Georgia Institute of Technology College of Sciences

In recent years, significant progress has been made in our understanding of the quantitative behavior of random matrices. Such results include delocalization properties of eigenvectors and tail estimates for the smallest singular value. A key ingredient in their proofs is a 'distance theorem', which is a small ball estimate for the distance between a random vector and subspace.  Building on work of Livshyts and Livshyts, Tikhomirov and Vershynin, we introduce a new distance theorem for inhomogeneous vectors and subspaces spanned by the columns of an inhomogeneous matrix. Such a result has a number of applications for generalizing results about the quantitative behavior of i.i.d. matrices to matrices without any identical distribution assumptions. To highlight this, we show that the smallest singular value estimate of Rudelson and Vershynin, proven for i.i.d. subgaussian rectangular matrices, holds true for inhomogeneous and heavy-tailed matrices.
This talk is partially based on joint work with Max Dabagia.

In this talk, we explore random trigonometric polynomials with dependent coefficients, moving beyond the typical assumption of independent or Gaussian-distributed coefficients. We show that, under mild conditions on the dependencies between the coefficients, the asymptotic behavior of the expected number of real zeros is still universal. This universality result, to our knowledge, is the first of its kind for non-Gaussian dependent settings. Additionally, we present an elegant proof, highlighting the robustness of real zeros even in the presence of dependencies. Our findings bring the study of random polynomials closer to models encountered in practice, where dependencies between coefficients are common.

Joint work with Jurgen Angst and Guillaume Poly.

Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with the name of John Tukey, is among the most popular. Tukey’s depth has found applications in robust statistics, graph theory, and the study of elections and social choice. 

We will give an introduction to the topic, describe the properties of Tukey’s depth, and introduce some remaining open questions as well as our recent progress towards the solutions.

The talk is based on a joint work with Yinan Shen.

In the 1990s, Arkadi Nemirovski asked the following question:

How hard is it to estimate a solution to unknown homogeneous linear difference equation with constant coefficients of order S, observed in the Gaussian noise on [0,N]?

The class of all such solutions, or "signals," is parametric---described by 2S complex parameters---but extremely rich: it includes the weighted sums of S exponentials, polynomials of degree S, harmonic oscillations with S arbitrary frequencies, and their algebraic combinations. Geometrically, this class is the union of all S-dimensional shift-invariant subspaces of the space of two-sided sequences, and of interest is the minimax risk on it with respect to the mean-squared error on [0,N]. I will present a recent result that shows this minimax risk to be O( S log(N) log(S)^2 ), improving over the state of the art by a polynomial in S factor, and coming within an O( log(S)^2 ) factor from the lower bound. It relies upon an approximation-theoretic construction related to minimal-norm interpolation over shift-invariant subspaces, in the spirit of the Landau-Kolmogorov problem, that I shall present in some detail. Namely, we will see that any shift-invariant subspace admits a bounded-support reproducing kernel whose spectrum has nearly the smallest possible Lp-energies for all p ≥ 1 at once.