arXiv subjects

Matt Jacobs

Publications and source records attributed to Matt Jacobs.

At least 19 recordsLinked to original sources

arXiv2026

The Signed Wasserstein Barycenter Problem

Barycenter problems encode important geometric information about a metric space. While these problems are typically studied with positive weight coefficients associated to each distance term, more general signed Wasserstein barycenter problems have recently drawn a great deal of interest. These mixed sign problems have appeared in statistical inference setting as a way to generalize least squares regression to measure valued outputs and have appeared in numerical methods to improve the accuracy of Wasserstein gradient flow solvers. Unfortunately, the presence of negatively weighted distance terms destroys the Euclidean convexity of the unsigned problem, resulting in a much more challenging optimization task. The main focus of this work is to study properties of the signed barycenter problem for a general transport cost with a focus on establishing uniqueness of solutions. In particular, when there is only one positive weight, we extend the uniqueness result of Tornabene et al. (2025) to any cost satisfying a certain convexity property. In the case of arbitrary weights, we introduce the dual problem in terms of Kantorovich potentials and provide a sufficient condition for a stationary solution of the dual problem to induce an optimal signed barycenter.

math.OC↗

arXiv2025

Guaranteeing Higher Order Convergence Rates for Accelerated Wasserstein Gradient Flow Schemes

In this paper, we study higher-order-accurate-in-time minimizing movements schemes for Wasserstein gradient flows. We introduce a novel accelerated second-order scheme, leveraging the differential structure of the Wasserstein space in both Eulerian and Lagrangian coordinates. For sufficiently smooth energy functionals, we show that our scheme provably achieves an optimal quadratic convergence rate. Under the weaker assumptions of Wasserstein differentiability and $\lambda$-displacement convexity (for any $\lambda\in \mathbb{R}$), we show that our scheme still achieves a first-order convergence rate and has strong numerical stability. In particular, we show that the energy is nearly monotone in general, while when the energy is $L$-smooth and $\lambda$-displacement convex (with $\lambda>0$), we prove the energy is non-increasing and the norm of the Wasserstein gradient is exponentially decreasing along the iterates. Taken together, our work provides the first fully rigorous proof of accelerated second-order convergence rates for smooth functionals and shows that the scheme performs no worse than the classical scheme JKO scheme for functionals that are $\lambda$-displacement convex and Wasserstein differentiable.

math.AP↗

arXiv2025

Asymptotic Linear Convergence of ADMM for Isotropic TV Norm Compressed Sensing

We prove an explicit local linear rate for ADMM solving the isotropic Total Variation (TV) norm compressed sensing problem in multiple dimensions, by analyzing the auxiliary variable in the equivalent Douglas-Rachford splitting on a dual problem. Numerical verification on large 3D problems and real MRI data will be shown. Though the proven rate is not sharp, it is close to the observed ones in numerical tests. The proven rate is not sharp, but it provides an explicit upper bound that appears close to the observed convergence rate in numerical experiments, although we do not claim this behavior holds in general.

math.OC↗

arXiv2025

On the singular limit of Brinkman's law to Darcy's law

In this paper we study singular limits of congestion-averse growth models, connecting different models describing the effect of congestion. These models arise in particular in the context of tissue growth. The main ingredient of our analysis is a family of energy evolution equations and their dissipation structures, which are novel and of independent interest. This strategy allows us to consider a larger family of pressure laws as well as proving the joint limit, from a compressible Brinkman's model to the incompressible Darcy's law, where the latter is a Hele-Shaw type free boundary problem.

math.AP↗

arXiv2024

An Optimal Transport Approach for Computing Adversarial Training Lower Bounds in Multiclass Classification

Despite the success of deep learning-based algorithms, it is widely known that neural networks may fail to be robust. A popular paradigm to enforce robustness is adversarial training (AT), however, this introduces many computational and theoretical difficulties. Recent works have developed a connection between AT in the multiclass classification setting and multimarginal optimal transport (MOT), unlocking a new set of tools to study this problem. In this paper, we leverage the MOT connection to propose computationally tractable numerical algorithms for computing universal lower bounds on the optimal adversarial risk and identifying optimal classifiers. We propose two main algorithms based on linear programming (LP) and entropic regularization (Sinkhorn). Our key insight is that one can harmlessly truncate the higher order interactions between classes, preventing the combinatorial run times typically encountered in MOT problems. We validate these results with experiments on MNIST and CIFAR-$10$, which demonstrate the tractability of our approach.

cs.LG↗

arXiv2023

Nonlocal Approximation of Slow and Fast Diffusion

Motivated by recent work on approximation of diffusion equations by deterministic interacting particle systems, we develop a nonlocal approximation for a range of linear and nonlinear diffusion equations and prove convergence of the method in the slow, linear, and fast diffusion regimes. A key ingredient of our approach is a novel technique for using the 2-Wasserstein and dual Sobolev gradient flow structures of the diffusion equations to recover the duality relation characterizing the pressure in the nonlocal-to-local limit. Due to the general class of internal energy densities that our method is able to handle, a byproduct of our result is a novel particle method for sampling a wide range of probability measures, which extends classical approaches based on the Fokker-Planck equation beyond the log-concave setting.

math.AP↗

arXiv2023

Optimal Transport of Linear Systems over Equilibrium Measures

We consider the optimal transport problem over convex costs arising from optimal control of linear time-invariant(LTI) systems when the initial and target measures are assumed to be supported on the set of equilibrium points of the LTI system. In this case, the probability measures are singular with respect to the Lebesgue measure, thus not considered in previous results on optimal transport of linear systems. This problem is motivated by applications, such as robotics, where the initial and target configurations of robots, represented by measures, are in equilibrium or stationary. Despite the singular nature of the measures, for many cases of practical interest, we show that the Monge problem has a solution by applying classical optimal transport results. Moreover, the problem is computationally tractable even if the state space of the LTI system is moderately high in dimension, provided the equilibrium set lives in a low dimensional space. In fact, for an important subclass of linear quadratic problems, such as control of the double integrator with linear quadratic cost, the optimal transport map happens to coincide with that of the Euclidean cost. We demonstrate our results by computing the optimal transport map for the minimum energy cost for a two dimensional double integrator, despite the fact that the state space is four dimensional due to position and velocity variables.

math.OC↗

arXiv2023

Free boundary regularity for tumor growth with nutrients and diffusion

In this paper, we study a tumor growth model where the growth is driven by nutrient availability and the tumor expands according to Darcy's law with a mechanical pressure resulting from the incompressibility of the cells. Our focus is on the free boundary regularity of the tumor patch that holds beyond topological changes. A crucial element in our analysis is establishing the regularity of the hitting time T, which records the first time the tumor patch reaches a given point. We achieve this by introducing a novel Hamilton-Jacobi-Bellman (HJB) interpretation of the pressure, which is of independent interest. The HJB structure is obtained by viewing the model as a limit of the Porous Media Equation (PME) and building upon a new variant of the AB estimate. Using the HJB structure, we establish a new Hopf-Lax type formula for the pressure variable. Combined with barrier arguments, the formula allows us to show that T is C^{\alpha}, where \alpha depends only on the dimension, which translates into a mild nondegeneracy of the tumor patch evolution. Building on this and obstacle problem theory, we show that the tumor patch boundary is regular in spacetime except on a set of Hausdorff dimension at most $d-\alpha$. On the set of regular points, we further show that the tumor patch is locally $C^{1,\alpha}$ in space-time. This conclusively establishes that instabilities in the boundary evolution do not amplify arbitrarily high frequencies.

math.AP↗

arXiv2023

It begins with a boundary: A geometric view on probabilistically robust learning

Although deep neural networks have achieved super-human performance on many classification tasks, they often exhibit a worrying lack of robustness towards adversarially generated examples. Thus, considerable effort has been invested into reformulating standard Risk Minimization (RM) into an adversarially robust framework. Recently, attention has shifted towards approaches which interpolate between the robustness offered by adversarial training and the higher clean accuracy and faster training times of RM. In this paper, we take a fresh and geometric view on one such method -- Probabilistically Robust Learning (PRL). We propose a mathematical framework for understanding PRL, which allows us to identify geometric pathologies in its original formulation and to introduce a family of probabilistic nonlocal perimeter functionals to rectify them. We prove existence of solutions to the original and modified problems using novel relaxation methods and also study properties, as well as local limits, of the introduced perimeters. We also clarify, through a suitable $\Gamma$-convergence analysis, the way in which the original and modified PRL models interpolate between risk minimization and adversarial training.

cs.LG↗

arXiv2023

On the existence of solutions to adversarial training in multiclass classification

We study three models of the problem of adversarial training in multiclass classification designed to construct robust classifiers against adversarial perturbations of data in the agnostic-classifier setting. We prove the existence of Borel measurable robust classifiers in each model and provide a unified perspective of the adversarial training problem, expanding the connections with optimal transport initiated by the authors in previous work and developing new connections between adversarial training in the multiclass setting and total variation regularization. As a corollary of our results, we prove the existence of Borel measurable solutions to the agnostic adversarial training problem in the binary classification setting, a result that improves results in the literature of adversarial training, where robust classifiers were only known to exist within the enlarged universal $\sigma$-algebra of the feature space.

cs.LG↗

arXiv2022

Lagrangian solutions to the Porous Media Equation and Reaction Diffusion Systems

In this paper, we construct global-in-time forward and backward Lagrangian flow maps along the pressure gradient generated by weak solutions of the Porous Media Equation. The main difficulty is that when the initial data has compact support, it is well-known that the pressure gradient is not a BV function. Thus, the theory of regular Lagrangian flows cannot be applied to construct the flow maps. To overcome this difficulty, we develop a new argument that combines Aronson-B\'enilan type estimates with the quantitative Lagrangian flow theory of Crippa and De Lellis to show that certain doubly logarithmic quantities measuring the stability of flow maps do not blow up fast enough to prevent compactness. Our arguments are sufficiently flexible to handle the Hele-Shaw limit and a multispecies generalization of the Porous Media Equation where the equation is replaced by a coupled hyperbolic-parabolic system of reaction diffusion equations. As one application of our flow maps, we are able to construct solutions where different species cannot mix together if they were separated at initial time.

math.AP↗

arXiv2022

The Multimarginal Optimal Transport Formulation of Adversarial Multiclass Classification

We study a family of adversarial multiclass classification problems and provide equivalent reformulations in terms of: 1) a family of generalized barycenter problems introduced in the paper and 2) a family of multimarginal optimal transport problems where the number of marginals is equal to the number of classes in the original classification problem. These new theoretical results reveal a rich geometric structure of adversarial learning problems in multiclass classification and extend recent results restricted to the binary classification setting. A direct computational implication of our results is that by solving either the barycenter problem and its dual, or the MOT problem and its dual, we can recover the optimal robust classification rule and the optimal adversarial strategy for the original adversarial problem. Examples with synthetic and real data illustrate our results.

cs.LG↗

arXiv2022

Tumor Growth with Nutrients: Regularity and Stability

In this paper we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the contact inhibition among the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics. In particular, we provide contraction estimates, exponential rate of asymptotic convergence, and boundary regularity of the tumor patch. These results are in sharp contrast to the models either with nutrient diffusion or with death rate in tumor cells.

math.AP↗

arXiv2021

Existence of solutions to reaction cross diffusion systems

Reaction cross diffusion systems are a two species generalization of the porous media equation. These systems play an important role in the mechanical modeling of living tissues and tumor growth. Due to their mixed parabolic-hyperbolic structure, even proving the existence of solutions to these equations is challenging. In this paper, we exploit the parabolic structure of the system to prove the strong compactness of the pressure gradient in L2. The key ingredient is the energy dissipation relation, which along with some compensated compactness arguments, allows us to upgrade weak convergence to strong convergence. As a consequence of the pressure compactness, we are able to prove the existence of solutions in a very general setting and pass to the Hele-Shaw/incompressible limit in any dimension.

math.AP↗

arXiv2021

A crystal symmetry-invariant Kobayashi--Warren--Carter grain boundary model and its implementation using a thresholding algorithm

One of the most important aims of grain boundary modeling is to predict the evolution of a large collection of grains in phenomena such as abnormal grain growth, coupled grain boundary motion, and recrystallization that occur under extreme thermomechanical loads. A unified framework to study the coevolution of grain boundaries with bulk plasticity has recently been developed by Admal et al. (2018), which is based on modeling grain boundaries as continuum dislocations governed by an energy based on the Kobayashi--Warren--Carter (KWC) model (Kobayashi et al., 1998, 2000). While the resulting unified model demonstrates coupled grain boundary motion and polygonization (seen in recrystallization), it is restricted to grain boundary energies of the Read--Shockley type, which applies only to small misorientation angles. In addition, the implementation of the unified model using finite elements inherits the computational challenges of the KWC model that originate from the singular diffusive nature of its governing equations. The main goal of this study is to generalize the KWC functional to grain boundary energies beyond the Read--Shockley-type that respect the bicrystallography of grain boundaries. The computational challenges of the KWC model are addressed by developing a thresholding method that relies on a primal dual algorithm and the fast marching method, resulting in an O(NlogN) algorithm, where N is the number of grid points. We validate the model by demonstrating the Herring angle relation, followed by a study of the grain microstructure evolution in a two-dimensional face-centered cubic copper polycrystal with crystal symmetry-invariant grain boundary energy data obtained from the lattice matching method of Runnels et al. (2016a,b).

cond-mat.mtrl-sci↗

arXiv2020

The back-and-forth method for Wasserstein gradient flows

We present a method to efficiently compute Wasserstein gradient flows. Our approach is based on a generalization of the back-and-forth method (BFM) introduced by Jacobs and L\'eger to solve optimal transport problems. We evolve the gradient flow by solving the dual problem to the JKO scheme. In general, the dual problem is much better behaved than the primal problem. This allows us to efficiently run large-scale simulations for a large class of internal energies including singular and non-convex energies.

math.NA↗

arXiv2020

Well-posedness and Regularity for a Polyconvex Energy

We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to the $H^1$ projection of measure-preserving maps. Our result introduces a new criteria on the uniqueness of the minimizer, based on the smallness of the lagrange multiplier. No estimate on the second derivatives of the pressure is needed to get a unique global minimizer. As an application, we construct a minimizing movement scheme to construct $L^r$ solutions of the Navier-Stokes equation for a short time interval.

math.AP↗

arXiv2020

The $L^1$-contraction principle in optimal transport

In this work, we use the JKO scheme to approximate a general class of diffusion problems generated by Darcy's law. Although the scheme is now classical, if the energy density is spatially inhomogeneous or irregular, many standard methods fail to apply to establish convergence in the continuum limit. To overcome these difficulties, we analyze the scheme through its dual problem and establish a novel $L^1$-contraction principle for the density variable. Notably, the contraction principle relies only on the existence of an optimal transport map and the convexity structure of the energy. As a result, the principle holds in a very general setting, and opens the door to using optimal-transport-based variational schemes to study a larger class of non-linear inhomogeneous parabolic equations.

math.AP↗