Julie Delon

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.

This paper serves as a user's guide to sampling strategies for sliced optimal transport. We provide reminders and additional regularity results on the Sliced Wasserstein distance. We detail the construction methods, generation time complexity, theoretical guarantees, and conditions for each strategy. Additionally, we provide insights into their suitability for sliced optimal transport in theory. Extensive experiments on both simulated and real-world data offer a representative comparison of the strategies, culminating in practical recommendations for their best usage.

In this study, we present a simplified implementation of a versatile framework for structured image abstraction, as outlined in [Faraj et al., A Generic Framework for the Structured Abstraction of Images, International Symposium on Non-Photorealistic Animation and Rendering, 2017]. The framework relies on the topographic map, a hierarchical and geometric representation composed of all shapes of an image, organized in a tree structure. Within this framework, abstract renderings of digital photographs can be generated by iteratively applying simple local operations like replacement, removal, or rotation of shapes. These operations give rise to a diverse spectrum of renderings, spanning from geometrical abstraction and painting-like effects to style transfer. We perform a brief analysis of the results produced by the method, highlighting its quality and limitations.

This paper presents a computational framework for the concise encoding of an ensemble of persistence diagrams, in the form of weighted Wasserstein barycenters [100], [102] of a dictionary of atom diagrams . We introduce a multi-scale gradient descent approach for the efficient resolution of the corresponding minimization problem, which interleaves the optimization of the barycenter weights with the optimization of the atom diagrams . Our approach leverages the analytic expressions for the gradient of both sub-problems to ensure fast iterations and it additionally exploits shared-memory parallelism. Extensive experiments on public ensembles demonstrate the efficiency of our approach, with Wasserstein dictionary computations in the orders of minutes for the largest examples. We show the utility of our contributions in two applications. First, we apply Wassserstein dictionaries to data reduction and reliably compress persistence diagrams by concisely representing them with their weights in the dictionary. Second, we present a dimensionality reduction framework based on a Wasserstein dictionary defined with a small number of atoms (typically three) and encode the dictionary as a low dimensional simplex embedded in a visual space (typically in 2D). In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.

This paper presents a novel method for restoring digital videos via a Deep Plug-and-Play (PnP) approach. Under a Bayesian formalism, the method consists in using a deep convolutional denoising network in place of the proximal operator of the prior in an alternating optimization scheme. We distinguish ourselves from prior PnP work by directly applying that method to restore a digital video from a degraded video observation. This way, a network trained once for denoising can be repurposed for other video restoration tasks. Our experiments in video deblurring, super-resolution, and interpolation of random missing pixels all show a clear benefit to using a network specifically designed for video denoising, as it yields better restoration performance and better temporal stability than a single image network with similar denoising performance using the same PnP formulation. Moreover, our method compares favorably to applying a different state-of-the-art PnP scheme separately on each frame of the sequence. This opens new perspectives in the field of video restoration.

Gromov–Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. We focus on the Gromov–Wasserstein distance with a ground cost defined as the squared Euclidean distance, and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also a solution of the linear Gromov–Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov–Wasserstein distance between Gaussian distributions.

Since the seminal work of Venkatakrishnan, Bouman, and Wohlberg [Proceedings of the Global Conference on Signal and Information Processing, IEEE, 2013, pp. 945--948] in 2013, Plug & Play (PnP) meth...