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Javier E. Santos

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4 papers

2 author rows

TMLR Journal 2025 Journal Article

Are We Really Learning the Score Function? Reinterpreting Diffusion Models Through Wasserstein Gradient Flow Matching

An Vuong
Michael Thompson McCann
Javier E. Santos
Yen Ting Lin

Diffusion models are commonly interpreted as learning the score function, i.e., the gradient of the log-density of noisy data. However, this learning target is a conservative vector field (i.e., a vector field that is the gradient of some function), a property not enforced by neural network architectures used in practice. We show numerically that trained diffusion networks violate both the integral and differential constraints that conservative vector fields must satisfy, indicating that the learned vector fields are not score functions of any density. Despite this, the models perform remarkably well as generative mechanisms. To explain this paradox, we propose a new theoretical perspective: diffusion training is better understood as \emph{flow matching} to the velocity field of a Wasserstein Gradient Flow (WGF), rather than as score learning for a reverse-time stochastic differential equation. Under this view, the "probability flow" arises naturally from the WGF framework, eliminating the need to invoke reverse-time SDE theory and clarifying why generative sampling remains successful, even when the neural vector field is not a true score. We further show that non-conservative errors from neural approximation do not necessarily harm density transport. Our results advocate adopting the WGF perspective as a principled, elegant, and theoretically grounded framework for understanding diffusion generative models.

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NeurIPS Conference 2025 Conference Paper

Benchmarking Large Language Models with Integer Sequence Generation Tasks

Daniel O'Malley
Manish Bhattarai
Nishath Ranasinghe
Erick Draayer
Javier E. Santos

We present a novel benchmark designed to rigorously evaluate the capabilities of large language models (LLMs) in mathematical reasoning and algorithmic code synthesis tasks. The benchmark comprises integer sequence generation tasks sourced from the Online Encyclopedia of Integer Sequences (OEIS), testing LLMs' abilities to accurately and efficiently generate Python code to compute these sequences without using lookup tables. Our comprehensive evaluation includes leading models from OpenAI (including the specialized reasoning-focused o-series), Anthropic, Meta, and Google across a carefully selected set of 1000 OEIS sequences categorized as easy'' or hard. '' Half of these sequences are classical sequences from the early days of OEIS and half were recently added to avoid contamination with the models' training data. To prevent models from exploiting memorized sequence values, we introduce an automated cheating detection mechanism that flags usage of lookup tables, validated by comparison with human expert evaluations. Experimental results demonstrate that reasoning-specialized models (o3, o3-mini, o4-mini from OpenAI, and Gemini 2. 5-pro from Google) achieve substantial improvements in accuracy over non-reasoning models, especially on more complex tasks. However, overall model performance on the hard sequences is poor, highlighting persistent challenges in algorithmic reasoning. Our benchmark provides important insights into the strengths and limitations of state-of-the-art LLMs, particularly emphasizing the necessity for further advancements to reliably solve complex mathematical reasoning tasks algorithmically.

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NeurIPS Conference 2025 Conference Paper

Discrete Spatial Diffusion: Intensity-Preserving Diffusion Modeling

Javier E. Santos
Agnese Marcato
Roman Colman
NIcholas Lubbers
Yen Ting Lin

Generative diffusion models have achieved remarkable success in producing high-quality images. However, these models typically operate in continuous intensity spaces, diffusing independently across pixels and color channels. As a result, they are fundamentally ill-suited for applications involving inherently discrete quantities such as particle counts or material units, that are constrained by strict conservation laws like mass conservation, limiting their applicability in scientific workflows. To address this limitation, we propose Discrete Spatial Diffusion (DSD), a framework based on a continuous-time, discrete-state jump stochastic process that operates directly in discrete spatial domains while strictly preserving particle counts in both forward and reverse diffusion processes. By using spatial diffusion to achieve particle conservation, we introduce stochasticity naturally through a discrete formulation. We demonstrate the expressive flexibility of DSD by performing image synthesis, class conditioning, and image inpainting across standard image benchmarks, while exactly conditioning total image intensity. We validate DSD on two challenging scientific applications: porous rock microstructures and lithium-ion battery electrodes, demonstrating its ability to generate structurally realistic samples under strict mass conservation constraints, with quantitative evaluation using state-of-the-art metrics for transport and electrochemical performance.

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ICML Conference 2023 Conference Paper

Blackout Diffusion: Generative Diffusion Models in Discrete-State Spaces

Javier E. Santos
Zachary R. Fox
Nicholas Lubbers
Yen Ting Lin

Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact (as opposed to variational) analysis. We relate the theory to the existing continuous-state Gaussian diffusion as well as other approaches to discrete diffusion, and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce “Blackout Diffusion”, which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the feasibility of our approach. Generalizing from specific (Gaussian) forward processes to discrete-state processes without a variational approximation sheds light on how to interpret diffusion models, which we discuss.

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