Bridge Matching with Generative Applications

In [Pel21] I introduce bridge matching (BM), a novel method for constructing a diffusion process which transports samples between two target distributions. The approach involves: (i) mixing a diffusion bridge over its endpoints with respect to the target distributions and; (ii) matching the marginal distributions of the process of step (i) through a diffusion process. Unlike denoising diffusions ([HJA20], [SSK⁺21]), BM can bridge two arbitrary distributions without being restricted to a simple distribution at one end.

My contribution precedes several others that rely on similar constructions such as [LGL22], [LVH⁺23], [LCB⁺23] and [TMH⁺23]. [SDC⁺23] shows in detail how these and additional proposals are equivalent to specific instances of BM. [Pel21] explores different diffusion dynamics including a Brownian motion scaled by σ, which corresponds to the I²SB proposal of [LVH⁺23], and yields the rectified flow of [LGL22] for σ=0. State-of-the-art image generation models such as Stable Diffusion 3 and FLUX.1 successfully employ this method.

BM (top) and Denoising Diffusion (bottom) CIFAR-10 generative models at different discretizations.

In [Pel23] I further study BM and introduce its iterative version, iterated bridge matching (I-BM, see also the concurrent work of [SDC⁺23]), which is shown to convergence toward a Schrödinger bridge / entropic optimal transport map. Contrary to the classical procedure employed in this setting, the iterative proportional fitting / Sinkhorn algorithm, I-BM achieves a valid bridge (coupling) between the target distribution at each iteration. Furthermore, I-BM exhibits improved robustness properties, which are especially relevant when approximating an optimal transport map. The theoretical findings are complemented by numerical experiments demonstrating the efficacy of BM. In generative applications, BM achieves accelerated training and superior sample quality at wider discretization intervals, as illustrated in the figure above.

Large Neural Networks' Limits

The seminal work by [Nea96] establishes a connection between infinitely wide neural networks (NN) and Gaussian processes (GP), assuming parameters that are identically and independently distributed (iid) with finite variance. This result has significant implications, such as facilitating the analysis of the trainability of very deep NNs and the design of efficient covariance kernels for GP inference in perceptual domains. In a series of joint works with Stefano Favaro and Sandra Fortini ([PFF20]-[FFP23a]-[FFP23b]), I investigate wide NN limits arising from iid parameters following heavy tailed α-stable distributions, demonstrating convergence toward α-stable processes. The convergence findings, applicable to the prior NN model or the NN at initialization, are complemented by an analysis of the training dynamics in the neural tangent kernel (NTK) regime.

Functions sampled from wide α-Stable NNs, left to right: α=2.0 (Gaussian), α=1.5, α=1.0, α=0.5.

Jointly with Stefano Favaro, in [PF20] I initiate the study of stochastic processes resulting from infinitely deep neural networks. To obtain non-degenerate limits, the NN must be a residual NN with identity mapping ([HZR⁺16]). Under suitable conditions, as depth grows unbounded, convergence towards diffusion processes is achieved, assuming iid parameters with a Gaussian distribution. These findings are expanded in [PF21] to include the convolutional setting, the doubly infinite limit in depth and width, and supplementary results regarding the training dynamics in the NTK regime.