Limit Points of Reflow with Minibatch Optimal Transport

Rectified flows, also called flow matching or stochastic interpolants, are generative models that learn a time-dependent vector field steering a probability curve between two probability distributions, usually referred to as latent and target distributions. Reflow accelerates inference by iteratively straightening the trajectories induced by this vector field. We study the asymptotic behavior of this iteration and characterize its limit points. First, we define weak rectified couplings which always exist. Next, when rectified flow updates are alternated with minibatch optimal transport steps of fixed batch size, we show that any limit is $N$-cyclically monotone, where $N$ is the batch size. Such $N$-cyclically monotone couplings enjoy favorable structural and stability properties such as rectifiability and straightness. Finally, restricting velocities to gradient fields and assuming additional support conditions, we prove that reflow limits coincide with the optimal transport map between the endpoint distributions.

A Stability Benchmark of Generative Regularizers for Inverse Problems

Generative (diffusion) priors demonstrate remarkable performance in addressing inverse problems in imaging. Yet, for scientific and medical imaging, it is crucial that reconstruction techniques remain stable and reliable under imperfect settings. Typical definitions of stability encompass the notion of ''convergent regularization'', robustness to out-of-distribution data, and to inaccuracies in the forward operator or noise model. We evaluate these properties numerically. Furthermore, we benchmark generative approaches against modern optimization-based methods inspired by the widely used variational techniques. Our results give insights for which settings and applications generative priors can deliver state-of-the-art reconstructions, and on those in which they fall short or may even be problematic.

Learning Regularization Functionals for Inverse Problems: A Comparative Study

In recent years, a variety of learned regularization frameworks for solving inverse problems in imaging have emerged. These offer flexible modeling together with mathematical insights. The proposed methods differ in their architectural design and training strategies, making direct comparison challenging due to non-modular implementations. We address this gap by collecting and unifying the available code into a common framework. This unified view allows us to systematically compare the approaches and highlight their strengths and limitations, providing valuable insights into their future potential. We also provide concise descriptions of each method, complemented by practical guidelines.

DeepInverse: A Python package for solving imaging inverse problems with deep learning

DeepInverse is an open-source PyTorch-based library for solving imaging inverse problems. The library covers all crucial steps in image reconstruction from the efficient implementation of forward operators (e.g., optics, MRI, tomography), to the definition and resolution of variational problems and the design and training of advanced neural network architectures. In this paper, we describe the main functionality of the library and discuss the main design choices.

On the Relation between Rectified Flows and Optimal Transport

This paper investigates the connections between rectified flows, flow matching, and optimal transport. Flow matching is a recent approach to learning generative models by estimating velocity fields that guide transformations from a source to a target distribution. Rectified flow matching aims to straighten the learned transport paths, yielding more direct flows between distributions. Our first contribution is a set of invariance properties of rectified flows and explicit velocity fields. In addition, we also provide explicit constructions and analysis in the Gaussian (not necessarily independent) and Gaussian mixture settings and study the relation to optimal transport. Our second contribution addresses recent claims suggesting that rectified flows, when constrained such that the learned velocity field is a gradient, can yield (asymptotically) solutions to optimal transport problems. We study the existence of solutions for this problem and demonstrate that they only relate to optimal transport under assumptions that are significantly stronger than those previously acknowledged. In particular, we present several counter-examples that invalidate earlier equivalence results in the literature, and we argue that enforcing a gradient constraint on rectified flows is, in general, not a reliable method for computing optimal transport maps.

Iterative Importance Fine-tuning of Diffusion Models

Diffusion models are an important tool for generative modelling, serving as effective priors in applications such as imaging and protein design. A key challenge in applying diffusion models for downstream tasks is efficiently sampling from resulting posterior distributions, which can be addressed using the $h$-transform. This work introduces a self-supervised algorithm for fine-tuning diffusion models by estimating the $h$-transform, enabling amortised conditional sampling. Our method iteratively refines the $h$-transform using a synthetic dataset resampled with path-based importance weights. We demonstrate the effectiveness of this framework on class-conditional sampling and reward fine-tuning for text-to-image diffusion models.

Plug-and-Play Half-Quadratic Splitting for Ptychography

Ptychography is a coherent diffraction imaging method that uses phase retrieval techniques to reconstruct complex-valued images. It achieves this by sequentially illuminating overlapping regions of a sample with a coherent beam and recording the diffraction pattern. Although this addresses traditional imaging system challenges, it is computationally intensive and highly sensitive to noise, especially with reduced illumination overlap. Data-driven regularisation techniques have been applied in phase retrieval to improve reconstruction quality. In particular, plug-and-play (PnP) offers flexibility by integrating data-driven denoisers as implicit priors. In this work, we propose a half-quadratic splitting framework for using PnP and other data-driven priors for ptychography. We evaluate our method both on natural images and real test objects to validate its effectiveness for ptychographic image reconstruction.

Generative Feature Training of Thin 2-Layer Networks

We consider the approximation of functions by 2-layer neural networks with a small number of hidden weights based on the squared loss and small datasets. Due to the highly non-convex energy landscape, gradient-based training often suffers from local minima. As a remedy, we initialize the hidden weights with samples from a learned proposal distribution, which we parameterize as a deep generative model. To train this model, we exploit the fact that with fixed hidden weights, the optimal output weights solve a linear equation. After learning the generative model, we refine the sampled weights with a gradient-based post-processing in the latent space. Here, we also include a regularization scheme to counteract potential noise. Finally, we demonstrate the effectiveness of our approach by numerical examples.

Fast Summation of Radial Kernels via QMC Slicing

The fast computation of large kernel sums is a challenging task, which arises as a subproblem in any kernel method. We approach the problem by slicing, which relies on random projections to one-dimensional subspaces and fast Fourier summation. We prove bounds for the slicing error and propose a quasi-Monte Carlo (QMC) approach for selecting the projections based on spherical quadrature rules. Numerical examples demonstrate that our QMC-slicing approach significantly outperforms existing methods like (QMC-)random Fourier features, orthogonal Fourier features or non-QMC slicing on standard test datasets.

Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\mathcal F_\nu := \text{MMD}_K^2(\cdot, \nu)$ towards given target measures $\nu$ on the real line, where we focus on the negative distance kernel $K(x,y) := -|x-y|$. In one dimension, the Wasserstein-2 space can be isometrically embedded into the cone $\mathcal C(0,1) \subset L_2(0,1)$ of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on $L_2(0,1)$. Based on the construction of an appropriate counterpart of $\mathcal F_\nu$ on $L_2(0,1)$ and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures $\nu$, this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of $\mathcal C(0,1)$. For certain $\mathcal F_\nu$-flows this implies that initial point measures instantly become absolutely continuous, and stay so over time. Finally, we illustrate the behavior of the flow by various numerical examples using an implicit Euler scheme and demonstrate differences to the explicit Euler scheme, which is easier to compute, but comes with limited convergence guarantees.