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Trajectory inference is a fundamental problem in many scientific domains: given a collection of unpaired snapshots of observations at discrete time points, the goal is to generate smooth trajectories that best resemble and interpolate the data. Existing algorithms exhibit computational challenges: they either rely on preprocessing subroutines to enforce smoothness or on simulation-based training objectives, both of which can be expensive. In order to overcome these limitations, we propose a new algorithm called Acceleration Matching (AM). Our approach consists of lifting the original interpolation problem to phase space and then regressing onto an explicit conditional acceleration field that induces random, smooth trajectories that agree with the prescribed marginals. Importantly, our resulting training algorithm only requires positional data, avoids trajectory simulation during training, and is devoid of expensive preprocessing. We provide ample numerical evidence suggesting that AM is competitive with or superior to existing algorithms on several benchmark problems in the literature. This is joint work with Bartolo Dazzini, Alain Durmus, and Giovanni Conforti.

Deep neural networks can learn powerful probability models of images, as demonstrated by the high-quality samples produced by score-based diffusion models. Yet it remains unclear how these networks capture complex global image statistics without suffering from the curse of dimensionality. I will present two complementary studies of this question in diffusion U-Nets. First, using a simplified multiscale U-Net, we show that coarse-to-fine conditioning in the decoder makes fine-scale image structure locally predictable: once conditioned on coarser-scale information, fine-scale features can be modeled using stationary local Markov models with small receptive fields. This transforms a high-dimensional modeling problem into a sequence of lower-dimensional conditional problems. Global structure, however, is handled differently at the U-Net bottleneck, where the spatial resolution is small enough for receptive fields to cover the entire image. In the second work, we investigate the representation that emerges in this bottleneck. We find that the middle block represents each image through a sparse subset of active channels, whose spatial averages form a nonlinear representation of the underlying clean image. Euclidean distances in this representation space reflect semantic similarity, even though the model is trained without external conditioning such as text or class labels. We further introduce a self-guided reconstruction method that uses a representation extracted by the unconditional model to condition its own stochastic synthesis, revealing the image features encoded in the bottleneck. Together, these results suggest that U-Nets manage image dimensionality by combining compact global representations with low-dimensional conditional models of local detail.

Diffusion model is a prevailing paradigm for generative AI, and this talk will briefly report two rigorous results that are centered around its performance.

First, I will quantify the generalization capability of the classical diffusion model, i.e. the one for Euclidean data. The question is, when the generative model is not memorizing the training data, what new samples will it generate? This question is not only pertinent to privacy and copyright considerations, but also important for understanding whether/how diffusion model creates new knowledge. The inductive bias of diffusion model’s generation will be examined, leading to a quantification of how diffusion model generalizes. This quantification is purely based on the empirical distribution without considering any population limit.

Then I will switch gear and describe a new test-time scaling method that improves masked diffusion model for discrete data, so that its output can maximize a given reward function, without any fine-tuning of the model weights. The method, MDM-VGB, is a discrete diffusion sampler that augments unmasking generation with principled reward-guided remasking. It is inspired by a recent success, VGB, that leverages the classical Jerrum-Sinclair backtracking Markov chain for the reward-tilted generation of autoregressive model; meanwhile, the Any-Order AutoRegressive nature of MDM-VGB allows more efficient error corrections. Besides strong empirical performance, we also prove that MDM-VGB is robust to process verifier noise, achieving a quadratic complexity, while popular test-time heuristics like best-of-N suffer from an exponential complexity due to accumulative errors.

A major bottleneck of standard auto-regressive large language models is that their inference process is inherently sequential, resulting in very long and costly inference times. To circumvent this, practitioners proposed a class of language models called diffusion language models, of which the masked diffusion model (MDM) is one of the most promising and successful. The MDM is able to sample out-of-order and, ostensibly, many tokens at once and in parallel. However, there is very limited rigorous understanding of how much parallel sampling these models can perform without noticeable degradation in their sampling performance. In this work, we give a new, exact characterization of the expected divergence between the true distribution and the sampled distribution, for any distribution and any unmasking schedule for the sampler, showing an elegant connection between MDM sampling and the classical theory of univariate function approximation.