We study Langevin dynamics with noise projected onto the directions orthogonal to an isometric group action. This mathematical model is introduced to shed new light on the effects of symmetry on stochastic gradient descent for over-parametrized models. Our main result identifies a novel form of implicit regularization: when the initial and target density are both invariant under the group action, Langevin dynamics with projected noise is equivalent in law to Langevin dynamics with isotropic diffusion but with an additional drift term proportional to the negative log volume of the group orbit. We prove this result by constructing a coupling of the two processes via a third process on the group itself, and identify the additional drift as the mean curvature of the orbits.

Geometric tempering is a popular approach to sampling from challenging multi-modal probability distributions by instead sampling from a sequence of distributions which interpolate, using the geometric mean, between an easier proposal distribution and the target distribution. In this paper, we theoretically investigate the soundness of this approach when the sampling algorithm is Langevin dynamics, proving both upper and lower bounds. Our upper bounds are the first analysis in the literature under functional inequalities. They assert the convergence of tempered Langevin in continuous and discrete-time, and their minimization leads to closed-form optimal tempering schedules for some pairs of proposal and target distributions. Our lower bounds demonstrate a simple case where the geometric tempering takes exponential time, and further reveal that the geometric tempering can suffer from poor functional inequalities and slow convergence, even when the target distribution is well-conditioned. Overall, our results indicate that the geometric tempering may not help, and can even be harmful for convergence.

The Schrödinger bridge is a stochastic process that finds the most likely coupling of two measures with respect to Brownian motion, and is equivalent to the popular entropically regularized optimal transport problem. Motivated by recent applications of the Schrödinger bridge to trajectory reconstruction problems, we study the problem of sampling from a Schrödinger bridge in high dimensions. We assume sample access to the marginals of the Schrödinger bridge process and prove that the natural plug-in sampler achieves a fast statistical rate of estimation for the population bridge in terms of relative entropy. This sampling procedure is given by computing the entropic OT plan between samples from each marginal, and joining a draw from this plan with a Brownian bridge. We apply this result to construct a new and computationally feasible estimator that yields improved rates for entropic optimal transport map estimation.

We propose a new method for smoothly interpolating probability measures using the geometry of optimal transport. To that end, we reduce this problem to the classical Euclidean setting, allowing us to directly leverage the extensive toolbox of spline interpolation. Unlike previous approaches to measure-valued splines, our interpolated curves (i) have a clear interpretation as governing particle flows, which is natural for applications, and (ii) come with the first approximation guarantees on Wasserstein space. Finally, we demonstrate the broad applicability of our interpolation methodology by fitting surfaces of measures using thin-plate splines.

We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric. Although the objective is geodesically non-convex, Riemannian gradient descent empirically converges rapidly, in fact faster than off-the-shelf methods such as Euclidean gradient descent and SDP solvers. This stands in stark contrast to the best-known theoretical results, which depend exponentially on the dimension. In this work, we prove new geodesic convexity results which provide stronger control of the iterates, yielding a dimension-free convergence rate. Our techniques also enable the analysis of two related notions of averaging, the entropically-regularized barycenter and the geometric median, providing the first convergence guarantees for these problems.

We study first order methods to compute the barycenter of a probability distribution $P$ over the space of probability measures with finite second moment. We develop a framework to derive global rates of convergence for both gradient descent and stochastic gradient descent despite the fact that the barycenter functional is not geodesically convex. Our analysis overcomes this technical hurdle by employing a Polyak-Ł{}ojasiewicz (PL) inequality and relies on tools from optimal transport and metric geometry. In turn, we establish a PL inequality when $P$ is supported on the Bures-Wasserstein manifold of Gaussian probability measures. It leads to the first global rates of convergence for first order methods in this context.

Motivated by the problem of sampling from ill-conditioned log-concave distributions, we give a clean non-asymptotic convergence analysis of mirror-Langevin diffusions as introduced in Zhang et al. (2020). As a special case of this framework, we propose a class of diffusions called Newton-Langevin diffusions and prove that they converge to stationarity exponentially fast with a rate which not only is dimension-free, but also has no dependence on the target distribution. We give an application of this result to the problem of sampling from the uniform distribution on a convex body using a strategy inspired by interior-point methods. Our general approach follows the recent trend of linking sampling and optimization and highlights the role of the chi-squared divergence. In particular, it yields new results on the convergence of the vanilla Langevin diffusion in Wasserstein distance.

We propose an algebraic framework for generalized graph Laplacians which unifies the study of resistor networks, the critical group, and the eigenvalues of the Laplacian and adjacency matrices. Given a graph with boundary $G$ together with a generalized Laplacian $L$ with entries in a commutative ring $R$, we define a generalized critical group $\Upsilon_R(G,L)$. We relate $\Upsilon_R(G,L)$ to spaces of harmonic functions on the network using the Hom, Tor, and Ext functors of homological algebra. We study how these algebraic objects transform under combinatorial operations on the network $(G,L)$, including harmonic morphisms, layer-stripping, duality, and symmetry. In particular, we use layer-stripping operations from the theory of resistor networks to systematize discrete harmonic continuation. This leads to an algebraic characterization of the graphs with boundary that can be completely layer-stripped, an algorithm for simplifying computation of $\Upsilon_R(G,L)$, and upper bounds for the number of invariant factors in the critical group and the multiplicity of Laplacian eigenvalues in terms of geometric quantities.