Empirical Bayes provides a powerful approach to learning and adapting to latent structure in data. Theory and algorithms for empirical Bayes have a rich literature for sequence models, but are less understood in settings where latent variables and data interact through more complex designs. In this work, we study empirical Bayes estimation of an i.i.d. prior in Bayesian linear models, via the nonparametric maximum likelihood estimator (NPMLE). We introduce and study a system of gradient flow equations for optimizing the marginal log-likelihood, jointly over the prior and posterior measures in its Gibbs variational representation using a smoothed reparametrization of the regression coefficients. A diffusion-based implementation yields a Langevin dynamics MCEM algorithm, where the prior law evolves continuously over time to optimize a sequence-model log-likelihood defined by the coordinates of the current Langevin iterate. We show consistency of the NPMLE as n,p→∞ under mild conditions, including settings of random sub-Gaussian designs when n≍p. In high noise, we prove a uniform log-Sobolev inequality for the mixing of Langevin dynamics, for possibly misspecified priors and non-log-concave posteriors. We then establish polynomial-time convergence of the joint gradient flow to a near-NPMLE if the marginal negative log-likelihood is convex in a sub-level set of the initialization.

This talk investigates the asymptotic behavior of the least singular value of a heavy-tailed random matrix model, random matrices with alpha-stable entries. We establish that the asymptotic distribution is the same as in the models with finite variance, for example when the entries of the matrix are Gaussian random variables, as the dimension of the matrices grows to infinity. The methods used to establish the result are based on the three step strategy, an important strategy developed in the last decade in the random matrix theory literature.

Dataset or feature alignment is a longstanding problem appearing in many areas including computer vision, natural language translation, and biostatistics. Here we show how a novel type of alignment problem arises in the matching of “untargeted” biological data where the concentrations of unlabeled biological molecules (features) are recorded over a collection of samples or patients. Partnering with biologists at the International Agency for Research on Cancer (IARC), we develop a practical and efficient tool for untargeted dataset alignment to be used in laboratory settings. Our approach aligns feature covariance matrices between datasets using the celebrated Gromov-Wasserstein (GW) algorithm from optimal transport. Motivated by the success of our approach, we investigate the statistical complexity of Gromov-Wasserstein for aligning empirical covariance matrices. Remarkably, we find that the GW algorithm achieves the same minimax optimal rates for this problem as a (quasi) maximum likelihood estimator, proving that it is statistically competitive. These results offer a new challenging setting for graph matching of Wishart (covariance) matrices and the first statistical rates of estimation for the Gromov-Wasserstein algorithm.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!

The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. It is known in certain cases that the marked length spectrum determines the metric up to isometry, and this is conjectured to be true in general. In this talk, we explore to what extent the marked length spectrum on a sufficiently large finite set approximately determines the metric.

The dynamics of active and passive filaments in viscous fluids is frequently used as a model for many complex fluids in biological systems such as: microtubules which are involved in intracellular transport and cell division; flagella and cilia which aid in locomotion. The numerical simulation of such systems is generally based on slender-body theory which give asymptotic approximations of the solution. However, these methods are low-order and cannot enforce no-slip boundary conditions to high-accuracy, uniformly over the boundary. Boundary-integral equation methods which completely resolve the fiber surface have so far been impractical due to the prohibitive cost of current layer-potential quadratures for such high aspect-ratio geometries. In this talk, I will present new quadrature schemes which make such computations possible and new integral equation formulations which lead to well-conditioned linear systems upon discretization. I will present numerical results to show the efficiency of our methods.

Estimation of tail probabilities in systems that involve uncertain parameters or random forcing is important when these unlikely events have severe consequences. Examples of such events are hurricanes, energy grid blackouts, or failure of engineered systems. After explaining the challenges of estimating rare event probabilities, I will make a connection between extreme event probability estimation and constrained optimization that is established by large deviation theory. The approach leads to practical methods to estimate small probabilities, and a novel class of challenging, large-scale PDE-constrained optimization problems. I will show examples governed by the shallow water equation where one is interested in estimated the probability of large tsunamis on shore, and the randomly forced Navier Stokes equations, where one is interested in the probability of large point strains.

In this talk, I will give a brief overview of the scattering transform for the mKdV equation from the perspective of the Riemann-Hilbert method, and introduce the robust inverse scattering transform. I will describe how we use this perspective to produce rational solutions to the mKdV equation of arbitrary order, already present at low order in the literature. By examining a limiting Riemann Hilbert problem, we also produce rational solutions of infinite order. We realize these solutions as limits of the finite order solutions, and describe some of their asymptotics. This is ongoing and joint work with Deniz Bilman, Elliot Blackstone, and Peter Miller. 

The Boson-Fermion correspondence has found connection to symmetric functions through its application for deriving soliton solutions of the KP equations. In this framework, the space of Young diagrams is the Fermionic Fock space, while the ring of symmetric functions is the Bosonic Fock space. Then the (second part of) BF correspondence asserts that the map sending a partition to its Schur function forms an isomorphism as H-modules, with H being the Heisenberg algebra. In this talk, we give a generalization of this correspondence into the context of Schubert calculus, wherein the space of infinite permutations plays the role of the fermionic space, and the ring of back-stable symmetric functions represents the bosonic space.