Investigating the optimal stochastic process beyond Gaussian for noise injection in a score-based generative model remains an open question. Brownian motion is a light-tailed process with continuous paths, which leads to a slow convergence rate for the Number of Function Evaluation (NFE). Recent studies have shown that diffusion models suffer from mode-collapse issues on imbalanced data.In order to overcome the limitations of Brownian motion, we introduce a novel score-based generative model referred to as Lévy-Itō Model (LIM).

In recent years, Rectified flow (RF) has gained considerable popularity largely due to its generation efficiency and state-of-the-art performance. In this paper, we investigate the degree to which RF automatically adapts to the intrinsic low dimensionality of the support of the target distribution to accelerate sampling. We show that, using a carefully designed choice of the time-discretization scheme and with sufficiently accurate drift estimates, the RF sampler enjoys an iteration complexity of order $O(k/\varepsilon)$ (up to log factors), where $\varepsilon$ is the precision in total variation distance and $k$ is the intrinsic dimension of the target distribution. In addition, we show that the denoising diffusion probabilistic model (DDPM) procedure is equivalent to a stochastic version of RF by establishing a novel connection between these processes and stochastic localization. Building on this connection, we further design a stochastic RF sampler that also adapts to the low-dimensionality of the target distribution under milder requirements on the accuracy of the drift estimates, and also with a specific time schedule. We illustrate with simulations on the synthetic data and text-to-image data experiments the improved performance of the proposed samplers implementing the newly designed time-discretization schedules.

In recent years, the neural stochastic differential equation (NSDE) has gained attention for modeling stochastic representations with great success in various types of applications.

Denoising diffusion models have recently emerged as the predominant paradigm for generative modelling on image domains. In addition, their extension to Riemannian manifolds has facilitated a range of applications across the natural sciences. While many of these problems stand to benefit from the ability to specify arbitrary, domain-informed constraints, this setting is not covered by the existing (Riemannian) diffusion model methodology. Recent work has attempted to address this issue by constructing novel noising processes based on the reflected Brownian motion and logarithmic barrier methods. However, the associated samplers are either computationally burdensome or only apply to convex subsets of Euclidean space. In this paper, we introduce an alternative, simple noising scheme based on Metropolis sampling that affords substantial gains in computational efficiency and empirical performance compared to the earlier samplers. Of independent interest, we prove that this new process corresponds to a valid discretisation of the reflected Brownian motion. We demonstrate the scalability and flexibility of our approach on a range of problem settings with convex and non-convex constraints, including applications from geospatial modelling, robotics and protein design.

The empirical success of deep learning is often attributed to scaling laws that predict consistent gains as model, data, and compute grow; however, large models can exhibit training instability and diminishing returns, suggesting that scaling laws describe what success looks like but not when and why scaling succeeds or fails. A central obstacle is the lack of a rigorous understanding of feature learning at large depth. While muP characterizes feature-learning dynamics in the infinite-width limit and enables hyperparameter transfer across width, its depth extension (depth-muP) breaks down for residual blocks with more than one internal layer. We derive Neural Feature Dynamics (NFD) for ResNets with single-layer residual blocks, characterizing feature learning via a coupled forward-backward stochastic system in the joint infinite-width and infinite-depth limit. In this regime, NFD identifies when scaling-law trends persist and explains diminishing returns. It also reveals a vanishing mechanism induced by the 1/sqrt(depth) residual scaling under which the gradient-independence assumption (GIA), known to fail during training at finite depth, becomes provably valid again at infinite depth, yielding an analytically tractable regime for end-to-end feature learning. Motivated by this insight, we study two-layer residual blocks and show that the same mechanism causes feature-learning collapse in the first internal layer at large depth, providing a structural explanation for the empirical failure of depth-muP. Based on this diagnosis, we propose a depth-aware learning-rate correction that counteracts the collapse and empirically restores depth-wise hyperparameter transfer, yielding stronger performance in deeper ResNets.

We introduce the first continuous-time score-based generative model that leverages fractional diffusion processes for its underlying dynamics. Although diffusion models have excelled at capturing data distributions, they still suffer from various limitations such as slow convergence, mode-collapse on imbalanced data, and lack of diversity. These issues are partially linked to the use of light-tailed Brownian motion (BM) with independent increments. In this paper, we replace BM with an approximation of its non-Markovian counterpart, fractional Brownian motion (fBM), characterized by correlated increments and Hurst index $H \in (0,1)$, where $H=0.5$

Neural SDEs combine many of the best qualities of both RNNs and SDEs, and as such are a natural choice for modelling many types of temporal dynamics. They offer memory efficiency, high-capacity function approximation, and strong priors on model space. Neural SDEs may be trained as VAEs or as GANs; in either case it is necessary to backpropagate through the SDE solve. In particular this may be done by constructing a backwards-in-time SDE whose solution is the desired parameter gradients. However, this has previously suffered from severe speed and accuracy issues, due to high computational complexity, numerical errors in the SDE solve, and the cost of reconstructing Brownian motion.

Studying the properties of stochastic noise to optimize complex non-convex functions has been an active area of research in the field of machine learning. Prior work~\citep{zhou2019pgd, wei2019noise} has shown that the noise of stochastic gradient descent improves optimization by overcoming undesirable obstacles in the landscape. Moreover, injecting artificial Gaussian noise has become a popular idea to quickly escape saddle points. Indeed, in the absence of reliable gradient information, the noise is used to explore the landscape, but it is unclear what type of noise is optimal in terms of exploration ability. In order to narrow this gap in our knowledge, we study a general type of continuous-time non-Markovian process, based on fractional Brownian motion, that allows for the increments of the process to be correlated. This generalizes processes based on Brownian motion, such as the Ornstein-Uhlenbeck process. We demonstrate how to discretize such processes which gives rise to the new algorithm ``fPGD''.

Many natural dynamic processes -- such as in vivo cellular differentiation or disease progression -- can only be observed through the lens of static sample snapshots. While challenging, reconstructing their temporal evolution to decipher underlying dynamic properties is of major interest to scientific research. Existing approaches enable data transport along a temporal axis but are poorly scalable in high dimension and require restrictive assumptions to be met. To address these issues, we propose Multi-Marginal temporal Schrödinger Bridge Matching (MMtSBM) from unpaired data, extending the theoretical guarantees and empirical efficiency of Diffusion Schrödinger Bridge Matching (arXiv:2303.16852) by deriving the Iterative Markovian Fitting algorithm to multiple marginals in a novel factorized fashion. Experiments show that MMtSBM retains theoretical properties on toy examples, achieves state-of-the-art performance on real-world datasets such as transcriptomic trajectory inference in 100 dimensions, and, for the first time, recovers couplings and dynamics in very high-dimensional image settings. Our work establishes multi-marginal Schrödinger bridges as a practical and principled approach for recovering hidden dynamics from static data.

Gradient optimization algorithms using epochs, that is those based on stochastic gradient descent without replacement (SGDo), are predominantly used to train machine learning models in practice. However, the mathematical theory of SGDo and related algorithms remain underexplored compared to their "with replacement" and "one-pass" counterparts. In this article, we propose a stochastic, continuous-time approximation to SGDo with additive noise based on a Young differential equation driven by a stochastic process we call an "epoched Brownian motion". We show its usefulness by proving the almost sure convergence of the continuous-time approximation for strongly convex objectives and learning rate schedules of the form $u_t = \frac{1}{(1+t)^β}, β\in (0,1)$. Moreover, we compute an upper bound on the asymptotic rate of almost sure convergence, which is as good or better than previous results for SGDo.