Georgia Institute of Technology College of Sciences

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.

Motifs (patterns of subgraphs), such as edges and triangles, encode important structural information about the geometry of a network. Consequently, counting motifs in a large network is an important statistical and computational problem. In this talk we will consider the problem of estimating motif densities and fluctuations of subgraph counts in an inhomogeneous random graph sampled from a graphon. We will show that the limiting distributions of subgraph counts can be Gaussian or non-Gaussian, depending on a notion of regularity of subgraphs with respect to the graphon. Using these results and a novel multiplier bootstrap for graphons, we will construct joint confidence sets for the motif densities. Finally, we will discuss various structure theorems and open questions about degeneracies of the limiting distribution and connections to quasirandom graphs.

Joint work with Anirban Chatterjee, Soham Dan, and Svante Janson

We will discuss a problem of estimation of functionals of the form $\tau_f(\Sigma):= {\rm tr} (f(\Sigma))$ of unknown covariance operator $\Sigma$ of a centered Gaussian random variable $X$ in a separable Hilbert space ${\mathbb H}$ based on i.i.d. observation $X_1,\dots, X_n$ of $X,$ where $f:{\mathbb R}\mapsto {\mathbb R}$ is a given function. A naive plug-in estimator $\tau_f(\hat \Sigma_n)$ based on the sample covariance operator $\hat \Sigma_n$ has a large bias and bias reduction methods are needed to construct estimators with better error rates. We develop estimators with reduced bias based on linear aggregation of several plug-in estimators with different sample sizes and obtain the error bounds for such estimators with explicit dependence on the sample size $n,$ the effective rank ${\bf r}(\Sigma)= \frac{tr(\Sigma)}{\|\Sigma\|}$ of covariance operator $\Sigma$ and the degree of smoothness of function $f.$

I will give essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from n samples. A version of this result for Banach spaces will also be presented. From this, we will derive an upper bound for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure on a Euclidean space and its symmetrized empirical distribution.