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5 papers in the last four weeks, with none the four weeks before. 0.1% of all new papers.

Jul 6Week of Sep 21

Latest papers 34

Oct 1, 2026cs.LG

Graph Representation via Elements of Discrete Morse and Cobordism Theories

Topology is, by its nature and design, suited to structure that is nonlinear, multiscale, and nonstationary - however, within machine learning, its use remains largely confined to topological data analysis. We advocate that tools from low-dimensional topology which have remained almost exclusively contained within the domain of pure mathematics (such as Morse theory) offer a strong, complementary, and yet virtually unexplored perspective on the hidden structure of data-generating processes and learning tasks built upon them. Here we introduce concepts from cobordism theory and harness tools from discrete Morse theory to improve the performance of graph diffusion models through our pipeline MG-Diff. Further, we derive theoretical guarantees and sufficient conditions so that under a positive decision-gap, the Morse-theoretic tools and their application for induced diffusion guidance are stable under small perturbations. Finally, we illustrate the utility of discrete Morse theory in application to graph diffusion models for spatio-temporal graph forecasting and graph regeneration, and argue that these applications are only a small window into the part of what low-dimensional topology can offer to the field of machine learning.

Jennifer Rozenblit, Chenguang Yang, Yuxin Liu +2

Department of Mathematics; University of Texas, Austin · Department of Statistics; University of California, Riverside · Department of Statistics; Virginia Tech

Topology Graph Representation Learning Graph Representations Discrete

Sep 30, 2026cs.CL

Can large language models unlock discrete data in ophthalmic diagnostic reports?

Objective: To assess the accuracy and efficiency of a large language model (LLM) using two prompt strategies to extract structured data from ophthalmic diagnostic PDF reports. Methods: Twenty deidentified reports across four types (Visual Field, OCT Glaucoma Overview, OCT retinal nerve fiber layer Single Exam, and OCT Thickness Map; n = 5 each) were processed using two GPT-4o-assisted pipelines and compared with a reconciled manual ground truth. Schema-Constrained used Structured Output mode with a predefined JSON Schema; Prompt-Only used a detailed instruction prompt followed by Python conversion to JSON. Outcomes were value accuracy, formatting accuracy, and extraction time. Results: Schema-Constrained value accuracy was 100.00% for Visual Field and RNFL Single Exam, 97.45% for Glaucoma Overview, and 98.00% for Thickness Map; Prompt-Only achieved 100.00% across all four report types. Formatting accuracy was 100.00% for Schema-Constrained across all report types and 100.00% for Prompt-Only except RNFL Single Exam (90.14%). Mean extraction time was 56.51 s per report for manual review versus 5.04 s for Schema-Constrained and 4.70 s for Prompt-Only, an approximately 92% reduction. Conclusions: In this small proof-of-concept dataset, general-purpose LLM-assisted pipelines extracted structured data from ophthalmic diagnostic PDFs with high accuracy and substantially reduced processing time. Prompt-Only achieved the highest value accuracy, while Schema-Constrained produced schema-compliant output with 100% formatting accuracy. These complementary strengths support further evaluation of hybrid, validation-aware workflows for research and clinical data abstraction.

Umair A. Zaidi, An-Lun Wu, Wei-Chun Lin +2

Division of Informatics, Clinical Epidemiology and Translational Data Science, Oregon Health & Science University, Portland, OR · Casey Eye Institute, Department of Ophthalmology, Oregon Health & Science University, Portland, OR · Department of Ophthalmology and Visual Sciences, University of Illinois Chicago, Chicago, IL

Ophthalmology Medical Report Generation Medical Vision-Language Models Extraction Schema Discrete

Sep 23, 2026cs.LG

Data-driven discrete-time deep recurrent neural network-based modeling for dissipative systems

Physical AI has gained increasing attention for its role in developing AI systems that better understand, predict, and control real-world dynamics. Achieving this requires AI models that not only achieve high prediction accuracy but also preserve fundamental physical properties of dynamical systems. In this paper, we propose a deep discrete-time dissipative recurrent neural network (DissipNet) that explicitly enforces dissipativity, a key property related to stability and energy dissipation, through structural weight constraints and a dedicated training algorithm. By construction, the proposed network is capable of learning dissipative dynamics while preserving their inherent stability, which is formally analyzed using Lyapunov theory. In contrast to Physics-Informed Neural Networks (PINNs), which incorporate governing equations into the training loss but do not guarantee preservation of internal analytical properties such as dissipativity or passivity, our approach provides explicit guarantees on stability at the model level. We demonstrate the effectiveness of the proposed method through several modeling applications, and compare its performance with a naive recurrent neural network (RNN) and a PINN-based model.

Tuan Luong, Hyungpil Moon

Department of Mechanical Engineering, Sungkyunkwan University, Suwon 16419, South Korea.

Parametric Physics-Informed Neural Network Recurrent Neural Networks Chaos Discrete Artificial Intelligence Systems

Sep 22, 2026math.ST

Statistical Rates for Entropic Optimal Transport in the Discrete to SubGaussian Regime

We study statistical rates in entropic optimal transport in the semi-discrete regime where one measure has finite support and the other is subGaussian. Our main result establishes parametric convergence rates for the empirical dual potentials to their population counterparts, with no dimension dependence in the leading term. Our result relies on tailored strong concavity analysis of the semi-dual objective, coupled with specialized bounds for the semi-discrete potentials. As a consequence, we obtain fast rates for downstream quantities derived from the optimal coupling. Chiefly, the empirical barycentric projection achieves a squared-error rate

n^{-1}

, matching the fully compact case and improving over the less favorable

n^{-1/2}

rate known for fully subGaussian settings. Altogether, these results may indicate a lower complexity adaptation phenomenon whereby the statistical complexity of the barycentric projection is governed by the discrete measure. As an application, we analyze Sinkhorn-EM, an EM-type algorithm in which the E-step is replaced by an entropic optimal transport problem. In a well-specified and balanced two-component Gaussian mixture model, we prove

\sqrt{n}

-consistency of the empirical iterates to their population counterparts for any fixed number of iterations, matching classical EM rates up to a

\sqrt{\log n}

factor. Simulations support the theory.

Tomas Gonzalez, Gonzalo Mena

Department of Machine Learning, Carnegie Mellon University · Department of Statistics & Data Science, Carnegie Mellon University

Optimal Transport Approach Non-Gaussianity Convergence Discrete

Sep 19, 2026cs.SD

Discrete vs. Continuous: A Comprehensive Study of Unified Audio Understanding in LALMs

Large Audio Language Models (LALMs) utilize either continuous features or discrete tokens, yet the optimal representation paradigm for general audio understanding remains debated. Existing benchmarks often focus on narrow domains or evaluate encoders outside LALM contexts. To address these gaps, we systematically evaluate continuous and discrete representations across speech, sound and music. Utilizing our UniARC framework with dual evaluation strategies across model scales from SmolLM2-135M to Llama-3-8B, we analyze the dynamic relationships of data volume, model capacity, and computational efficiency. Our results reveal the pivotal role of semantic constraints in tokenization for audio understanding and demonstrate that scaling backbones fail to compensate for information loss in audio representation, especially in data-limited tasks. These findings offer practical guidance for balancing semantic density, fidelity, and efficiency in future LALMs.

Jing Peng, Zichao Nie, Zhisheng Zhang +2

Shenzhen International Graduate School, Tsinghua University, Shenzhen, China

Large Audio Language Models Audio Understanding Audio Understanding Discrete

Sep 8, 2026stat.ML

Optimal estimation for Functional Linear Regression with Noisy Discretized Data

In this paper, we consider the scalar-on-function linear regression model under a realistic sampling scheme in which the functional covariates are observed on a regular grid and contaminated by additive noise. We propose a two-step estimation procedure: first, the underlying curves are reconstructed from the discrete noisy observations using a Fourier-based projection method; second, the slope function is estimated by a penalized least-squares criterion over finite-dimensional trigonometric spaces, with data-driven selection of the model dimension. We establish oracle-type inequalities for the prediction error, both with respect to the reconstructed curves and to the true latent curves. Under regularity assumptions on the slope function and polynomial decay of the eigenvalues of the covariate, we derive convergence rates for the prediction error and show that our estimator attains the minimax rate when the number of grid points is sufficiently large. Finally, the proposed method is illustrated on simulated data and on a real meteorological dataset.

Sixtine Sphabmixay

Université Paris Cité, CNRS, MAP5, F-75006 Paris, France

Covariates Noisy Data Discrete

Sep 2, 2026cs.LG

Frontier LLMs are effective batch optimizers: Assessing reasoning models in continuous and discrete settings

Frontier large language models (LLMs) have become attractive priors for optimization due to their large-scale pretraining that enables them to navigate a variety of optimization settings. However, the effectiveness of modern reasoning LLMs in batch optimization settings remains underexplored. Here we investigate the performance of the current generation of frontier LLMs as batch optimizers in both continuous and discrete settings. We find that while LLMs are competitive zero-shot batch optimizers for numerical test functions, their performance is brittle compared to classical non-LLM optimization approaches. However, LLM priors are significantly better in semantically rich settings, indicating that their batch optimization behavior is highly effective when navigating and reasoning over the discrete spaces most similar in structure to their pretraining data.

Frank Hu, Shriram Chennakesavalu, David Graff

Prescient Design, Genentech South San Francisco, CA, USA

Adaptive Optimizers Optimization Modeling Frontiers Large Reasoning Models Prior Knowledge Integration Discrete

Jul 29, 2026cs.LG

Latent-Kernel Discrete Flow Maps for Few-Step Generation

Discrete diffusion and flow-matching models denoise a sequence over many steps, but to keep each step cheap, they factorize the transition across positions and decide every token independently. This makes few-step generation challenging for text when the target couples two positions, such as a subject and a verb that must agree. An independent update commits to them separately, and many function evaluations are spent repairing the mismatch. Existing few-step methods buy back the lost correlation by distilling or rectifying a slow teacher, and so inherit the teacher's quality ceiling. We ask instead whether a model can express correlated steps natively, and answer with Latent-Kernel Discrete Flow Maps (LKF), a from-scratch flow-map kernel that is a mixture of M factorized components tied by a single shared latent. Conditioned on the latent, each component is cheap, and the mixture is summed over the latent in closed form for small M. We show that a single step places mass on correlated completions with the same sampling time complexity as a factorized model, since one latent is drawn per sequence and reused across the entire denoising trajectory. We also show that the Masked Diffusion Language Model (MDLM) is a special case of our LKF model at M=1. The experiments for unconditional text generation on the One-Billion-Word (LM1B) and WikiText-103 benchmarks show that our LKF model learns strongly heterogeneous components and improves generative perplexity by 2.1x to 3.3x over the likelihood baselines without losing diversity. The gain grows with M, and at M=8, it surpasses distilled and rectified few-step samplers. The source code is available at: https://github.com/mansoor181/lkf.git

Mansoor Ahmed, Yue-Tsz Fan, Hemanth Venkateswara +1

Georgia State University, Atlanta, GA, USA · Georgia Institute of Technology, Atlanta, GA, USA

Latent Flow Few-Step Generation Masked Diffusion Language Models Discrete Exploratory Factor Analysis

Jul 2, 2026stat.ML

Born Discrete, Made Smooth: Variational Formulation of Shallow Neural Networks

Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics. In this work, we propose a paradigm shift by replacing the discrete training problem of shallow neural networks with a well-posed continuum variational surrogate. We identify a family of

λ

-convex functionals over parameter densities in weighted Sobolev spaces and prove that these variational problems are globally well-posed, stable, and exhibit unexpected almost

C^3

regularity. Unlike existing Wasserstein-based or Mean-Field approaches, which often face limited regularity and discretization challenges, our formulation provides direct access to elliptic regularity and convex analysis. This allows us to prove that the optimal parameter density can be obtained by solving a single linear system, bypassing iterative optimization entirely. We establish explicit generalization error controls at a rate of

1/α

relative to the regularization parameter, and prove that finite-width networks of size

N

achieve the continuum optimum at an

O(1/N)

rate. This perspective bridges the gap between the Neural Tangent Kernel (NTK) and feature-learning regimes, providing a principled framework for understanding over-parameterization through the lens of variational calculus.

Matej Benko, Pierre Bousquet, Iwona Chlebicka +1

Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology · Technická 2896/2, 616 69, Brno, Czech Republic · Université de Toulouse, INSA Toulouse, CNRS, IMT, F-31062 Toulouse Cedex 9, France +2

Variational Neural Network Variational Formulation Regularization Discrete

Jul 1, 2026cs.AI

Discrete Diffusion Language Models for Interactive Radiology Report Drafting

Diffusion language models, which generate text by denoising a token canvas bidirectionally instead of emitting tokens left to right, have become competitive with autoregressive (AR) generation. Medical foundation models, however, remain almost entirely autoregressive. We adapt a mixture-of-experts diffusion language model, DiffusionGemma-26B, and benchmark it against its same-size AR sibling Gemma-4-26B under an identical LoRA recipe on medical visual question answering datasets, scored by a verbosity-robust LLM judge. Diffusion matches or exceeds AR on all of them, and the finetuned model (3.8B active) is competitive with frontier vision-language models; its decoding is also 3.5-4.4x faster. Beyond this parity, the diffusion model offers a drafting capability AR lacks: any-order infill. Because the canvas is denoised bidirectionally, a radiologist can fix report fragments and have the model fill the text between them, an operation inherent to diffusion but not to autoregression, which is subpar at it. This suits real reports, which are often terse or inconsistent across clinicians and institutions.

Max Van Puyvelde, Halil Ibrahim Gulluk, Wim Van Criekinge +1

Department of Biomedical Data Science, Stanford University School of Medicine · Department of Mathematical Modelling, Statistics & Bioinformatics, Ghent University · Department of Electrical Engineering, Stanford University

Radiology Report Generation Diffusion Language Models Drafts Decoding Discrete

Jun 26, 2026cs.LG

HybridCodec: Modeling Discrete and Continuous Representations for Efficient Speech Language Models

Discrete audio representations have become increasingly popular for building multimodal text-audio systems and integrating audio capabilities into Large Language Models (LLMs). However, numerous studies report performance degradation on various downstream tasks due to information loss during discretization. To address this, we propose a novel approach combining temporally compressed discrete tokens with dimensionality-reduced continuous residuals. Our framework consists of a hybridized discrete-continuous focal modulation codec and a hybrid Transformer. This architecture performs autoregressive inference in the discrete domain, coupled with non-autoregressive prediction and continuous residual upsampling. Experimental results show that our approach significantly improves the retention of speaker characteristics compared to discrete-only methods, while simultaneously reducing the number of required autoregressive steps.

Artem Ploujnikov, Francesco Verdini, Samir Sadok +1

Mila, Quebec AI Institute, Canada · Concordia University, Canada · Sapienza University of Rome, Italy +1

Discrete Speech Representations Hybrid Transformer Affectcodec Discrete Representation Space

Jun 25, 2026math.OC

Mean-Field PhiBE: Continuous-Time Mean-Field Reinforcement Learning from Discrete-Time Data

This paper addresses model-free continuous-time mean-field control in a setting where the population dynamics evolve continuously according to an unknown McKean-Vlasov stochastic differential equation, while only discrete-time transition data are available. In the model-based formulation, policy evaluation is naturally described by a stationary Hamilton-Jacobi-Bellman equation on

\mathcal P_2(\mathbb R^d)

, but this equation involves the drift and diffusion coefficients of the controlled McKean-Vlasov dynamics, which are not identifiable when only discrete-time data are available. On the other hand, a direct reduction to a time-discrete Bellman equation avoids the non-identifiability issue but loses the differential equation structure. To bridge these two viewpoints, we introduce a Mean-Field-PhiBE (MF-PhiBE), which incorporates discrete-time transition information into a continuous-time PDE on the Wasserstein space. The MF-PhiBE replaces the unknown infinitesimal drift and covariance in the policy-evaluation equation by one-step estimators computed from data, while preserving the generator structure of the McKean-Vlasov HJB equation. We also derive a policy-gradient theorem for entropy-regularized randomized feedback policies, expressing the actor direction through an action-wise infinitesimal advantage and the score of the policy. Combining these two ingredients yields a model-free actor-critic method. We prove a first-order consistency estimate showing that the value induced by an optimal MF-PhiBE policy approximates the optimal continuous-time value with an error of order

Δt

. In the linear-quadratic case, we show our approximation achieves second-order accuracy with only one-step data. Numerical experiments on an LQR benchmark and a crowd-aversion problem illustrate the proposed framework.

Erhan Bayraktar, Martin Hernandez, Qinxin Yan +1

Department of Mathematics, University of Michigan, Ann Arbor, MI, USA. · Department of Statistics and Data Science, University of California, Los Angeles, CA, USA. · Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, USA.

Stochastic Optimal Control Model-Free Mean-Field Limit Bellman Equation Fokker--Planck Equation Optimal Policies+2

Jun 12, 2026cs.LG

OmniPMNet: Bridging discrete and gridded PM10 forecasts via omni-query neural processes

Forecasting particulate matter (PM10) requires both station-scale accuracy and continuous spatial fields, especially during severe dust storms. Chemical transport models (CTMs) provide gridded forecasts but retain local biases, whereas graph neural networks (GNNs) track monitoring sites well at short lead times but do not produce gridded outputs. Here we present OmniPM-Net, a Convolutional Conditional Neural Process (ConvCNP)-based fusion model that reconciles these two forecast types within a shared spatial representation. A terrain-aware Gaussian set convolution lifts irregular GNN station forecasts onto a regular grid, where a multi-scale Spatial Source Attention (SSA) module blends them with Copernicus Atmosphere Monitoring Service (CAMS) forecasts; a shared omni-query readout then decodes this representation into consistent PM10 predictions at either stations or grid cells over a 108 h horizon. Evaluated across 1,618 air-quality monitoring stations throughout China over the full year of 2024, OmniPM-Net matches the station-level accuracy of the stronger GNN baseline (mean absolute error 21.14 versus 22.00 ug/m3) and reduces the CAMS mean absolute error by 30%, while simultaneously delivering the gridded fields that the discrete GNN cannot. Its clearest gains are in the high-concentration tail, where the 90th-percentile MAE falls by 9% relative to the GNN and 25% relative to CAMS, and during dust episodes, where it improves categorical detection skill while tracking the evolving spatial trajectory.

Shuangshuang He, Shuo Wang

ColorfulClouds Technology Co., Ltd., 100875 Beijing, China. · School of Systems Science, Beijing Normal University, 100875 Beijing, China. · D-ITET, ETH Zurich, 8092 Zurich, Switzerland.

Air Pollution Spatio-Temporal Forecasting Forecast Discrete Omnichem

Jun 9, 2026cs.CV

IDEAL: In-DEpth ALignment Makes A Discrete Representation AutoEncoder

Built on pretrained vision foundation models (VFMs), representation autoencoders (RAEs) have recently emerged as a promising approach for constructing semantically rich latent spaces for image generation. However, their reconstruction quality often remains suboptimal, largely because deep VFM representations do not preserve sufficient fine-grained visual detail. This limitation becomes even more severe after discretization, where missing low-level information is difficult to recover. In fact, we observe that shallow VFM features retain considerably richer local appearance and structural detail, which complements the high-level semantics carried by deep features used in existing RAEs. Motivated by this complementary property, we propose Ideal, an In-depth Alignment framework for discrete representation autoencoding. By jointly aligning quantized tokens with both shallow and deep VFM features, Ideal enables the resulting discrete visual tokens to preserve both visual fidelity and rich semantics. Extensive experiments demonstrate that Ideal yields superior reconstruction performance, achieving 0.61 rFID on ImageNet and outperforming the previous best method by 0.28. When used for autoregressive image generation, Ideal further produces a gFID of 1.89, establishing a new state of the art for autoregressive image generation.

Yitong Chen, Zijie Diao, Junke Wang +5

Institute of Trustworthy Embodied AI, Fudan University · Shanghai Innovation Institute · University of Maryland, College Park

Autoregressive Image Generation Autoencoder Architectures Visual Fidelity Recent Vision Foundation Models Discrete

Jun 9, 2026cs.LG

KG-SoftMAP: Soft Knowledge-Graph Priors for Bayesian Network Structure Learning from Sparse Discrete Data

Learning Bayesian network (BN) structure from sparse discrete data is hard: when each instance records only a few variables, most variable pairs lack the joint observations needed for reliable scoring, and data-only methods recover little structure. Imperfect domain knowledge, expressible as a weighted directed knowledge graph (KG), is often available. KG-SoftMAP encodes such a KG as a finite-strength, confidence-weighted edge prior and maximizes a MAP objective that adds this logit-form prior to the BDeu score. With an informative but imperfect KG, KG-SoftMAP recovers partial directed structure even at observation rate rho=0.05, with directed F1 (DF1) of 0.19-0.32 across benchmarks. At higher observation rates within this sparse grid, DF1 reaches 0.44-0.66 at rho=0.20 and 0.46-0.64 at rho=0.40. Across the same three rates, KG-SoftMAP without the KG prior averages DF1 0.00, 0.19, and 0.21. Stress tests that corrupt, remove, or blur the KG signal, together with checks on LLM-extracted graphs beyond canonical benchmarks, show that recovery rises and falls with KG quality. On three real sparse educational datasets without ground-truth DAGs, we evaluate prediction, calibration, and KG-consistency. On Short Answer Feedback (SAF), KG-SoftMAP+VE reaches Fail-class F1 0.75 versus 0.78 for logistic regression while also providing an inspectable concept graph, calibrated Fail probabilities, and posterior queries from partially observed concept evidence. The remaining datasets sharpen the operating picture: weak heuristic KG signal leaves prediction unchanged, while an independent expert ontology moves the learned graph toward expert relatedness.

Guoliang Xu, James E. Corter

Columbia University

Bayesian Networks Graph Representation Learning Discrete

Jun 5, 2026cs.CL

Geometry of Semantic Space: Comparative Study of Discrete and Continuous Models

This work examines the semantic geometry underlying NLP models. We compare supervised vector embeddings, such as CamemBERT, with lexical co-occurrence graphs that encode semantic relations more directly. While transformer-based embeddings achieve strong performance, their induced geometries often display unsatisfactory distributions. In contrast, graph-based models reveal a clearer and more human-readable organization of meaning. We have implemented a methodology that allows us to perform a comparative analysis either based on the structure of the graphs or based on the topology of the embeddings induced by these two approaches. The results of the comparison -- applied to the French "Great National Debate" corpus a collection of citizen contributions to the public debate -- show a similar local topology but a very different overall structure and topology. Theses findings suggest complementary perspectives between deep supervised models and graph-based models, considering a new pathway to guide neural architectures toward more stable and interpretable convergence with graphs structures.

Gabriel Bounias, Sabine Ploux

ISC-PIF (Institut des Systemes Complexes de Paris IdF), CNRS, France · CAMS (Centre d’analyse et de mathématique sociales), CNRS & EHESS, Paris, France

Contextual Embeddings Semantic Relationships Semantic Geometry Token Co-Occurrence Graphs Comparison Discrete

Jun 2, 2026math.OC

Bregman meets Lévy: Stochastic mirror descent with heavy-tailed noise in continuous and discrete time

We study the robustness of stochastic mirror descent (SMD) under heavy-tailed noise, focusing on whether the method retains its convergence guarantees when run with infinite-variance stochastic gradient input. To address this question in a principled manner, we begin by introducing a continuous-time model of SMD as a stochastic differential equation (SDE) driven by a centered Lévy noise process with finite

p

-th order moments,

1 < p \leq 2

. This scheme -- which we call the Lévy mirror flow (LMF) -- arises naturally as the scaling limit of SMD in the presence of heavy-tailed noise. In particular, when

p < 2

-- the heavy noise regime -- the trajectories of LMF generically exhibit jump discontinuities of arbitrary magnitude which, if frequent enough, lead to infinite variance. Nonetheless, despite this highly singular behavior, we show that LMF attains

ε

-optimality within

\mathcal{O}(ε^{-p/(p-1)})

time in the convex case, and within

\mathcal{\tilde O}(ε^{-1/(p-1)})

time for (relatively) strongly convex objectives. These guarantees provide a transparent characterization of the impact of frequent long jumps on the convergence of the process, and percolate to a series of matching discrete-time guarantees for several variants of SMD under heavy-tailed noise.

Pierre-Louis Cauvin, Panayotis Mertikopoulos

Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG, 38000 Grenoble, France.

Mirror Descent Stochastic Gradient Descent Stochastic Differential Equations Convergence Bregman Divergences Discrete

May 28, 2026cs.LG

Masked Diffusion Modeling for Anomaly Detection

Anomaly detection aims to identify samples that deviate from the nominal data distribution and is central to many safety-critical applications. However, developing effective anomaly detection methods for categorical, mixed-type, and discrete sequence data remains challenging and relatively underexplored. Masked diffusion models provide a natural way to model such data by learning to recover masked values from the remaining visible context. In this paper, we propose Masked Diffusion for Anomaly Detection (MaskDiff-AD), a forward-only method based on masked diffusion models trained only on nominal data. Given a test sample, MaskDiff-AD constructs anomaly scores from the difficulty of reconstructing randomly masked coordinates, yielding a content-sensitive score that operates directly on discrete state spaces while avoiding reverse-time sampling. We also develop a non-parametric variant of MaskDiff-AD and provide theoretical guarantees by characterizing Type-I and Type-II errors under a fixed detection threshold. Experiments on fourteen categorical and mixed-type tabular datasets from ADBench and UADAD, as well as four text anomaly detection datasets from NLP-ADBench, show that MaskDiff-AD achieves competitive performance against classical, diffusion-based, and recent tabular/text anomaly detection baselines. Notably, MaskDiff-AD achieves the best overall average rank, outperforming all twelve tabular baseline methods.

Lixing Zhang, Yuchen Liang, Liyan Xie

University of Minnesota · Ohio State University

Graph Anomaly Detection Anomaly Score Masked Diffusion Language Models Discrete

May 22, 2026cs.LG

When Good Equations Get Bad Scores: Improving Symbolic Regression Through Better Parameter Optimization

Symbolic Regression (SR) plays a central role in scientific knowledge discovery by distilling mathematical equations from observational data. Most existing SR methods function within a bi-level optimization framework: an outer loop that searches for the discrete equation structure, and an inner loop that optimizes the continuous parameters of that structure. Crucially, parameter-fitting quality directly determines a structure's score and thus the outer-loop search. However, nonlinear operators make the inner loop highly non-convex, and budget-driven reliance on fast local solvers (e.g., BFGS) often yields poor local minima and underestimated scores for correct structures. This ``Good Structure, Bad Score'' phenomenon becomes a key bottleneck, degrading efficiency and misguiding the search away from the true equation. To resolve this, we propose SAGE-Fit (Structure-Aware and Semantics-Guided Evaluator for Symbolic Regression), an SR-native fitting framework that exploits the dual native priors of symbolic expressions. By capitalizing on the structural and semantic priors unique to SR, we design tailored modules for each property, thereby effectively mitigating this optimization bottleneck. Extensive experiments demonstrate that our approach, as a plug-and-play module, significantly enhances evaluation fidelity and universally improves the performance of various SR systems.

Boxiao Wang, Kai Li, Zhiwei Chen +5

Institute of Automation, Chinese Academy of Sciences

Symbolic Regression Unified Path-Integral Update Structure-Aware Bad Discrete

May 14, 2026cs.CL

Factorization-Error-Free Discrete Diffusion Language Model via Speculative Decoding

Discrete diffusion language models improve generation efficiency through parallel token prediction, but standard

X_0

prediction methods introduce factorization errors by approximating the clean token posterior with independent token-wise distributions. This paper proposes Factorization-Error-Free Discrete Diffusion Language Modeling (FeF-DLLM), which replaces independent clean-token prediction with an exact prefix-conditioned factorization of the clean posterior to better preserve token dependencies. To reduce the sequential cost introduced by prefix conditioning, FeF-DLLM further incorporates speculative decoding within diffusion denoising, accelerating inference while maintaining the parallel prediction and re-masking properties of DLLMs. Theoretically, we prove that FeF-DLLM generates from the true joint distribution and derive its expected acceleration ratio. Experiments on GSM8K, MATH, HumanEval, and MBPP demonstrate that our method improves accuracy by an average of 5.04 percentage points while achieving an average inference speedup of

3.86\times

.

Xun Fang, Yunchen Li, Hang Yuan +1

East China Normal University · Beijing Zhongguancun Academy

Diffusion Language Models Speculative Decoding Diffusion Models Denoising Diffusion Probabilistic Model Discrete

May 11, 2026cs.RO

VRA: Grounding Discrete-Time Joint Acceleration in Voltage-Constrained Actuation

Discrete-time joint acceleration constraints are widely used to enforce position and velocity limits. However, under voltage-constrained electric actuators, kinematically admissible accelerations may be physically unrealizable, exposing a missing execution-level abstraction. We propose Voltage-Realizable Acceleration (VRA), a joint-level acceleration interface that grounds kinematic acceleration in voltage-constrained actuator physics by restricting commanded accelerations to voltage-realizable constraints. Hardware experiments on electric actuators and a wheel-legged quadruped show that VRA removes unrealizable accelerations, restores consistent near-constraint execution, and reduces constraint-induced oscillations.

Lingwei Zhang, Jiaming Wang, Tianlin Zhang +5

Department of Mechanical and Automation Engineering; Hong Kong Embodied AI Lab, The Chinese University of Hong Kong

Actuator Acceleration Standard Traffic Speed Benchmarks Discrete Grounding

May 10, 2026eess.AS

Kinetic-Optimal Scheduling with Moment Correction for Metric-Induced Discrete Flow Matching in Zero-Shot Text-to-Speech

Metric-induced discrete flow matching (MI-DFM) exploits token-latent geometry for discrete generation, but its practical use is limited by two issues: heuristic schedulers requiring hyperparameter search, and finite-step path-tracking error from its first-order continuous-time Markov chain (CTMC) solver. We address both issues. First, we derive a kinetic-optimal scheduler for prescribed scalar-parameterized probability paths, and instantiate it for MI-DFM as a training-free numerical schedule that traverses the path at constant Fisher-Rao speed. Second, we introduce a finite-step moment correction that adjusts the jump probability while preserving the CTMC jump destination distribution. We validate the resulting method, GibbsTTS, on codec-based zero-shot text-to-speech (TTS). Under controlled comparisons with a unified architecture and large-scale dataset, GibbsTTS achieves the best objective naturalness and is preferred in subjective evaluations over masked discrete generative baselines. Additionally, in comparison with the evaluated state-of-the-art TTS systems, GibbsTTS shows strong speaker similarity, achieving the highest similarity on three of four test sets and ranking second on the fourth. Project page: https://ydqmkkx.github.io/GibbsTTSProject

Dong Yang, Yiyi Cai, Haoyu Zhang +2

1The University of Tokyo · 2Independent Researcher

Flow-Matching Text-To-Speech Speaker Similarity Discrete Schedulers

May 10, 2026cs.LG

Discrete Langevin-Inspired Posterior Sampling

We study posterior sampling for inverse problems in discrete state spaces using discrete diffusion models as generative priors. While continuous diffusion models have become widely used for inverse problems, their discrete counterparts remain comparatively underexplored. Existing discrete posterior samplers often rely on continuous relaxations of discrete variables, Gibbs-style updates, or mechanisms specialized to particular corruption processes, which can limit scalability or generality. We propose

Δ

LPS, a Discrete Langevin-Inspired Posterior Sampler that uses gradient information to identify promising discrete moves without leaving the discrete state space. The resulting approach enables efficient parallel updates across all token dimensions and is agnostic to the training paradigm of the discrete diffusion prior, including masked and uniform-state diffusion. We evaluate our method on image restoration tasks across MNIST, CIFAR, and FFHQ, as well as spatial mapping, covering linear, nonlinear, and blind inverse problems. Across these settings, we improve over recent discrete diffusion posterior samplers and are competitive with strong continuous diffusion-based inverse solvers. Our results suggest that fully discrete, gradient-informed posterior samplers offer a scalable and general path toward solving inverse problems over discrete representations.

Chaitanya Amballa, Sattwik Basu, Jorge Vančo Sampedro +1

University of Illinois Urbana-Champaign

Posterior Sampling Bayesian Inverse Problems Discrete

May 8, 2026cs.LG

Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency

Recovering continuous-time dynamics from discrete observations is difficult because local supervision (e.g., pointwise regression targets, derivative approximations, or equation residuals) loses fidelity as the observation interval grows. We replace local supervision with a global structural constraint: any flow representing autonomous dynamics must satisfy the semi-group property under time translation. We train a time-conditioned secant velocity field whose deviation from this property, which we call Symmetry Rupture, serves two purposes. As a training regularizer, it confines the hypothesis space to flows that compose consistently across temporal scales. As an inference oracle, it lets the solver select the largest step size that preserves internal consistency, replacing the local truncation error that conventional adaptive solvers depend on. On the diffusion-reaction benchmark under time-informed inference, our method reduces rollout RMSE by 87% while using 5x fewer function evaluations than a Neural ODE baseline. In the more demanding direct auto-regressive setting, where the model must predict distant future frames without intermediate temporal cues, our adaptive solver allocates compute based on local geometric complexity -- maintaining the lowest rollout RMSE on two of three PDE benchmarks while baselines either diverge or require up to an order of magnitude more function evaluations to remain stable.

Yuxiang Luo, Andrew Perrault

Department of Computer Science and Engineering The Ohio State University

Neural Ordinary Differential Equations Velocity Field Discrete Symmetry

May 7, 2026cs.LG

Continuous First, Discrete Later: VQ-VAEs Without Dimensional Collapse

While many approaches to improve VQ-VAE performance focus on codebook size and utilization, the effect of dimensional collapse, where trained VQ-VAE representations live in an extremely low-dimensional subspace (1-2% of full rank), remains unaddressed. We show theoretically and empirically that dimension collapse causes a hard loss lower bound that various codebook improvement techniques fail to surpass. Our analytic framework extends the sequential learning effect of Saxe et al. [2014] by introducing ideas from rate-distortion theory and explains how the latent collapse is caused by the VQ suppressing lower-variance directions. Our theory justifies a simple solution: a "warm-up phase" that trains the model as an (unquantized) autoencoder before introducing VQ. On both synthetic experiments and large-scale image (VQGAN) and audio (WavTokenizer) VQ-VAEs, we show that AE Warm-Up successfully restores representation dimension, leading to lower reconstruction and perceptual loss at the same training budget. Across codebook sizes

K \in

{

2^{10}, 2^{14}, 2^{16}

}, AE warm-up raises VQGAN codebook effective dimension from 3-5 to 17-19 and reduces rFID by 17-35%; on WavTokenizer at

K \in

{

2^{13}, 2^{14}

}, it raises codebook dimension from 4 to 17-19 and improves PESQ by 11-14%. We empirically characterize how warm-up duration governs the achievable final loss. In agreement with experiment, our theoretical analysis predicts downstream performance as a function of warm-up length, enabling an adaptive criterion for switching from AE Warm-up to VQ-VAE training.

Xinyu Zhao, Nikita Karagodin, Hamed Hassani +3

MIT

Vector-Quantized Variational Autoencoder Codebooks Warm Starts Collapse Discrete

May 7, 2026cs.CE

Discrete Elastic Ribbons: A Unified Discrete Differential Geometry Framework for One-Dimensional Energy Models

Elastic ribbons, slender structures whose length (

L

), width (

W

), and thickness (

b

) satisfy

L \gg W \gg b

, exhibit mechanical behaviors intermediate between one-dimensional rods (

L \gg W, b

) and two-dimensional plates (

L, W \gg b

). In quadratic Kirchhoff-type rod-based frameworks, such as Discrete Elastic Rods (DER), the governing equilibrium equations are independent of width, and therefore these models cannot capture width-dependent mechanical effects. Reduced centerline-based ribbon models attempt to capture width dependence via coupled bending-twisting energies. However, their relative accuracy remain unclear due to the absence of a unified simulation framework. In this work, we formulate a framework grounded in discrete differential geometry where the energy is expressed as functions of coupled bending-twisting strain measures along the centerline, rather than a linear sum of quadratic bending and twisting energies in DER. We derive analytical gradients and Hessians of the energy that enable implicit time integration. Within this unified setting, we compare five ribbon models: Kirchhoff, Sadowsky, Wunderlich, Sano, and Audoly. As a benchmark, a straight ribbon is longitudinally constrained into a pre-buckled arch and subjected to transverse displacement, inducing a supercritical pitchfork bifurcation. Predicted bifurcation thresholds are compared against shell-based finite element simulations, with the Sano model providing the closest agreement in capturing width-dependent shifts. Our high-performance JAX-based implementation achieves

\mathcal{O}(N)

per-iteration cost and also confirms that Sano model introduces negligible per-iteration overhead relative to standard DER.

Shivam Kumar Panda, M Khalid Jawed

University of California, Los Angeles, 420 Westwood Plaza, Los Angeles, 90095, CA, United States of America

Cosserat Rod Theory Finite Element Method Dynamic Rope Manipulation Discrete

May 3, 2026cs.LG

Skipping the Zeros in Diffusion Models for Sparse Data Generation

Diffusion models (DMs) excel on dense continuous data, but are not designed for sparse continuous data. They do not model exact zeros that represent the deliberate absence of a signal. As a result, they erase sparsity patterns and perform unnecessary computation on mostly zero entries. With Sparsity-Exploiting Diffusion (SED), we model only non-zero values, preserving sparsity. SED delivers computational savings while maintaining or improving generation quality by skipping zeros during training and inference. Across physics and biology benchmarks, SED matches or surpasses conventional DMs and domain-specific baselines, while vision experiments provide intuitive insights into the limitations of dense DMs and the benefits of SED.

Phil Sidney Ostheimer, Mayank Nagda, Andriy Balinskyy +6

1RPTU University Kaiserslautern-Landau · 2Heidelberg Univer · University of California, Irvine.

Diffusion Models Sparsity Discrete Data Generation

May 1, 2026cs.LG

Beyond Continuity: Simulation-free Reconstruction of Discrete Branching Dynamics from Single-cell Snapshots

Inferring cellular trajectories from destructive snapshots is complicated by the challenges of stochasticity and non-conservative mass dynamics such as cell proliferation and apoptosis. Existing unbalanced Optimal Transport (OT) methods treat mass as a continuous fluid, performing inference at the population level. However, this macroscopic view often fails to capture the discrete, jump-like nature of birth-death events at single-cell resolution, which is essential for understanding lineage branching and fate decisions. We present Unbalanced Schrödinger Bridge (USB), a simulation-free framework for learning underlying dynamics that effectively integrates both stochastic and unbalanced effects which also models the discrete, jump-like birth-death dynamics at single-cell resolution. Theoretically, USB provides a tractable solution to the Branching Schrödinger Bridge (BSB) problem, offering a rigorous microscopic interpretation where individual cells undergo both Brownian motion and discrete birth-death jumps. Technically, the method implements an efficient solver by introducing a simulation-free training objective that effectively scales to high-dimensional omics data. Empirically, we demonstrate on both simulated and real-world datasets that USB not only achieves trajectory reconstruction performance better than or comparable to deterministic baselines but also uniquely enables realistic discrete simulation of birth-death dynamics at single-cell resolution.

Junda Ying, Yuxuan Wang, Bowen Yang +2

Beijing International Center for Mathematical Research, Peking University · Center for Data Science, Peking University · School of Mathematical Sciences, Peking University +4

Schrödinger Bridges Population Dynamics Discrete

Apr 27, 2026cs.RO

DiscreteRTC: Discrete Diffusion Policies are Natural Asynchronous Executors

Unlike chatbots, physical AI must act while the world keeps evolving. Therefore, the inter-chunk pause of synchronous executors are fatal for dynamic tasks regardless of how fast the inference is. Asynchronous execution -- thinking while acting -- is therefore a structural requirement, and real-time chunking (RTC) makes it viable by recasting chunk transitions as inpainting: freezing committed actions and consistently generating the remainder. However, RTC with flow-matching policy is structurally suboptimal: its inpainting comes from inference-time corrections rather than the base policy, yielding little pre-training benefit, specific fine-tuning, heuristic guidance, and extra computation that inflates the latency. In this work, we observe that discrete diffusion policies, which generate actions by iteratively unmasking, are natural asynchronous executors that resolve all limitations at once: they are fine-tuning free since inpainting is their native operation, while early stopping further provides adaptive guidance and reduces inference cost. We propose DiscreteRTC, which replaces external corrections with native unmasking, and show on dynamic simulated benchmarks and real-world dynamic manipulation tasks that it achieves higher success rates than continuous RTC and other baselines. In summary, DiscreteRTC is simpler to implement with 0 lines of additional code to enable async inpainting, faster at inference with only ~0.7 computation compared with generating actions from scratch, and better at execution with 65% higher success rate in real-world hockey defend task compared with flow-matching RTC, and 30% higher compared with training-time flow-matching RTC. More visualizations are on https://outsider86.github.io/DiscreteRTCSite/.

Pengcheng Wang, Kaiwen Hong, Chensheng Peng +4

UC Berkeley1 · UIUC2 · UT Austin3 +1

Diffusion Policies Asynchronous Execution Action Chunks Fast Inference Discrete Artificial Intelligence Systems+2

Apr 22, 2026stat.ML

Properties and limitations of geometric tempering for gradient flow dynamics

We consider the problem of sampling from a probability distribution

π

. It is well known that this can be written as an optimisation problem over the space of probability distributions in which we aim to minimise the Kullback--Leibler divergence from

π

. We consider the effect of replacing

π

with a sequence of moving targets

(π_t)_{t\ge0}

defined via geometric tempering on the Wasserstein and Fisher--Rao gradient flows. We show that convergence occurs exponentially in continuous time, providing novel bounds in both cases. We also consider popular time discretisations and explore their convergence properties. We show that in the Fisher--Rao case, replacing the target distribution with a geometric mixture of initial and target distribution never leads to a convergence speed up both in continuous time and in discrete time. Finally, we explore the gradient flow structure of tempered dynamics and derive novel adaptive tempering schedules.

Francesca Romana Crucinio, Sahani Pathiraja

ESOMAS, University of Turin, Italy · Collegio Carlo Alberto, Turin, Italy · School of Mathematics & Statistics, UNSW Sydney, Australia

Wasserstein Gradient Flows Convergence Kullback-Leibler Divergence Discrete

Mar 21, 2026cs.LG

Achieving $\widetilde{O}(1/ε)$ Sample Complexity for Bilinear Systems Identification under Bounded Noises

This paper studies finite-sample set-membership identification for discrete-time bilinear systems under bounded symmetric log-concave disturbances. Our analysis considers trajectory-dependent regressors and allows marginally stable dynamics with polynomial mean-square state growth. We prove that the diameter of the feasible parameter set shrinks with sample complexity

\widetilde{\mathcal O}(1/ε)

where

ε

is the estimation error. Simulation supports the theory and illustrates the advantage of the proposed estimator for uncertainty quantification.

Hongyu Yi, Chenbei Lu, Jing Yu

Department of Electrical and Computer Engineering, University of Washington, Seattle, WA, USA. · Cornell University AI for Science Institute, Cornell University, Ithaca, NY, USA.

System Identification Optimal Sample Complexity State Estimation Log-Concave Distributions State Space Models Discrete

Aug 3, 2025cs.CV

Beyond Discrete Samples: High Information Density Replay for Efficient Lifelong Person Re-Identification

Lifelong Person Re-Identification (LReID) typically resists catastrophic forgetting by replaying historical samples, rehearsing domain distributions, or distilling previous model knowledge. Among these, data replay is favored for its simplicity and efficiency, as it fundamentally relies on storing discrete raw images. Although often claimed to be efficient, repeatedly training on an accumulating replay buffer with complex selection strategies across sequential domains is actually highly inefficient. Furthermore, this discrete selection severely restricts historical data coverage and results in low information density, inevitably leading to poor generalization on evolving domains and causing these methods to gradually fall behind other approaches. In this paper, we rethink LReID replay and shift the paradigm from sample selection to information compression, proposing a High Information Density Replay (HiDeR) framework. Rather than saving sparse instances, we continually consolidate historical data into a compact, fixed-budget memory. Specifically, we introduce a complexity aware allocation mechanism to dynamically assign memory quotas based on intra-class variance, alongside a metric guided condensation objective that directly preserves essential identity topologies. Furthermore, since highly compressed synthetic samples exhibit artifact styles unsuitable for current domain training, we introduce a cross modality adaptation strategy. By bidirectionally translating styles between synthetic and real samples, this strategy bridges the modality gap to mitigate optimization conflicts, while also enriching stylistic diversity for better generalization. Extensive experiments demonstrate that our framework outperforms state-of-the-art methods in retaining historical knowledge and improving overall generalization, while substantially reducing the cumulative replay cost.

Mingyu Wang, Haojie Liu, Zhiyong Li +3

College of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China

Re-Identification Discrete

Jun 17, 2025math.AT

Topological data analysis using persistent discrete homology

We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods. In particular, we provide empirical evidence that persistent discrete homology is more noise-resistant than persistent homology of the Vietoris-Rips complex for data coming from non-metric settings.

Chris Kapulkin, Nathan Kershaw

Persistent Homology Topology Discrete Data Manifold Metric Spaces Empirical Study

Apr 15, 2025math.CA

Limits of Discrete Energy of Families of Increasing Sets

The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points,

\{x_n\}

, ``fills in'' a set

E \subset \mathbb{R}^d

in an appropriate sense and investigate the degree to which the discrete analog to the Riesz energy of these sets can be used to bound the Hausdorff dimension of

E

. We also discuss applications to data science and Erdős/Falconer type problems.

Hari Sarang Nathan

Fractal Dimension Discrete Energy Resources