](images/b9ae95e97f143f07090bbd4c7085eedbb1adbef04f7a541f491fd2c23b2cf4be.jpg)  
+Figure 8. Adjusted rand index vs. latent space dimensions
+
+![](images/dcce7fdba784ab9c6894d87aaa872293a199e007d652efd7d1dcecada0979953.jpg)

2025/Multilayer Matrix Factorization via Dimension-Reducing Diffusion Variational Inference/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:283fab59cdf73651b0d3fe381d50aa57f84ecbe60e32eca760b99504ba4cfd02
+size 2476742

2025/Multimodal Medical Code Tokenizer/932dc8e4-b1cb-4cd5-a163-6ce8fd1ac8f6_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8c007e57aafb7ecf8343fe293c39fdfab4ef85e4655c361053c3436d9a55b8b0
+size 1769066

2025/Multimodal Medical Code Tokenizer/full.md

@@ -0,0 +1,425 @@
+Xiaorui Su<sup>1</sup> Shvat Messica<sup>1</sup> Yepeng Huang<sup>1</sup> Ruth Johnson<sup>1</sup> Lukas Fesser<sup>1</sup> Shanghai Gao<sup>1</sup> Faryad Sahneh<sup>2</sup> Marinka Zitnik<sup>1</sup>
+
+# Abstract
+
+Foundation models trained on patient electronic health records (EHRs) require tokenizing medical data into sequences of discrete vocabulary items. Existing tokenizers treat medical codes from EHRs as isolated textual tokens. However, each medical code is defined by its textual description, its position in ontological hierarchies, and its relationships to other codes, such as disease co-occurrences and drug-treatment associations. Medical vocabularies contain more than 600,000 codes with critical information for clinical reasoning. We introduce MEDTOK, a multimodal medical code tokenizer that uses the text descriptions and relational context of codes. MEDTOK processes text using a language model encoder and encodes the relational structure with a graph encoder. It then quantizes both modalities into a unified token space, preserving modality-specific and cross-modality information. We integrate MEDTOK into five EHR models and evaluate it on operational and clinical tasks across in-patient and out-patient datasets, including outcome prediction, diagnosis classification, drug recommendation, and risk stratification. Swapping standard EHR tokenizers with MEDTOK improves AUPRC across all EHR models, by  $4.10\%$  on MIMIC-III,  $4.78\%$  on MIMIC-IV, and  $11.32\%$  on EHRShot, with the largest gains in drug recommendation. Beyond EHR modeling, we demonstrate using MEDTOK tokenizer with medical QA systems. Our results demonstrate the potential of MEDTOK as a unified tokenizer for medical codes, improving tokenization for medical foundation models.
+
+![](images/a426e6a8372197b07f03365055fe2aea0caefee0dd3585bf69c7da6d7f39a8df.jpg)  
+Figure 1. MEDTOK is a multimodal tokenizer of medical codes that combines text descriptions of codes with graph-based representations of dependencies between codes derived from clinical ontologies and standard medical terminologies. MEDTOK is a general-purpose tokenizer that can be integrated into any transformer-based model or system that requires tokenization.
+
+# 1. Introduction
+
+Electronic health records (EHRs) are the backbone of modern healthcare, capturing a person's health state with increasing precision across diverse modalities. Structured EHR data, encoded through standardized medical codes, support a wide range of applications, from personalized risk prediction (Goldstein et al., 2016; Yu et al., 2024b) and disease trajectory modeling (Jensen et al., 2017; Heumos et al., 2024) to emulation of clinical trials (Katsoulakis et al., 2024; Kraljevic et al., 2024). The cornerstone of structured EHRs is medical coding systems, which assign standardized alphanumeric codes to various aspects of patient health, including diseases, procedures, medications, and laboratory tests. These codes come from widely used terminologies such as ICD-9, ICD-10, SNOMED CT, CPT, and ATC, among others (Foley et al., 1992; Organization et al., 1988; Organization, 2004; Donnelly et al., 2006; Dotson, 2013; Miller & Britt, 1995). Although essential for interoperability, medical codes introduce challenges for models, particularly in the tokenization process, which transforms structured EHR data into token sequences that foundation
+
+models can process.
+
+Transformer-based models for structured EHRs (Poulain & Beheshti, 2024; Yang et al., 2023b; Jiang et al., 2023b; Renc et al., 2024) rely on tokenizers to map raw data into discrete vocabulary items. However, standard tokenization strategies inherited from general-purpose language models fail to capture the complexity of medical codes, leading to six key challenges: (1) Scalability of medical vocabularies - Medical coding systems contain over 600,000 unique codes, far exceeding standard tokenizer capacities. Treating each code as a separate token leads to inefficient vocabulary expansion, increasing memory demands and fragmenting rare codes (e.g., splitting "ICD9: 250.0" into arbitrary subwords). (2) Loss of hierarchical and relational structure - Many coding systems encode structured dependencies, such as ATC codes, which classify drugs based on pharmacological and chemical properties (Miller & Britt, 1995). Standard tokenizers, relying only on co-occurrence statistics, fail to capture hierarchical relationships, losing dependencies like disease co-occurrences and drug contraindications. (3) Redundancy across coding systems - Identical clinical concepts often appear under different codes across terminologies (e.g., ICD vs. SNOMED). Standard tokenization treats them as separate tokens, creating redundancy and complicating cross-system data integration. (4) Inefficiency in token storage - Expanding vocabulary sizes to accommodate medical codes results in bloated embedding tables that degrade computational efficiency, particularly for low-resource codes that appear infrequently but still require dedicated tokens. (5) Sparse and inconsistent usage - Many medical codes are rarely used or inconsistently documented, making it difficult for standard tokenizers to learn meaningful representations. Low-frequency codes suffer from poor embeddings, reducing performance on underrepresented conditions. (6) Lack of multimodal representations - Existing methods (Jiang et al., 2023b; Zhu et al., 2024; Xu et al., 2024) treat medical codes as isolated textual tokens, discarding graph-based relationships that encode essential links between diagnoses, treatments, and medications. A robust tokenizer must integrate both textual and relational information to fully represent medical codes.
+
+Several models attempt to enrich the representations of medical codes by incorporating external knowledge from Large Language Models (LLMs) (Jiang et al., 2023b; Zhu et al., 2024; Xu et al., 2024). Methods like GraphCare and RAM-EHR prompt LLMs to generate structured knowledge triplets of medical codes or summarize retrieved knowledge. Although effective in specific tasks, these approaches suffer from limited generalizability, a heavy reliance on knowledge generated by LLMs, and a lack of a unified framework for handling various medical coding systems. Despite advances in medical representation learning, a unified tokenizer that integrates textual and structured relational knowledge across
+
+coding systems remains an open challenge.
+
+Present work. We introduce MEDTOK (https://github.com/mims-harvard/MedTok), a multimodal medical code tokenizer that integrates textual descriptions and graph-based dependencies from biomedical ontologies (Figure 1). Unlike standard tokenization methods that treat medical codes as isolated textual tokens, MEDTOK captures both semantic meaning and structured relationships by encoding multiple modalities into a unified token space. MEDTOK operates in three stages. Multimodal encoding first extracts text embeddings from medical code descriptions and graph-based representations from biomedical knowledge graphs using separate encoders. Next, vector quantization maps both modalities into a shared token space, generating distinct text-informed and graph-informed token embeddings while preserving cross-modality interactions. Finally, optimization for expressivity ensures that token representations capture hierarchical relationships, semantic equivalence across different coding systems, and dependencies such as comorbidities and drug interactions.
+
+We integrate MEDTOK into five EHR models and evaluate it in clinical and operational tasks that span the inpatient (MIMIC-III, MIMIC-IV) and outpatient (EHRShot) settings. These tasks include disease prediction, operational outcome modeling, drug recommendation, patient risk stratification, and operational outcomes. Our key contributions are:
+
+- Multimodal tokenization of medical codes - MEDTOK tokenizer jointly encodes both textual descriptions and graph-based representations of medical codes, enabling richer and structured embeddings.  
+- Improved cross-system generalization - By incorporating ontological knowledge, MEDTOK bridges semantic gaps between different coding systems.  
+- Demonstrated performance gains – Replacing standard EHR tokenizers with MEDTOK improves AUPRC by  $4.10\%$  on MIMIC-III,  $4.78\%$  on MIMIC-IV, and  $11.30\%$  on EHRShot, with the largest gains in drug recommendation tasks. MEDTOK is a general purpose tokenizer that can be integrated into any transformer-based model or system that requires tokenization. Beyond EHR models, we demonstrate its applicability in medical question-answering systems, further highlighting the benefit of optimized tokenization of structured medical data.
+
+# 2. Related work
+
+Domain-specific tokenizers. Tokenizers tailored for specific domains have been employed to process various types of data, including language (Sennrich et al., 2016; Kudo & Richardson, 2018; Song et al., 2021; Wang et al., 2024b; Minixhofer et al., 2024), images (Zhou et al., 2022; Yu et al., 2022; 2024a; Zha et al., 2024), videos (Choudhury et al.,
+
+2024), graphs (Perozzi et al., 2024; Yang et al., 2024a), and molecular and material sciences (Fu et al., 2024; Tahmid et al., 2024; Qiao et al., 2024). While these tokenizers perform well within their respective domains, they are not directly applicable to medical codes, which contains specialized medical semantics. Medical codes reside in relation contexts and are accompanied by textual descriptions. Directly using the tokenizers for languages risks flattening the relationships among codes and failing to preserve the biomedical information. This will lead to fragmented tokenization of medical codes, resulting in loss of contextual information during encoding. Meanwhile, visual tokenizer typically focus on local pixel-level relationships, which are insufficient for capturing the complex semantics inherent in medical codes. Graph tokenizers are designed to encode structured information from graphs into a discrete token, then enabling LLMs to process relational and topological knowledge effectively. However, graph tokenizers may suffer from information loss when applied to graphs in other domains, making them less flexible and efficient for large, dynamic, and cross-domain graphs. In contrast, our MED-TOK tokenizer explicitly incorporates the relevant medical semantics by integrating textual descriptions with graph-based relational contexts.
+
+Vector-quantized tokenizers. Tokenization strategies often vary according to the problem domain and data modality where recent work has highlighted the benefits of discrete tokenization (Du et al., 2024). This process involves partitioning the input according to a finite set of tokens, often held in a codebook (this concept is independent of medical coding despite the similar name), and the quantization process involves learning a mapping from input data to the optimal set of tokens according to a pre-defined objective such as reconstruction loss (Van Den Oord et al., 2017). Recent work has highlighted the ability of vector quantized (VQ-based) tokenization to effectively compress semantic information (Gu et al., 2024). This approach is particularly successful for tokenizing inputs with an inherent semantic structure such as graphs (Yang et al., 2023a; Wang et al., 2024c), speech (Zeghidour et al., 2021; Baevski et al., 2019), and time (Yu et al., 2021) as well as complex tasks like recommendation retrieval (Wang et al., 2024d; Rajput et al., 2023; Sun et al., 2024) and image synthesis (Zhang et al., 2023; Yu et al., 2021). Another significant advantage to VQ-based tokenization is the natural integration of multiple modalities. By learning a shared latent space across modalities, each modality can jointly modeled using a common token vocabulary (Agarwal et al., 2025; Yu et al., 2023). TokenFlow leverages a dual-codebook design that allows for correlations across modalities through a dual encoder (Qu et al., 2024).
+
+Structured EHR, transformer-based, and foundation models. Structured EHR models leverage patient records
+
+to learn representations for clinical prediction and operational healthcare tasks. These models differ from medical LLMs (Singhal et al., 2025; Tu et al., 2024; Singhal et al., 2023), which are typically trained on free-text clinical notes (Jiang et al., 2023a) and biomedical literature rather than structured EHR data. BEHRT (Li et al., 2020) applies deep bidirectional learning to predict future medical events, encoding disease codes, age, and visit sequences using self-attention. TransformEHR (Yang et al., 2023b) adopts an encoder-decoder transformer with visit-level masking to pretrain on EHRs, enabling multitask prediction. GT-BEHRT (Poulain & Beheshti, 2024) models intra-visit dependencies as a graph, using a graph transformer to learn visit representations before processing patient-level sequences with a transformer encoder. Other models enhance EHR representations with external knowledge. GraphCare (Jiang et al., 2023b) integrates LLMs and biomedical knowledge graphs to construct patient-specific graphs processed via a Bi-attention Augmented Graph Neural Network. MulT-EHR (Chan et al., 2024) introduces multi-task heterogeneous graph learning with causal denoising to address data heterogeneity and confounding effects. ETHOS (Renc et al., 2024) tokenizes patient health timelines for transformer-based pretraining, achieving zero-shot performance. While these models focus on learning patient representations, MEDTOK serves a different role as a medical code tokenizer. It can be integrated into any structured EHR, transformer-based, or other foundation model, improving how medical codes are tokenized before being processed. Unlike these models, which rely on predefined tokenization schemes, MEDTOK optimizes the tokenization process itself.
+
+# 3. Approach
+
+MEDTOK is a multimodal medical tokenizer that leverages both text descriptions and relational contexts of medical codes. MEDTOK operates as a tokenization function  $f(\cdot)$  that maps a medical code  $m \in \mathcal{M}$  to a sequence of elements  $\mathcal{T}$  in the vocabulary  $\mathcal{V}$  with a size of  $N$  by leveraging both its textual definition  $\mathcal{D}(m)$  and a subgraph  $\mathcal{G}(m)$  extracted from a biomedical knowledge graph  $G$ . Here,  $\mathcal{M}$  is a set of 617,490 medical codes from eight medical coding systems: ICD-9, ICD-10-CM, ICD-10-PCS, SNOMED CT, ATC, NDC, CPT, and RxNORM.
+
+Problem definition. Our goal is to train a multimodal tokenizer  $f(\cdot)$  so that  $\mathcal{T} = f(\mathcal{D}(m),\mathcal{G}(m))$ , where  $\mathcal{T} = [t_1,t_2,\dots ,t_T]$  and  $t_i\in \mathcal{V},1\leq i\leq T$ . Then the generated  $\mathcal{T}$  for medical code  $m$  could be integrated to any EHR-based models  $h(\cdot)$  and LMs or LLMs  $p(\cdot)$  to perform predictive or generative tasks.
+
+Figure 2 illustrates the architecture of MEDTOK, which takes both the medical code description and contextual
+
+![](images/6cde95326b9a3dfcd8716b4eb2c147d0ca269c7cdfdb180bca2264d93163e92d.jpg)  
+Figure 2. MEDTOK is a general multimodal tokenizer of medical codes that can be integrated into any transformer-based model or a system that requires tokenization. 'X-attn' denotes a cross-attention module.
+
+knowledge from biomedical KGs as input. MEDTOK takes two steps, multimodal tokenization and token packing.
+
+# 3.1. Multimodal tokenization
+
+Given a medical code  $m$ , paired with its description  $t$  and its biological subgraph  $G$ , MEDTOK first adopts the text encoder, denoted as  $\mathrm{E}_t$  and the graph encoder, denoted as  $\mathrm{E}_g$ , to generate two embeddings: the text semantic embedding  $\mathbf{x}_t \in \mathbb{R}^{d_t}$  for  $t$  and the graph-level embedding  $\mathbf{x}_g \in \mathbb{R}^{d_g}$  for  $G$ . These embeddings are computed as  $\mathbf{x}_t = \mathrm{E}_t(t)$  and  $\mathbf{x}_g = \mathrm{E}_g(G)$ , where  $\mathrm{E}_t$  and  $\mathrm{E}_g$  represent the text and graph encoders, respectively.
+
+Modality-specific embeddings. MEDTOK then adopts two linear projectors:  $f_{t}:\mathbb{R}^{d_{t}}\to \mathbb{R}^{d}$  and  $f_{g}:\mathbb{R}^{d_{g}}\to \mathbb{R}^{d}$ , to generate modality-specific embeddings  $\mathbf{e}_t^s\in \mathbb{R}^d$  and  $\mathbf{e}_g^s\in \mathbb{R}^d$ , respectively, where  $\mathbf{e}_t^s = f_t(\mathbf{x}_t)$ ,  $\mathbf{e}_g^s = f_g(\mathbf{x}_g)$ , and  $d$  is the dimension of specific embeddings.
+
+Cross-modality embeddings. Moreover, MEDTOK incorporates a cross-attention module to derive cross-modality embeddings  $\mathbf{e}_t^c\in \mathbb{R}^d$  and  $\mathbf{e}_g^c\in \mathbb{R}^d$ , Specifically, the embedding  $\mathbf{e}_t^c$  is computed as:
+
+$$
+\mathbf {e} _ {t} ^ {c} = \operatorname {s o f t m a x} \left(\frac {\mathbf {W} _ {q} ^ {t} \mathbf {x} _ {t} \left(\mathbf {W} _ {k} ^ {g} \mathbf {x} _ {g}\right) ^ {T}}{\sqrt {d}}\right) \left(\mathbf {W} _ {v} ^ {g} \mathbf {x} _ {g}\right) \tag {1}
+$$
+
+where  $\mathbf{W}_q^t\in \mathbb{R}^{d\times d_t}$ $\mathbf{W}_k^g\in \mathbb{R}^{d\times d_g}$  , and  $\mathbf{W}_v^g\in \mathbb{R}^{d\times d_g}$  represent the query, key and value weight matrix. Similarly, the embedding  $\mathbf{e}_g^c$  is given by:
+
+$$
+\mathbf {e} _ {g} ^ {c} = \operatorname {s o f t m a x} \left(\frac {\mathbf {W} _ {q} ^ {g} \mathbf {x} _ {g} \left(\mathbf {W} _ {k} ^ {t} \mathbf {x} _ {t}\right) ^ {T}}{\sqrt {d}}\right) \left(\mathbf {W} _ {v} ^ {t} \mathbf {x} _ {t}\right) \tag {2}
+$$
+
+where  $\mathbf{W}_q^g\in \mathbb{R}^{d\times d_g},\mathbf{W}_k^t\in \mathbb{R}^{d\times d_t}$  and  $\mathbf{W}_v^t\in \mathbb{R}^{d\times d_t}$  represents the query, key, and value weight matrix.
+
+Tokenization. After generating modality-specific and cross-modality embeddings, for each embedding, MEDTOK quantizes the embedding into  $K$  tokens by querying a unified codebook  $\mathbf{C} \in \mathbb{R}^{N \times d}$ . The  $K$  tokens are identified by the top  $K$  nearest vectors in the codebook.
+
+In detail, for any modality-specific or cross-modality embedding  $\mathbf{e}_{\cdot}$ , its quantized tokens  $\mathcal{I}(\mathbf{e}_{\cdot})$  is formulated by:
+
+$$
+\mathcal {I} (\mathbf {e} _ {:}) = \operatorname {a r g m i n} _ {K} \left(\left\{\operatorname {d i s t} \left(\mathbf {e} _ {:}, \mathbf {C} _ {i}\right) \right\} _ {i = 1} ^ {N}\right) \tag {3}
+$$
+
+where  $\mathrm{dist}(:,:)$  denotes the Euclidean distance,  $|\mathcal{I}(\mathbf{e}_z)| = K$ , and  $\mathbf{C}_i = \mathbf{C}[i,:]$ . Then MEDTOK assigns a weight to each token  $k\in \mathcal{I}(\mathbf{e}_z)$  based on the distance between  $\mathbf{e}_z$  and its corresponding vector  $\mathbf{C}_k = \mathbf{C}[k,:]$ . These weighted tokens are then summed together to obtain the quantized vector for  $\mathbf{e}_z$ , denoted as  $\hat{\mathbf{e}}_z$ , which is given by:
+
+$$
+\hat {\mathbf {e}} _ {:} = \sum_ {k \in \mathcal {I} (\mathbf {e} _ {:})} - \operatorname {s o f t m a x} \left(\operatorname {d i s t} \left(\mathbf {e} _ {:}, \mathbf {C} _ {k}\right)\right) * \mathbf {C} _ {k} \tag {4}
+$$
+
+Following vector quantization conventions, we employ a straight-through gradient estimator:  $\mathbf{e}_{:} = \mathrm{sg}[\mathbf{e}_{:} - \hat{\mathbf{e}}_{:}] + \hat{\mathbf{e}}_{:}$  where  $\mathrm{sg}[\cdot ]$  denotes the stop-gradient operation. The codebook learning objective is  $\mathcal{L}(\mathbf{e}_{:,}\hat{\mathbf{e}}_{:}) = \| \mathrm{sg}[\hat{\mathbf{e}}_{:}] - \mathbf{e}_{:}\| _2^2 +$ $\alpha *\| \hat{\mathbf{e}}_{:} - \mathrm{sg}[\mathbf{e}_{:}]\|_{2}^{2},$  where  $\alpha$  is a hyperparameter.
+
+To preserve the distinctiveness of modality-specific and cross-modality embeddings, MEDTok divides the entire codebook into three regions: a text-specific region, a graph-specific region, and a shared region. The shared region
+
+includes the graph-shared and text-shared region and is shared with both two modalities. MEDTOK then queries distinct regions of the codebook to generate their respective tokens and quantized vectors, which are represented by:  $(\mathcal{I}(\mathbf{e}_t^s),\mathcal{I}(\mathbf{e}_g^s),\mathcal{I}(\mathbf{e}_t^c),\mathcal{I}(\mathbf{e}_g^c))$  and  $(\hat{\mathbf{e}}_t^s,\hat{\mathbf{e}}_g^s,\hat{\mathbf{e}}_t^c,\hat{\mathbf{e}}_g^c)$ . The final codebook objective is as follows:
+
+$$
+\mathcal {L} _ {C} = \mathcal {L} (\mathbf {e} _ {t} ^ {s}, \hat {\mathbf {e}} _ {t} ^ {s}) + \mathcal {L} (\mathbf {e} _ {g} ^ {s}, \hat {\mathbf {e}} _ {g} ^ {s}) + \mathcal {L} (\mathbf {e} _ {t} ^ {c}, \hat {\mathbf {e}} _ {t} ^ {c}) + \mathcal {L} (\mathbf {e} _ {g} ^ {c}, \hat {\mathbf {e}} _ {g} ^ {c}) (5)
+$$
+
+# 3.2. Token packing
+
+Unlike image-text paired data, where modalities have substantial overlap, the two modalities considered here (text and graph representations of medical codes) are more distinct but also highly complementary. Text provides the clinical definitions and describes each medical code in natural language. In contrast, the graph representation encodes domain-specific relationships such as disease co-occurrence, hierarchical groupings, and other medical ontologies. These relationships are not fully captured by text alone and introduce structured, expert-driven knowledge that is critical for many clinical and scientific applications. By considering both modalities, MEDTOK generates a representation that captures the shared information and also preserves information that is unique to each modality. MEDTOK achieves this by extracting modality-specific features during the tokenization process, rather than relying on standard approaches that may blend or discard valuable distinctions. This ensures that MEDTOK's tokens reflect both clinical language and the structured relationships present in the graph.
+
+We pack tokens  $(\mathcal{I}(\mathbf{e}_t^s),\mathcal{I}(\mathbf{e}_g^s),\mathcal{I}(\mathbf{e}_t^c),\mathcal{I}(\mathbf{e}_g^c))$  by optimizing both modality-shared and modality-specific information between these tokens and their corresponding quantized vectors. To capture modality-shared information, we focus on the tokens  $\mathcal{I}(\mathbf{e}_t^c)$  and  $\mathcal{I}(\mathbf{e}_g^c)$ . The objective uses Kullback-Leibler divergence to align their distance matrices,  $\mathrm{dist}(\mathbf{e}_t^c,\mathbf{C})$  and  $\mathrm{dist}(\mathbf{e}_g^c,\mathbf{C})$ , such that they follow a similar distribution:  $\mathcal{L}_{\mathrm{KL}} = D_{\mathrm{KL}}(\mathrm{softmax}(-\mathrm{dist}(\mathbf{e}_t^c,\mathbf{C}))\parallel$ $\mathrm{softmax}(-\mathrm{dist}(\mathbf{e}_g^c,\mathbf{C})))$ . Next, we optimize the quantized vectors  $\hat{\mathbf{e}}_t^c$  and  $\hat{\mathbf{e}}_g^c$  to maximize the information they carry about the other modality, while minimizing redundancy with their own modality. Specifically, we solve:
+
+$$
+\hat {\mathbf {e}} _ {t} ^ {c *} = \arg \max  _ {\hat {\mathbf {e}} _ {t} ^ {c}} \left(I \left(\hat {\mathbf {e}} _ {t} ^ {c}; \mathbf {e} _ {g} ^ {c}\right) - \beta \cdot I \left(\hat {\mathbf {e}} _ {t} ^ {c}; \mathbf {e} _ {t} ^ {c} \mid \mathbf {e} _ {g} ^ {c}\right)\right) \tag {6}
+$$
+
+$$
+\hat {\mathbf {e}} _ {g} ^ {c *} = \arg \max  _ {\hat {\mathbf {e}} _ {g} ^ {c}} \left(I \left(\hat {\mathbf {e}} _ {g} ^ {c}; \mathbf {e} _ {t} ^ {c}\right) - \beta \cdot I \left(\hat {\mathbf {e}} _ {g} ^ {c}; \mathbf {e} _ {g} ^ {c} \mid \mathbf {e} _ {t} ^ {c}\right)\right) \tag {7}
+$$
+
+For modality-specific information, MEDTOK optimizes tokens  $\mathcal{I}(\mathbf{e}_t^s)$  and  $\mathcal{I}(\mathbf{e}_g^s)$  by ensuring that the quantized vectors  $\hat{\mathbf{e}}_t^s$  and  $\hat{\mathbf{e}}_g^s$  retain maximal information about their respective modalities while minimizing shared information between modalities. The optimal solutions are given by:
+
+$$
+\hat {\mathbf {e}} _ {t} ^ {s *} = \arg \max _ {\hat {\mathbf {e}} _ {t} ^ {s}} \left(I (\hat {\mathbf {e}} _ {t} ^ {s}, \hat {\mathbf {e}} _ {g} ^ {c}; \mathbf {e} _ {t} ^ {s}) - \lambda \cdot I (\mathbf {e} _ {t} ^ {s}; \hat {\mathbf {e}} _ {g} ^ {c *})\right) \tag {8}
+$$
+
+$$
+\hat {\mathbf {e}} _ {g} ^ {s *} = \arg \max  _ {\hat {\mathbf {e}} _ {g} ^ {s}} \left(I \left(\hat {\mathbf {e}} _ {g} ^ {s}, \hat {\mathbf {e}} _ {t} ^ {c}; \mathbf {e} _ {g} ^ {s}\right) - \lambda \cdot I \left(\mathbf {e} _ {g} ^ {s}; \hat {\mathbf {e}} _ {g} ^ {c *}\right)\right) \tag {9}
+$$
+
+Based on the derivation of Wang et al., the loss for packing shared information across two modalities is formulated by:  $\mathcal{L}_{token}^{c} = \mathcal{L}_{\mathrm{InfoNCE}}(\hat{\mathbf{e}}_{t}^{c},\hat{\mathbf{e}}_{g}^{c}) + \mathcal{L}_{\mathrm{InfoNCE}}(\hat{\mathbf{e}}_{g}^{c},\hat{\mathbf{e}}_{t}^{c}) - 2\beta \mathbb{E}_{\mathbf{e}_{t}^{c},\mathbf{e}_{g}^{c}}(\mathbf{e}_{t}^{c}\cdot \mathbf{e}_{g}^{c})$  where  $\mathcal{L}_{\mathrm{InfoNCE}}$  denotes the InfoNCE loss. Additionally, the loss for packing specific information across two modalities is formulated by:  $\mathcal{L}_{token}^{s} = \mathcal{L}_{\mathrm{InfoNCE}}(\hat{\mathbf{e}}_{t}^{c},\tilde{\mathbf{e}}_{t}^{c}) + \lambda \mathcal{L}_{\mathrm{orthogonal}}(\hat{\mathbf{e}}_{t}^{c},\mathbf{e}_{t}^{c}) + \mathcal{L}_{\mathrm{InfoNCE}}(\hat{\mathbf{e}}_{g}^{c},\tilde{\mathbf{e}}_{g}^{c}) + \lambda \mathcal{L}_{\mathrm{orthogonal}}(\hat{\mathbf{e}}_{g}^{c},\mathbf{e}_{g}^{c}),$  where  $\mathcal{L}_{\mathrm{orthogonal}}$  denotes the orthogonal loss.
+
+We combine modality-shared and modality-specific losses into an overall token packing loss as:  $\mathcal{L}_{token} = \mathcal{L}_{token}^c + \mathcal{L}_{token}^s$ , where  $\beta$  and  $\lambda$  are hyperparameters set to be equal. This approach allows MEDTOK to leverage both modality-shared and modality-specific information.
+
+# 3.3. Training and inference for MEDTOK
+
+During the training stage, MEDTOK is trained by the sum of codebook loss  $\mathcal{L}_C$ , KL divergence loss  $\mathcal{L}_{KL}$ , token packing loss  $\mathcal{L}_{token}$ , where  $\mathcal{L} = \mathcal{L}_C + \mathcal{L}_{KL} + \mathcal{L}_{token}$ . After pre-training, MEDTOK can be integrated into any model or pipeline dealing with medical codes, providing unified medical tokens for downstream tasks.
+
+# 4. Experiments
+
+Medical coding systems. We collected a total of 617,490 medical codes from eight commonly used coding systems: ICD-9 (Organization et al., 1988), ICD-10-CM (Fung et al., 2020), ICD-10-PCS (Averill et al., 2001), SNOMED CT (Donnelly et al., 2006), ATC (Miller & Britt, 1995), NDC (Palmer, 2006), CPT (Dotson, 2013), and RxNORM (Nelson et al., 2011), as shown in Table 1. These codes cover various events, including procedures, diagnoses, and medications. Each code is paired with a textual description from official documents and a subgraph from PrimeKG (Chandak et al., 2023). Details are available in Appendix A.
+
+<table><tr><td>Code system</td><td>Count</td><td>Code system</td><td>Count</td></tr><tr><td>SNOMED</td><td>303,325</td><td>ICD9</td><td>18,365</td></tr><tr><td>ICD10-CM</td><td>81,184</td><td>CPT</td><td>10,602</td></tr><tr><td>RxNorm</td><td>81,151</td><td>ATC</td><td>6,659</td></tr><tr><td>ICD10-PCS</td><td>61,644</td><td>NDC</td><td>54,560</td></tr></table>
+
+Table 1. Summary of the dataset's code systems distribution.
+
+Patient EHR datasets. We used three publicly available EHR datasets: MIMIC-III (Johnson et al., 2016), MIMIC-IV (Johnson et al., 2024), and EHRShot (Wornow et al., 2023). MIMIC-III and MIMIC-IV are in-patient datasets with medical records for ICU patients, while EHRShot is a dataset containing longitudinal medical records that include both out-patients and ICU/ED patients. MIMIC datasets
+
+include NDC medications and ICD-9 / ICD-10 codes for diagnoses and procedures. In contrast, EHRShot mainly uses RxNorm codes for medications, SNOMED codes for diagnoses, and CPT, SNOMED, ICD-9, and ICD-10 codes for procedures. Table 2 summarizes the statistics of three EHR datasets.
+
+<table><tr><td></td><td>#patients</td><td>#visits</td><td>#visits/patient</td><td>#events/patient</td></tr><tr><td>MIMIC-III</td><td>35,707</td><td>44,399</td><td>1.24</td><td>51.14</td></tr><tr><td>MIMIC-IV</td><td>123,488</td><td>232,263</td><td>1.88</td><td>70.33</td></tr><tr><td>EHRShot</td><td>6,739</td><td>921,499</td><td>136.74</td><td>6182.17</td></tr></table>
+
+Table 2. Statistics of EHR datasets.
+
+Baselines. We consider two tokenizers and five EHR-based models based on EHR. The first type of tokenizer is text-based (e.g., bert-base-uncased (Devlin et al., 2018)), while the second is graph-based (e.g., VQGraph (Yang et al., 2024b)). Five EHR-based models are ETHOS (Renc et al., 2024), GT-BEHRT (Poulain & Beheshti, 2024), MulT-EHR (Chan et al., 2024), TransformEHR (Yang et al., 2023b), and BEHRT (Li et al., 2020). Details on implementation can be found in the Appendix B.
+
+Evaluation setup. We consider two evaluation setups:
+
+- In-patient evaluation: This setting combines the MED-TOK tokenizer with patient prediction models, using two in-patient datasets that include individuals admitted to a hospital. The evaluation encompasses five tasks: ① mortality prediction (MT), ② readmission prediction (RA), ③ length-of-stay prediction (LOS), ④ phenotype prediction (Pheno), and ⑤ drug recommendation (DrugRec). The first three tasks focus on predicting a patient's future health status using their historical medical records. Phenotype prediction involves the identification of the phenotype of a patient's disease based on their medical history. We identified 24 phenotypes for diseases in MIMIC-III and MIMIC-IV, as follows (Harutyunyan et al., 2019). Drug recommendation aims to suggest appropriate medications for a patient, considering their historical medical records and the diseases identified during their current visit. For drug recommendation, we focus on five specific drug candidates, including Vancomycin, Levofloxacin, Heparin Sodium, Metoprolol, and Atorvastatin, rather than considering the entire range of available medications. AUPRC is adopted to evaluate the model's performance on the above classification tasks.  
+- Out-patient evaluation: We evaluate MEDTOK together with patient prediction models on a dataset of patients who are not admitted to a hospital and consider two categories of tasks: ① Operational Outcomes (OO), and ② new diagnosis assignments (ND), following (Wornow et al., 2023). The OO includes MT, RA, and prolonged LOS. The new diagnosis assignments are used to predict the first diagnosis of a disease. Details are in Appendix C.
+
+# 4.1. MEDTok tokenizer with in-patient EHR models
+
+Table 3 presents the AUPRC values for each baseline and their integration with our MEDTOK for five tasks in two in-patient datasets. Compared to baselines that treat each medical code as an individual token, integrating our MEDTOK consistently improves performance across all five tasks, achieving an average improvement of  $3.29\%$  on MIMIC-III and  $2.67\%$  on MIMIC-IV. This improvement comes from more informative tokens generated by MEDTOK, which strengthen the EHR-based models. Among five tasks, MEDTOK demonstrates the most significant impact on drug recommendation tasks, highlighting the value of incorporating prior knowledge into our tokenizer.
+
+To further assess the effectiveness of MEDTOK, we compare it against two tokenization methods: the text-based BERT tokenizer and the graph-based VQGraph tokenizer. Figure 3 presents the performance of each tokenizer when integrated with a Transformer-based EHR model (TransformEHR) across five tasks on two in-patient datasets. MEDTOK consistently outperforms BERT and VQGraph in all tasks and datasets, demonstrating the superiority of its tokenization strategy.
+
+![](images/2af5661d96902e503ffbf8fe1d7d86ca9723829111dfd5b75fec2a7f42525cb4.jpg)  
+Figure 3. The AUPRC values of three types of tokenizers on inpatient and out-patient datasets, where OO means Operational Outcomes and ND means assignment of new diagnoses.
+
+# 4.2. MEDTok tokenizer with out-patient EHR models
+
+Table 4 presents the AUPRC values for each baseline and its integration with MEDTOK across two task types on the out-patient EHRShot dataset. The results reveal that our tokenizer has the most significant impact on mortality prediction in Operational Outcomes, achieving an average improvement of  $11.32\%$ . It also significantly improves the detection of new diagnoses of Hyperlipidemia, with an average improvement of  $6.00\%$ . As shown in Figure 3, a comparison of three types of tokenizers further demonstrates the effectiveness of MEDTOK in integrating both graph and textual modalities. Additionally, when comparing performance across two in-patient datasets, we observe that MEDTOK is particularly beneficial for longitudinal data.
+
+Multimodal Medical Code Tokenizer  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Task 1: MT+</td><td colspan="2">Task 2: RA(&lt;15 days)+</td><td colspan="2">Task 3: LOS*</td><td colspan="2">Task 4: Pheno°</td><td colspan="2">Task 5: DrugRec°</td></tr><tr><td>MIMIC-III AUPRC</td><td>MIMIC-IV AUPRC</td><td>MIMIC-III AUPRC</td><td>MIMIC-IV AUPRC</td><td>MIMIC-III AUPRC</td><td>MIMIC-IV AUPRC</td><td>MIMIC-III AUPRC</td><td>MIMIC-IV AUPRC</td><td>MIMIC-III AUPRC</td><td>MIMIC-IV AUPRC</td></tr><tr><td>ETHOS</td><td>0.617 (0.010)</td><td>0.282 (0.001)</td><td>0.421 (0.007)</td><td>0.648 (0.005)</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>0.104 (0.008)</td><td>0.131 (0.005)</td></tr><tr><td>+ MEDTOK</td><td>0.634 (0.020)</td><td>0.412 (0.030)</td><td>0.463 (0.017)</td><td>0.690 (0.007)</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>0.170 (0.014)</td><td>0.240 (0.012)</td></tr><tr><td>GT-BEHRT</td><td>0.160 (0.037)</td><td>0.028 (0.004)</td><td>0.612 (0.058)</td><td>0.586 (0.070)</td><td>0.230 (0.010)</td><td>0.103 (0.001)</td><td>0.423 (0.002)</td><td>0.493 (0.005)</td><td>0.715 (0.002)</td><td>0.736 (0.007)</td></tr><tr><td>+ MEDTOK</td><td>0.193 (0.046)</td><td>0.034 (0.005)</td><td>0.623 (0.052)</td><td>0.609 (0.064)</td><td>0.287 (0.039)</td><td>0.114 (0.003)</td><td>0.459 (0.028)</td><td>0.512 (0.006)</td><td>0.740 (0.004)</td><td>0.783 (0.010)</td></tr><tr><td>MulT-EHR</td><td>0.136 (0.021)</td><td>0.120 (0.003)</td><td>0.574 (0.008)</td><td>0.515 (0.007)</td><td>0.176 (0.018)</td><td>0.118 (0.032)</td><td>0.460 (0.012)</td><td>0.498 (0.001)</td><td>0.523 (0.008)</td><td>0.445 (0.027)</td></tr><tr><td>+ MEDTOK</td><td>0.156 (0.025)</td><td>0.141 (0.013)</td><td>0.585 (0.016)</td><td>0.565 (0.002)</td><td>0.198 (0.011)</td><td>0.136 (0.030)</td><td>0.480 (0.002)</td><td>0.504 (0.001)</td><td>0.571 (0.006)</td><td>0.465 (0.003)</td></tr><tr><td>TransformEHR</td><td>0.207 (0.012)</td><td>0.042 (0.012)</td><td>0.527 (0.030)</td><td>0.518 (0.012)</td><td>0.132 (0.021)</td><td>0.119 (0.001)</td><td>0.469 (0.022)</td><td>0.507 (0.007)</td><td>0.533 (0.030)</td><td>0.612 (0.046)</td></tr><tr><td>+ MEDTOK</td><td>0.246 (0.044)</td><td>0.058 (0.007)</td><td>0.568 (0.036)</td><td>0.525 (0.017)</td><td>0.159 (0.031)</td><td>0.121 (0.002)</td><td>0.513 (0.024)</td><td>0.518 (0.012)</td><td>0.580 (0.035)</td><td>0.661 (0.092)</td></tr><tr><td>BEHRT</td><td>0.163 (0.037)</td><td>0.028 (0.003)</td><td>0.529 (0.053)</td><td>0.514 (0.015)</td><td>0.232 (0.015)</td><td>0.112 (0.003)</td><td>0.587 (0.004)</td><td>0.493 (0.006)</td><td>0.539 (0.013)</td><td>0.778 (0.014)</td></tr><tr><td>+ MEDTOK</td><td>0.220 (0.025)</td><td>0.032 (0.006)</td><td>0.574 (0.040)</td><td>0.515 (0.005)</td><td>0.251 (0.030)</td><td>0.137 (0.004)</td><td>0.603 (0.008)</td><td>0.504 (0.006)</td><td>0.558 (0.006)</td><td>0.792 (0.007)</td></tr><tr><td>Improvement (%)</td><td>+3.32%</td><td>3.54%</td><td>3.00%</td><td>2.46%</td><td>3.13%</td><td>1.40%</td><td>2.90%</td><td>1.18%</td><td>4.10%</td><td>4.78%</td></tr></table>
+
++: imbalanced binary classification; *: multi-class classification, macro-averaged; o: multi-label classification; N/A indicates that the model was not configured for this task.
+
+Table 3. The results of MEDTOK with all baseline models across five tasks on two in-patient datasets.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">Task 1: Operational Outcomes (OO)</td><td colspan="4">Task 2: Assignment of New Diagnoses (ND)</td></tr><tr><td>Long LOS AUPRC</td><td>RA (&lt;15 days) AUPRC</td><td>MT AUPRC</td><td>Hypertension AUPRC</td><td>Hyperlipidemia AUPRC</td><td>Pancreatic Cancer AUPRC</td><td>Acute MI AUPRC</td></tr><tr><td>ETHOS</td><td>NA</td><td>0.079 (0.017)</td><td>0.102 (0.018)</td><td>0.166 (0.020)</td><td>0.155 (0.031)</td><td>0.056 (0.006)</td><td>0.093 (0.011)</td></tr><tr><td>+ MEDTOK</td><td>NA</td><td>0.128 (0.025)</td><td>0.339 (0.010)</td><td>0.175 (0.019)</td><td>0.163 (0.025)</td><td>0.056 (0.013)</td><td>0.104 (0.017)</td></tr><tr><td>GT-BEHRT</td><td>0.714 (0.021)</td><td>0.115 (0.012)</td><td>0.239 (0.012)</td><td>0.303 (0.018)</td><td>0.239 (0.007)</td><td>0.044 (0.008)</td><td>0.015 (0.008)</td></tr><tr><td>+ MEDTOK</td><td>0.739 (0.025)</td><td>0.154 (0.013)</td><td>0.444 (0.015)</td><td>0.360 (0.012)</td><td>0.441 (0.005)</td><td>0.074 (0.010)</td><td>0.031 (0.015)</td></tr><tr><td>MuIT-EHR</td><td>0.539 (0.025)</td><td>0.125 (0.014)</td><td>0.397 (0.016)</td><td>0.218 (0.005)</td><td>0.243 (0.005)</td><td>0.022 (0.008)</td><td>0.017 (0.003)</td></tr><tr><td>+ MEDTOK</td><td>0.571 (0.015)</td><td>0.188 (0.021)</td><td>0.444 (0.012)</td><td>0.226 (0.006)</td><td>0.254 (0.021)</td><td>0.037 (0.015)</td><td>0.028 (0.014)</td></tr><tr><td>TransformEHR</td><td>0.652 (0.023)</td><td>0.197 (0.016)</td><td>0.344 (0.030)</td><td>0.376 (0.018)</td><td>0.305 (0.021)</td><td>0.053 (0.006)</td><td>0.025 (0.006)</td></tr><tr><td>+ MEDTOK</td><td>0.675 (0.018)</td><td>0.243 (0.016)</td><td>0.379 (0.034)</td><td>0.413 (0.026)</td><td>0.333 (0.018)</td><td>0.082 (0.012)</td><td>0.052 (0.017)</td></tr><tr><td>BEHRT</td><td>0.582 (0.032)</td><td>0.332 (0.022)</td><td>0.389 (0.018)</td><td>0.233 (0.027)</td><td>0.251 (0.019)</td><td>0.036 (0.008)</td><td>0.013 (0.031)</td></tr><tr><td>+ MEDTOK</td><td>0.723 (0.028)</td><td>0.397 (0.036)</td><td>0.431 (0.017)</td><td>0.287 (0.018)</td><td>0.302 (0.015)</td><td>0.057 (0.012)</td><td>0.036 (0.015)</td></tr><tr><td>Improvement (%)</td><td>+5.52%</td><td>+5.24%</td><td>+11.32%</td><td>+3.30%</td><td>+6.00%</td><td>+1.90%</td><td>+1.76%</td></tr></table>
+
+Table 4. The results of MEDTOK with all baseline models across two tasks on the EHRShot dataset.
+
+# 4.3. Ablation studies
+
+To comprehensively understand the contributions of various components in MEDTOK, we conduct ablation studies on: (1) the effect of input modalities, and (2) the effect of shared and modality-specific optimization strategies. To ensure that our analysis is not confounded by architectural differences, we integrate MEDTOK with a standardized Transformer-based backbone (e.g., TransformEHR). This setup allows us to attribute performance differences directly to modalities and optimization strategies, rather than model architecture.
+
+Multimodal learning in MEDTOK. To evaluate the impact of the two modalities (text, graph) used in MEDTOK—medical code definitions and biological subgraphs derived from a biomedical knowledge graph—we assess its performance by removing the text and graph modalities separately. As shown in Figure 4, MEDTOK, when leveraging both modalities, achieves the best performance across all tasks on three datasets. By comparing the performance of MEDTOK without the graph modality and MEDTOK without the text modality, we observe that both modalities contribute significantly to EHR-based prediction tasks. The graph modality benefits drug recommendation and new disease detection tasks, while the text modality proves essential for readmission prediction on MIMIC-III and operational
+
+outcomes in EHRShot. These findings emphasize the importance of incorporating the underlying information linked to medical codes.
+
+Effects of modality-shared and modality-specific information on MEDTOK. MEDTOK is built on two modalities, and we have analyzed the impact of each modality in Figure 4. The results show that the model performs best when using both modalities. Additionally, MEDTOK optimizes tokens by maximizing shared information between modalities while preserving modality-specific information. To assess the contribution of each loss component, we conducted ablation studies by retraining MEDTOK with different loss function combinations. We then apply the pretrained MEDTOK to all tasks across datasets to obtain its performances. The results (average AUPRC across all tasks) in Table 5 demonstrate that both shared and specific information optimization enhance performance, with the full optimization achieving the best results across all datasets. The vector quantization loss  $\mathcal{L}_C$  is the basic loss for tokenization. By optimizing both shared and specific information across two modalities, the performances are improved by  $3.9\%$ ,  $5.7\%$ , and  $9.1\%$  on three datasets, respectively. The experimental results also show that shared and specific information contribute more to out-patient datasets.
+
+![](images/6db7c51b8b823e006242004e3afff200ecebaeebb2477bc4e20e0a26412a6ef8.jpg)  
+Figure 4. The AUPRC values obtained by removing the text and graph modalities across all tasks on two in-patient datasets and one out-patient dataset.
+
+![](images/6443e051fa7ea23c889fe230cd1ada3b4d30154df807ae8a8c640ac12fdead53.jpg)
+
+![](images/aa9002d338cee82eaaa2c829da0de28e707c37745f1b2ac5d836bec08506059c.jpg)
+
+<table><tr><td>Optimization</td><td>MIMIC III</td><td>MIMIC IV</td><td>EHRShot</td></tr><tr><td>LC</td><td>0.373</td><td>0.387</td><td>0.287</td></tr><tr><td>LC + Lc token + KL</td><td>0.379</td><td>0.409</td><td>0.314</td></tr><tr><td>LC + Ls token</td><td>0.382</td><td>0.402</td><td>0.366</td></tr><tr><td>L</td><td>0.412</td><td>0.444</td><td>0.378</td></tr></table>
+
+$\mathcal{L} = \mathcal{L}_C + \mathcal{L}_{token}^c$  and  $\mathcal{L}_{KL} + \mathcal{L}_{token}^{s}$
+
+Table 5. The averaged AUPRC across all tasks with different loss function combinations.
+
+# 4.4. Hyperparameter sensitivity analysis
+
+We next investigate the impact of hyperparameters on MedTok's performance across all datasets. MedTok is trained with three key hyperparameters: two weighting coefficients,  $\lambda$  and  $\beta$ , which control the contribution of shared and specific information loss components, and a codebook size parameter,  $N$ , which determines the number of discrete tokens available for representation.
+
+Effects of loss weight  $\lambda$  and  $\beta$ . To ensure that MedTok treats shared and specific information equally, we set  $\lambda = \beta$ , where  $\lambda$  and  $\beta$  are the weighting coefficients for the shared and specific information loss terms, respectively. The average AUPRC scores across all tasks are presented in below Figure 5A. The results demonstrate the influence of hyperparameter choices on model performance across the three datasets. Based on our findings, we recommend setting  $\lambda = \beta = 0.1$  for in-patient settings and  $\lambda = \beta = 0.01$  for out-patient settings.
+
+Effects of codebook size  $N$ . We further evaluate the impact of the codebook size on the performance of MEDTOK by training it with varying sizes and assessing its effectiveness across three distinct datasets integrated with TransformEHR. Figure 5B presents the results for various codebook sizes across all tasks on the three datasets. The performance trends observed on MIMIC-III and MIMIC-IV are quite consistent, demonstrating a clear pattern where increasing the codebook size enhances the model's performance. Specifically, the most stable average performance is achieved when
+
+![](images/31e02b7d5fc9ac563f80c4f954af929f71c9f011b35b302189e7bc5b61c1f968.jpg)  
+Figure 5. A, The AUPRC values of MEDTOK with different weighting coefficients  $\lambda, \beta$ ; B, The AUPRC values of MEDTOK with different codebook size  $N$ .
+
+![](images/cc864cc9a7a6bf9aaa8d3099bb95698b6643a70dbc923cfd2d482bdc31a94149.jpg)
+
+the codebook size is set to  $N = 12,000$ , indicating that this size strikes an optimal balance between sufficient coverage of the medical vocabulary and avoiding overfitting.
+
+In contrast, when analyzing the performance on EHRShot, a dataset consisting of patients with longer visit histories than those in MIMIC-III and MIMIC-IV, we observe that MED-TOK benefits from a larger codebook size. For EHRShot, the highest average performance is achieved when the codebook size is increased to  $N = 24,000$ . This suggests that for datasets with more extensive patient visit histories, a larger codebook may be more effective in capturing the underlying complexity of the medical information, thus improving the model's predictive capabilities.
+
+# 4.5. Using MEDTok tokenizer for medical QA
+
+MEDTOK demonstrates strong performance in EHR-based tasks, as shown in Tables 3-4. To further assess its capabilities, we explore its effectiveness in a generation task, specifically multiple-choice medical question answering (MedicalQA), where the goal is to select the correct answer to a given clinical question (Singhal et al., 2023).
+
+We evaluate whether MEDTok enhances few-shot learning in MedicalQA by integrating its tokenized representations with three LLMs (LLaMA3.1-8B (Dubey et al., 2024), Qwen2.5-7B (Hui et al., 2024), MMedLM (Qiu et al., 2024)). MEDTok-generated tokens are used as prefix to
+
+![](images/9b4bbd78d9490dfb74df623987cc00c4a13cbec2ceea85e3bffa5bd3e9ac75f0.jpg)  
+Figure 6. The accuracy of LLMs vs. MEDTOK+LLMs on three medical QA datasets.
+
+![](images/5249e21991d3eae5656f885f5a90fa5a55c5073abd596dbb74b5311d538364a0.jpg)
+
+![](images/080646286600d87cf1c6217ebb16de6cc332fa3b87157d54c776d66c13a2274c.jpg)
+
+kens, which provide structured medical context before the main input, allowing the LLM to incorporate medical codes.
+
+For this evaluation, we use three medical QA datasets, including MMLU (Hendrycks et al., 2021), MedDDx (Su et al., 2024), and AfrimedQA (Olatunji et al., 2024). In addition, MedDDx dataset contains questions at three difficulty levels: Basic, Intermediate, and Expert.
+
+The process consists of three steps: (1) Disease code mapping – Extract disease mentions from each question and retrieve their corresponding medical codes. (2) Tokenization via MEDTOK – Convert medical codes into structured tokens using MEDTOK. (3) Prefix token fine-tuning – Finetune LLMs using MEDTOK tokens as prefix inputs before the question text. We fine-tune the model on the MedMCQA dataset (Pal et al., 2022), and then evaluate the fine-tuned model on MMLU, MedDDx, and AfrimedQA.
+
+The results in Figure 6 show an accuracy improvement across all datasets compared with LLMs, suggesting that MEDTOK can enhance medical QA when used as a structured representation of medical codes and integrated with an LLM through prefix tuning.
+
+# 4.6. Interpretability of MEDTOK
+
+To this end, we selected a subset of patients classified as high risk for Hyperlipidemia by MEDTOK+TransformEHR, where these patients have no records of Hyperlipidemia before. We then counted the tokens assigned to these patients and identified those appearing more than 100 times, as shown in Figure 7. We then mapped these frequent tokens to medical codes, with the most overlapping codes being Rosuvastatin  $5\mathrm{mg}$  Oral Tablet (RxNorm 2669980), Burn of skin (SNOMED CT 147087003), Type 2 diabetes mellitus without complication (disorder) (SNOMED CT 373555004), and Hyperlipidemia (SNOMED CT 285605009). They are closely related to Hyperlipidemia. Rosuvastatin corresponds to medications commonly prescribed for lipid disorders. The other three medical codes represent clinical diagnoses or findings associated with hyperlipidemia-related cardiovascular risk. It suggests that MEDTOK captures key medical concepts related to Hyperlipidemia, supporting its predictive capability.
+
+![](images/cde3d41749d8c3b73cc242545f378bc976ab46704db6ea10868311c6d8b7bf86.jpg)  
+Figure 7. Top 100 frequent token IDs appearing in patients at high risk of Hyperlipidemia.
+
+# 5. Conclusion
+
+Tokenizing medical codes is a critical yet challenging step in developing foundation models for EHRs. Existing tokenizers treat medical codes as isolated textual units, failing to capture their structured relationships within large-scale medical ontologies. With more than 600,000 codes that span multiple terminologies, standard tokenization methods struggle to scale while preserving the rich semantic and relational context necessary for downstream clinical and operational tasks. We introduced MEDTOK, a multimodal tokenizer of medical codes that integrates textual definitions and relational ontologies of medical codes to create a unified token representation. MEDTOK applies vector quantization to encode both modalities in a structured token space, preserving cross-modality relationships. We integrated MEDTOK with five EHR models, evaluating its impact across inpatient (MIMIC-III, MIMIC-IV) and outpatient (EHRShot) settings, as well as in fine-tuning a medical question-answering system. Our results establish MEDTOK as a generalizable tokenizer for medical codes, shedding light on how optimizing the tokenization process can benefit medical foundation models.
+
+# Impact statement
+
+This work presents a tokenizer for medical codes designed to assist other models in better encoding semantic knowledge. While our tokenizer complements other models, it does not directly raise ethical concerns. Instead, it enhances the trustworthiness of these models by providing text and graph-based information as references for tokenization. In future work, we will explore ways to better integrate our tokenizer with other models to further improve both their performance and trustworthiness.
+
+# Acknowledgement
+
+We sincerely appreciate the valuable discussions with Intae Moon and Zhenglun Kong, whose insights and feedback have contributed to the development of this work. We gratefully acknowledge the support of NIH R01-HD108794, NSF CAREER 2339524, US DoD FA8702-15-D-0001, ARPA-H BDF program, awards from Chan Zuckerberg Initiative, Bill & Melinda Gates Foundation INV-079038, Amazon Faculty Research, Google Research Scholar Program, AstraZeneca Research, Roche Alliance with Distinguished Scientists, Sanofi iDEA-iTECH, Pfizer Research, John and Virginia Kaneb Fellowship at Harvard Medical School, Biswas Computational Biology Initiative in partnership with the Milken Institute, Harvard Medical School Dean's Innovation Fund for the Use of Artificial Intelligence, Harvard Data Science Initiative, and Kempner Institute for the Study of Natural and Artificial Intelligence at Harvard University. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funders.
+
+# Conflict of interest
+
+Faryad Sahneh is a Sanofi employee and may hold shares and/or stock options in the company. Other authors declare no conflict of interest.
+
+# References
+
+Agarwal, N., Ali, A., Bala, M., Balaji, Y., Barker, E., Cai, T., Chattopadhyay, P., Chen, Y., Cui, Y., Ding, Y., et al. Cosmos world foundation model platform for physical ai. arXiv preprint arXiv:2501.03575, 2025.  
+Averill, R. F., Mullin, R. L., Steinbeck, B. A., Goldfield, N. I., and Grant, T. M. Development of the icd-10 procedure coding system (icd-10-pcs). Topics in health information management, 21(3):54-88, 2001.  
+Baevski, A., Schneider, S., and Auli, M. vq-wav2vec: Self-supervised learning of discrete speech representations. arXiv preprint arXiv:1910.05453, 2019.
+
+Balsa-Canto, E., Brush, M. H., Carbon, S., et al. Mondo: Unifying diseases for the world, by the world. *Nucleic Acids Research*, 51(D1):D1182–D1190, 2023. doi: 10.1093/nar/gkac1062.  
+Bodenreider, O. The unified medical language system (umls): integrating biomedical terminology. *Nucleic Acids Research*, 32(Database issue):D267–D270, 2004. doi: 10.1093/nar/gkh061.  
+Chan, T. H., Yin, G., Bae, K., and Yu, L. Multi-task heterogeneous graph learning on electronic health records. Neural Networks, 180:106644, 2024.  
+Chandak, P., Huang, K., and Zitnik, M. Building a knowledge graph to enable precision medicine. Scientific Data, 10(1):67, 2023.  
+Choudhury, R., Zhu, G., Liu, S., Niinuma, K., Kitani, K. M., and Jeni, L. Don't look twice: Faster video transformers with run-length tokenization. (arXiv:2411.05222), November 2024. doi: 10.48550/arXiv.2411.05222. URL http://arxiv.org/abs/2411.05222. arXiv:2411.05222.  
+Devlin, J., Chang, M., Lee, K., and Toutanova, K. BERT: pre-training of deep bidirectional transformers for language understanding. CoRR, abs/1810.04805, 2018. URL http://arxiv.org/abs/1810.04805.  
+Donnelly, K. et al. Snomed-ct: The advanced terminology and coding system for ehealth. Studies in health technology and informatics, 121:279, 2006.  
+Dotson, P. Cpt® codes: what are they, why are they necessary, and how are they developed?, 2013.  
+Du, T., Wang, Y., and Wang, Y. On the role of discrete tokenization in visual representation learning. arXiv preprint arXiv:2407.09087, 2024.  
+Dubey, A., Jauhri, A., Pandey, A., Kadian, A., Al-Dahle, A., Letman, A., Mathur, A., Schelten, A., Yang, A., and Fan, A. The llama 3 herd of models, 2024. URL https://arxiv.org/abs/2407.21783.  
+Foley, S. M., Daley, J., Hughes, J., Fisher, E. S., Heeren, T., et al. Comorbidities, complications, and coding bias: does the number of diagnosis codes matter in predicting in-hospital mortality? Jama, 267(16):2197-2203, 1992.  
+Fu, C., Li, X., Olson, B., Ji, H., and Ji, S. Fragment and geometry aware tokenization of molecules for structure-based drug design using language models. (arXiv:2408.09730), August 2024. doi: 10.48550/ arXiv.2408.09730. URL http://arxiv.org/abs/2408.09730. arXiv:2408.09730.
+
+Fung, K. W., Xu, J., and Bodenreider, O. The new international classification of diseases 11th edition: a comparative analysis with icd-10 and icd-10-cm. Journal of the American Medical Informatics Association, 27(5): 738-746, 2020.  
+Goldstein, B. A., Navar, A. M., and Pencina, M. J. Risk prediction with electronic health records: The importance of model validation and clinical context. JAMA Cardiology, 1(9):976, December 2016. ISSN 2380-6583. doi: 10.1001/jamacardio.2016.3826.  
+Gu, Y., Wang, X., Ge, Y., Shan, Y., and Shou, M. Z. Rethinking the objectives of vector-quantized tokenizers for image synthesis. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 7631-7640, 2024.  
+Harutyunyan, H., Khachatrian, H., Kale, D. C., Ver Steeg, G., and Galstyan, A. Multitask learning and benchmarking with clinical time series data. Scientific data, 6(1):96, 2019.  
+Hendrycks, D., Burns, C., Basart, S., Zou, A., Mazeika, M., Song, D., and Steinhardt, J. Measuring massive multitask language understanding. Proceedings of the International Conference on Learning Representations (ICLR), 2021.  
+Heumos, L., Ehmele, P., Treis, T., Upmeier Zu Belzen, J., Roellin, E., May, L., Namsaraeva, A., Horlava, N., Shitov, V. A., Zhang, X., Zappia, L., Knoll, R., Lang, N. J., Hetzel, L., Virshup, I., Sikkema, L., Curion, F., Eils, R., Schiller, H. B., Hilgendorff, A., and Theis, F. J. An open-source framework for end-to-end analysis of electronic health record data. Nature Medicine, 30(11):3369-3380, November 2024. ISSN 1078-8956, 1546-170X. doi: 10.1038/s41591-024-03214-0.  
+Hui, B., Yang, J., Cui, Z., Yang, J., Liu, D., Zhang, L., Liu, T., Zhang, J., Yu, B., Lu, K., et al. Qwen2. 5-coder technical report. arXiv preprint arXiv:2409.12186, 2024.  
+Jensen, K., Soguero-Ruiz, C., Oyvind Mikalsen, K., Lindsetmo, R.-O., Kouskoumvekaki, I., Girolami, M., Olav Skrovseth, S., and Augestad, K. M. Analysis of free text in electronic health records for identification of cancer patient trajectories. Scientific Reports, 7(1):46226, April 2017. ISSN 2045-2322. doi: 10.1038/srep46226.  
+Jiang, L. Y., Liu, X. C., Nejatian, N. P., Nasir-Moin, M., Wang, D., Abidin, A., Eaton, K., Riina, H. A., Laufer, I., Punjabi, P., et al. Health system-scale language models are all-purpose prediction engines. Nature, 619(7969): 357-362, 2023a.  
+Jiang, P., Xiao, C., Cross, A., and Sun, J. Graphcare: Enhancing healthcare predictions with personalized knowledge graphs. arXiv preprint arXiv:2305.12788, 2023b.
+
+Johnson, A., Pollard, T., and Mark, R. MIMIC-III Clinical Database (version 1.4), 2016. URL https://doi.org/10.13026/C2XW26. Available at: https://doi.org/10.13026/C2XW26.  
+Johnson, A., Bulgarelli, L., Pollard, T., Gow, B., Moody, B., Horng, S., Celi, L. A., and Mark, R. MIMIC-IV (version 3.1), 2024. URL https://doi.org/10.13026/kpb9-mt58.  
+Katsoulakis, E., Wang, Q., Wu, H., Shahriyari, L., Fletcher, R., Liu, J., Achenie, L., Liu, H., Jackson, P., Xiao, Y., Syeda-Mahmood, T., Tuli, R., and Deng, J. Digital twins for health: a scoping review. npj Digital Medicine, 7 (1):77, March 2024. ISSN 2398-6352. doi: 10.1038/s41746-024-01073-0.  
+Kraljevic, Z., Bean, D., Shek, A., Bendayan, R., Hemingway, H., Yeung, J. A., Deng, A., Balston, A., Ross, J., Idowu, E., Teo, J. T., and Dobson, R. J. B. Foresight—a generative pretrained transformer for modelling of patient timelines using electronic health records: a retrospective modelling study. The Lancet Digital Health, 6(4):e281-e290, April 2024. ISSN 25897500. doi: 10.1016/S2589-7500(24)00025-6.  
+Kudo, T. and Richardson, J. Sentencepiece: A simple and language independent subword tokenizer and detokenizer for neural text processing. (arXiv:1808.06226), August 2018. doi: 10.48550/arXiv.1808.06226. URL http://arxiv.org/abs/1808.06226.arXiv:1808.06226.  
+Li, Y., Rao, S., Solares, J. R. A., Hassaine, A., Ramakrishnan, R., Canoy, D., Zhu, Y., Rahimi, K., and Salim-Khorshidi, G. Behrt: transformer for electronic health records. Scientific reports, 10(1):7155, 2020.  
+Miller, G. and Britt, H. A new drug classification for computer systems: the atc extension code. International journal of bio-medical computing, 40(2):121-124, 1995.  
+Minixhofer, B., Ponti, E. M., and Vulic, I. Zero-shot tokenizer transfer. (arXiv:2405.07883), May 2024. doi: 10.48550/arXiv.2405.07883. URL http://arxiv.org/abs/2405.07883.arXiv:2405.07883.  
+Nelson, S. J., Zeng, K., Kilbourne, J., Powell, T., and Moore, R. Normalized names for clinical drugs: Rxnorm at 6 years. Journal of the American Medical Informatics Association, 18(4):441-448, 2011.  
+Olatunji, T., Nimo, C., Owodunni, A., Abdullahi, T., Ayodele, E., Sanni, M., Aka, C., Omofoye, F., Yuehghoh, F., Faniran, T., et al. Afrimed-qa: A pan-african, multispecialty, medical question-answering benchmark dataset. arXiv preprint arXiv:2411.15640, 2024.  
+Organization, W. H. International Statistical Classification of Diseases and related health problems: Alphabetical index, volume 3. World Health Organization, 2004.
+
+Organization, W. H. et al. International classification of diseases—ninth revision (icd-9). Weekly Epidemiological Record=Relevé épidemiologique hebdomadaire, 63(45): 343–344, 1988.  
+Pal, A., Umapathi, L. K., and Sankarasubbu, M. Medmcqa: A large-scale multi-subject multi-choice dataset for medical domain question answering. In Conference on health, inference, and learning, pp. 248-260. PMLR, 2022.  
+Palmer, E. What is ndc? Pension reform: Issues and prospects for non-financial defined contribution (NDC) schemes, pp. 17-34, 2006.  
+Perozzi, B., Fatemi, B., Zelle, D., Tsitsulin, A., Kazemi, M., Al-Rfou, R., and Halcrow, J. Let your graph do the talking: Encoding structured data for llms. (arXiv:2402.05862), February 2024. doi: 10.48550/ arXiv.2402.05862. URL http://arxiv.org/abs/2402.05862. arXiv:2402.05862.  
+Poulain, R. and Beheshti, R. Graph transformers on ehrs: Better representation improves downstream performance. In The Twelfth International Conference on Learning Representations, 2024.  
+Qiao, L., Ye, P., Ren, Y., Bai, W., Liang, C., Ma, X., Dong, N., and Ouyang, W. Model decides how to tokenize: Adaptive dna sequence tokenization with mxdna. (arXiv:2412.13716), December 2024. doi: 10.48550/ arXiv.2412.13716. URL http://arxiv.org/abs/2412.13716. arXiv:2412.13716.  
+Qiu, P., Wu, C., Zhang, X., Lin, W., Wang, H., Zhang, Y., Wang, Y., and Xie, W. Towards building multilingual language model for medicine. Nature Communications, 15(1):8384, 2024.  
+Qu, L., Zhang, H., Liu, Y., Wang, X., Jiang, Y., Gao, Y., Ye, H., Du, D. K., Yuan, Z., and Wu, X. Tokenflow: Unified image tokenizer for multimodal understanding and generation. arXiv preprint arXiv:2412.03069, 2024.  
+Rajput, S., Mehta, N., Singh, A., Hulikal Keshavan, R., Vu, T., Heldt, L., Hong, L., Tay, Y., Tran, V., Samost, J., et al. Recommender systems with generative retrieval. Advances in Neural Information Processing Systems, 36: 10299-10315, 2023.  
+Renc, P., Jia, Y., Samir, A. E., Was, J., Li, Q., Bates, D. W., and Sitek, A. Zero shot health trajectory prediction using transformer. NPJ Digital Medicine, 7(1):256, 2024.  
+Sennrich, R., Haddow, B., and Birch, A. Neural machine translation of rare words with subword units. June 2016. doi: 10.48550/arXiv.1508.07909. URL http://arxiv.org/abs/1508.07909.
+
+Singhal, K., Azizi, S., Tu, T., Mahdavi, S. S., Wei, J., Chung, H. W., Scales, N., Tanwani, A., Cole-Lewis, H., Pfohl, S., et al. Large language models encode clinical knowledge. Nature, 620(7972):172-180, 2023.  
+Singhal, K., Tu, T., Gottweis, J., Sayres, R., Wulczyn, E., Amin, M., Hou, L., Clark, K., Pfohl, S. R., Cole-Lewis, H., et al. Toward expert-level medical question answering with large language models. Nature Medicine, pp. 1-8, 2025.  
+Song, X., Salcianu, A., Song, Y., Dopson, D., and Zhou, D. Fast wordpiece tokenization. (arXiv:2012.15524), October 2021. doi: 10.48550/arXiv.2012.15524. URL http://arxiv.org/abs/2012.15524.arXiv:2012.15524.  
+Su, X., Wang, Y., Gao, S., Liu, X., Giunchiglia, V., Clevert, D.-A., and Zitnik, M. Knowledge graph based agent for complex, knowledge-intensive qa in medicine. arXiv preprint arXiv:2410.04660, 2024.  
+Sun, W., Yan, L., Chen, Z., Wang, S., Zhu, H., Ren, P., Chen, Z., Yin, D., Rijke, M., and Ren, Z. Learning to tokenize for generative retrieval. Advances in Neural Information Processing Systems, 36, 2024.  
+Tahmid, M. T., Shahgir, H. S., Mahbub, S., Dong, Y., and Bayzid, M. S. Birna-bert allows efficient rna language modeling with adaptive tokenization. November 2024. doi: 10.1101/2024.07.02.601703. URL https://www.biorxiv.org/content/10.1101/2024.07.02.601703v3.  
+Tu, T., Azizi, S., Driess, D., Schaekermann, M., Amin, M., Chang, P.-C., Carroll, A., Lau, C., Tanno, R., Ktena, I., et al. Towards generalist biomedical ai. NEJM AI, 1(3): AIoa2300138, 2024.  
+Van Den Oord, A., Vinyals, O., et al. Neural discrete representation learning. Advances in neural information processing systems, 30, 2017.  
+Wang, C., Gupta, S., Zhang, X., Tonekaboni, S., Jegelka, S., Jaakkola, T., and Uhler, C. An information criterion for controlled disentanglement of multimodal data. arXiv preprint arXiv:2410.23996, 2024a.  
+Wang, D., Li, Y., Jiang, J., Ding, Z., Jiang, G., Liang, J., and Yang, D. Tokenization matters! degrading large language models through challenging their tokenization. (arXiv:2405.17067), May 2024b. doi: 10.48550/ arXiv.2405.17067. URL http://arxiv.org/abs/2405.17067. arXiv:2405.17067.  
+Wang, L., Hassani, K., Zhang, S., Fu, D., Yuan, B., Cong, W., Hua, Z., Wu, H., Yao, N., and Long, B. Learning graph quantized tokenizers for transformers. arXiv preprint arXiv:2410.13798, 2024c.
+
+Wang, W., Bao, H., Lin, X., Zhang, J., Li, Y., Feng, F., Ng, S.-K., and Chua, T.-S. Learnable item tokenization for generative recommendation. In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management, pp. 2400-2409, 2024d.  
+Wishart, D. S., Feunang, Y. D., Guo, A. C., Lo, E. J., Marcu, A., Grant, J. R., Sajed, T., Johnson, D., Li, C., Sayeeda, Z., et al. Drugbank 5.0: a major update to the drugbank database for 2018. *Nucleic Acids Research*, 46(D1): D1074–D1082, 2018. doi: 10.1093/nar/gkx1037.  
+Wornow, M., Thapa, R., Steinberg, E., Fries, J. A., and Shah, N. EHRSHOT: An EHR benchmark for few-shot evaluation of foundation models. In Thirty-seventh Conference on Neural Information Processing Systems Datasets and Benchmarks Track, 2023.  
+Xu, R., Shi, W., Yu, Y., Zhuang, Y., Jin, B., Wang, M. D., Ho, J., and Yang, C. RAM-EHR: Retrieval augmentation meets clinical predictions on electronic health records. In Ku, L.-W., Martins, A., and Srikumar, V. (eds.), Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pp. 754-765, Bangkok, Thailand, August 2024. Association for Computational Linguistics. doi: 10.18653/v1/2024.acl-short.68.  
+Yang, L., Tian, Y., Xu, M., Liu, Z., Hong, S., Qu, W., Zhang, W., Cui, B., Zhang, M., and Leskovec, J. Vqgraph: Rethinking graph representation space for bridging gnns and mlps. arXiv preprint arXiv:2308.02117, 2023a.  
+Yang, L., Tian, Y., Xu, M., Liu, Z., Hong, S., Qu, W., Zhang, W., Cui, B., Zhang, M., and Leskovec, J. Vqgraph: Rethinking graph representation space for bridging gnns and mlps. (arXiv:2308.02117), March 2024a. doi: 10. 48550/arXiv.2308.02117. URL http://arxiv.org/abs/2308. 02117.arXiv:2308.02117.  
+Yang, L., Tian, Y., Xu, M., Liu, Z., Hong, S., Qu, W., Zhang, W., CUI, B., Zhang, M., and Leskovec, J. Vqgraph: Rethinking graph representation space for bridging gnns and mlps. In International Conference on Learning Representations, 2024b.  
+Yang, Z., Mitra, A., Liu, W., Berlowitz, D., and Yu, H. Transformehr: transformer-based encoder-decoder generative model to enhance prediction of disease outcomes using electronic health records. Nature communications, 14(1):7857, 2023b.  
+Yu, J., Li, X., Koh, J. Y., Zhang, H., Pang, R., Qin, J., Ku, A., Xu, Y., Baldridge, J., and Wu, Y. Vector-quantized image modeling with improved vqgan. arXiv preprint arXiv:2110.04627, 2021.
+
+Yu, J., Li, X., Koh, J. Y., Zhang, H., Pang, R., Qin, J., Ku, A., Xu, Y., Baldridge, J., and Wu, Y. Vector-quantized image modeling with improved vqgan. (arXiv:2110.04627), June 2022. doi: 10.48550/arXiv.2110.04627. URL http://arxiv.org/abs/2110.04627. arXiv:2110.04627.  
+Yu, L., Lezama, J., Gundavarapu, N. B., Versari, L., Sohn, K., Minnen, D., Cheng, Y., Birodkar, V., Gupta, A., Gu, X., et al. Language model beats diffusion-tokenizer is key to visual generation. arXiv preprint arXiv:2310.05737, 2023.  
+Yu, L., Lezama, J., Gundavarapu, N. B., Versari, L., Sohn, K., Minnen, D., Cheng, Y., Birodkar, V., Gupta, A., Gu, X., Hauptmann, A. G., Gong, B., Yang, M.-H., Essa, I., Ross, D. A., and Jiang, L. Language model beats diffusion - tokenizer is key to visual generation. (arXiv:2310.05737), March 2024a. doi: 10.48550/arXiv.2310.05737. URL http://arxiv.org/abs/2310.05737.arXiv:2310.05737.  
+Yu, Z., Zhang, C., Wang, Y., Tang, W., Wang, J., and Ma, L. Predict and interpret health risk using ehr through typical patients. In ICASSP 2024 - 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 1506-1510, 2024b. doi: 10.1109/ICASSP48485.2024.10447313.  
+Zeghidour, N., Luebs, A., Omran, A., Skoglund, J., and Tagliasacchi, M. Soundstream: An end-to-end neural audio codec. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 30:495-507, 2021.  
+Zha, K., Yu, L., Fathi, A., Ross, D. A., Schmid, C., Katabi, D., and Gu, X. Language-guided image tokenization for generation. (arXiv:2412.05796), December 2024. doi: 10.48550/arXiv.2412.05796. URL http://arxiv.org/abs/2412.05796.arXiv:2412.05796.  
+Zhang, J., Zhan, F., Theobalt, C., and Lu, S. Regularized vector quantization for tokenized image synthesis. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 18467-18476, 2023.  
+Zhou, J., Wei, C., Wang, H., Shen, W., Xie, C., Yuille, A., and Kong, T. ibot: Image bert pre-training with online tokenizer. January 2022. doi: 10.48550/arXiv.2111.07832. URL http://arxiv.org/abs/2111.07832.  
+Zhu, Y., Ren, C., Wang, Z., Zheng, X., Xie, S., Feng, J., Zhu, X., Li, Z., Ma, L., and Pan, C. Emerge: Enhancing multimodal electronic health records predictive modeling with retrieval-augmented generation. In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management, CIKM '24, pp. 3549-3559, New York, NY, USA, 2024. Association for Computing Machinery. ISBN 9798400704369. doi: 10.1145/3627673.3679582.
+
+# A. Data preprocessing
+
+# A.1. A multimodal text-graph dataset of medical codes
+
+The medical codes dataset consists of medical codes, their descriptions, and associated knowledge subgraphs, encompassing eight commonly used health coding systems: ICD-9-CM (procedures and diagnoses), ICD-10-CM, ICD-10-PCS, NDC (National Drug Codes), SNOMED CT, ATC (Anatomical Therapeutic Chemical Classification), CPT (Current Procedural Terminology), and RxNorm. All code lists were obtained from official sources. Specifically, ICD-9 and ICD-10 (CM and PCS) were sourced from the CMS website; NDC codes from the U.S. Food and Drug Administration (FDA) database; and CPT (Level I HCPCS) from the Physician Fee Schedule (PFS) Relative Value Files at CMS. SNOMED CT, RxNorm (active codes only), and ATC were downloaded via the National Library of Medicine (NLM), part of the National Institutes of Health (NIH).
+
+# A.1.1. BUILDING A KNOWLEDGE GRAPH OF MEDICAL CODES
+
+In the final dataset, each medical code is linked to a knowledge graph capturing relevant medical insights and relationships. We constructed these subgraphs in two steps: mapping each code to one or more nodes in the PrimeKG knowledge graph (Chandak et al., 2023); and extracting node-centered subgraphs to represent the code's associated knowledge and connections. To facilitate mapping, we leveraged several external resources, notably the UMLS database (Bodenreider, 2004) and MONDO Disease Ontology files (Balsa-Canto et al., 2023). Medical codes were first mapped to Concept Unique Identifiers (CUIs) in the UMLS database, then linked to PrimeKG nodes via a custom UMLS-to-PrimeKG file. Because PrimeKG includes MONDO annotations, we also aligned medical codes to MONDO terms using the mondo.owl file, thus achieving direct integration with PrimeKG nodes. Additionally, a custom entity linker was employed to enhance coverage by translating medical codes into descriptive text (via PyHealth's MedCode InnerMap) and matching these descriptions to PrimeKG node names. When exact matches were unavailable, we resorted to an NLP-based linker (SciSpacy with UMLS) to measure semantic similarity. For drug codes, the rxnav.nlm.nih.gov API was used to map RxNorm codes to ATC identifiers, which were then associated with DrugBank entities (Wishart et al., 2018) through a predefined ATC-to-DrugBank mapping.
+
+# A.1.2. ASSEMBLING TEXT DEFINITIONS OF MEDICAL CODES
+
+Initially, each medical code's description was taken from its official source. For medication codes (e.g., NDC) where the original text was sparse, additional details were derived from attributes such as trade name, proprietary name, and pharmacological classification. These preliminary definitions were then refined and enriched using GPT-4 (turbo), with prompts tailored to each coding system but sharing a common goal of elaborating on clinical uses (for drugs), procedural steps (for procedures), or mechanistic and clinical context (for diagnoses).
+
+# B. Implementation details
+
+# B.1. Experimental environments
+
+MEDTOK is training on a machine equipped with 4 NVIDIA H100. All experiments were conducted with 1 NVIDIA H100.
+
+We implement MEDTOK using Python 3.9.19, PyTorch 2.3.1, Transformers 4.43.1, and Tokenizers 0.19.1. All LMs and LLMs adopted in this study are retrieved from Hugging Face, except for OpenAI models.
+
+# B.2. Details in MEDTOK training
+
+MEDTOK is trained on 4 NVIDIA H100 GPUs by using the loss defined in Section 3.2. During the training stage, we set the training step as 3000 with a global batch size of 1024, the dimension of quantized vectors is 64. In terms of the models' weights, we freeze the text encoder in MEDTOK and the graph encoder is trainable during the training stage.
+
+# B.3. Implementation details of baseline models
+
+All results presented in this study were obtained using the same machine on which the MEDTOK was trained.
+
+ETHOS (Renc et al., 2024) experiments were conducted using the authors' original repository. For each experimental setting, three models were trained on the MIMIC-IV dataset with different random seeds, and their predictions were averaged during inference to ensure robustness. In the "MEDTOK + ETHOS" configuration, the original vocabulary was extended to
+
+incorporate MEDTOK's tokens for diagnoses, procedures, and prescriptions. The lab measurements were excluded from the analysis. Training and dataset splitting on MIMIC-IV adhered to the methodology outlined in the ETHOS paper. During inference, the number of generated tokens was limited to 2048, and the timeline duration was adjusted based on the specific task: fifteen days for readmission, two weeks for mortality, and up to six months for other tasks. Each model was executed five times, and the resulting predictions were averaged to produce a continuous output, as described in the ETHOS study. Inference on the MIMIC-III dataset was performed on the entire dataset, excluding BMI, ICU stay tables, blood pressure, and lab data. For the EHRShot dataset, inference was conducted on the full dataset for mortality and disease-related tasks, and on randomly selected, stratified samples of ten thousand instances for other tasks.
+
+As for the other baselines adopted in this work, we first downloaded their code and deploy these models on our working machine. For BEHRT (Li et al., 2020) and GT-BEHRT (Poulain & Beheshti, 2024), we re-trained it in an end-to-end way and integrated the tokens for time, visit, and patient's info as that in their original work. For MulT-EHR (Chan et al., 2024), we first pre-train it on MIMIC-III, MIMIC-IV, and EHRShot, respectively, to get the embedding of medical codes, and next fine-tune it on multi-task learning. For the "MEDTok+" experiments, we use our token embeddings to initialize the nodes or tokens the original work adopted and then train or pre-train the model. It should be noted that we adopt a unified epoch number for all baselines, which is 50.
+
+# C. Task definitions and data preparation under in-patient setting
+
+# C.1. Mortality prediction
+
+Task definition. Mortality (MT) prediction estimates the mortality label of the subsequent visit for each sample, with the last sample dropped. Formally,
+
+$$
+f: (v _ {1}, v _ {2}, \ldots , v _ {t - 1}) \to y [ v _ {t} ],
+$$
+
+where  $y[v_{t}]\in \{0,1\}$  is a binary label indicating the patient's survival status recorded in visit  $v_{t}$ .
+
+# C.2. Readmission prediction
+
+Task definition. Readmission prediction checks if the patient will be readmitted to the hospital within  $\sigma$  days. Formally,  $f:(v_{1},v_{2},\ldots ,v_{t - 1})\to y[\tau (v_{t}) - \tau (v_{t - 1})]$ , where  $y\in \{0,1\}$  and  $\tau (v_{t})$  denotes the encounter time of visit  $v_{t}$ . Specifically,
+
+$$
+y \big [ \tau (v _ {t}) - \tau (v _ {t - 1}) \big ] = \left\{ \begin{array}{l l} 1 & \text {i f} \tau (v _ {t}) - \tau (v _ {t - 1}) \leq \sigma , \\ 0 & \text {o t h e r w i s e}. \end{array} \right.
+$$
+
+In our study, we set  $\sigma = 15$  days.
+
+# C.3. Length-of-Stay (LOS) prediction
+
+Task definition. Length-of-Stay (LOS) prediction follows the formulation of (Harutyunyan et al., 2019), estimating ICU stay length for each visit. Formally,  $f:(v_{1},v_{2},\ldots ,v_{t})\to y[v_{t}]$ , where  $y[v_{t}]\in \mathbb{R}^{1\times C}$  is a one-hot vector indicating its class among  $C$  possible categories. We define 10 classes,  $\{0,1,\dots ,7,8,9\}$ , representing the following durations: 0 for one day or less, 1-7 for within one week, 8 for one to two weeks, and 9 for at least two weeks.
+
+# C.4. Phenotype prediction
+
+Task definition. Phenotype prediction aims to classify which acute care conditions are present in a given patient record:  $f:(v_{1},v_{2},\dots,v_{t})\to y[v_{t}]$ , where  $y[v_{t}]\in \mathbb{R}^{1\times C}$  is a one-hot vector indicating its class among  $C$  possible categories. This task is a multilabel classification problem with macro-averaged AUC-ROC being the main metric.
+
+# C.5. Drug recommendation
+
+Task definition. Drug recommendation aims to recommend drugs for a patient according to the patient's visit history and diagnosis in current visit:  $f:(v_{1},v_{2},\dots,v_{t})\to y[v_{t}]$ , where  $y[v_{t}]\in \mathbb{R}^{1\times C}$  is a one-hot vector indicating its class among  $C$  possible categories. This task is a multilabel classification problem with macro-averaged AUC-ROC being the main metric.
+
+Data preprocessing. In this study, we adopted a data preprocessing approach similar to that used in previous research (Harutyunyan et al., 2019), which defined 25 acute care conditions. Each diagnosis code was mapped to one of these 25 phenotype
+
+categories. Finally, we got 24 diagnosis codes. Since ICD-9 codes in MIMIC-III are associated with hospital visits rather than specific ICU stays, we linked diagnoses to ICU stays using the hospital admission identifier. To reduce ambiguity, we excluded hospital admissions involving multiple ICU stays, ensuring that each diagnosis corresponded to a single ICU stay per admission. It's important to note that our phenotype classification was retrospective; we analyzed the complete ICU stay before predicting the presence of specific diseases.
+
+# C.6. Out-patient setting
+
+Under this setting, we adopt two types of tasks in EHRShot: operational outcomes prediction and assignment of new diagnoses. In the field of operational outcomes, we follow the same task definitions in long length of stay prediction, which only considers if a patient stay in the hospital for less than 7 days or more than 7 days. In terms of the readmission task, we set the time window as 15 days, which is the same as that under in-patient setting. We also add another operational outcome task, which is mortality prediction. The definition of mortality prediction is the same as that under the in-patient setting.

2025/Multimodal Medical Code Tokenizer/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ea5466c037603fff50c7a62107e6d46e49729a40646fcfd4d2f36080c96fd5ef
+size 633221

2025/Multinoulli Extension_ A Lossless Yet Effective Probabilistic Framework for Subset Selection over Partition Constraints/2f74a846-6168-4a9b-8587-04417ae152bc_origin.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:db1e512ca0a73941dc5c86e30f5fb1b31926cdd8d9bc6147c9ca1c4bf963b06b
+size 910306

2025/Multinoulli Extension_ A Lossless Yet Effective Probabilistic Framework for Subset Selection over Partition Constraints/images.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:26f8b716988f1aed581c0e44a44fea1d380c3b423e068ce3585e98c5af295784
+size 2517606