Departmental Colloquium

I will survey some work on using noetherian properties of algebraic structures to prove stability results for functions and vector spaces coming from combinatorial representation theory and classical algebraic geometry. Some examples include Kronecker coefficients from the representation theory of the symmetric groups and Segre embeddings of products of projective spaces. No prior knowledge of representation theory or algebraic geometry is assumed and all of the terminology will be defined in the talk.

In this talk I will give an overview of work with Eric Friedlander, Julia Pevtsova and Andrei Suslin on module of constant Jordan type. Given a nilpotent linear operator on a vector space, the Jordan type is the partition of the dimension that describes the Jordan canonical form of the operator. Given two commuting nilpotent operators X and Y, we can ask about the configuration of the Jordan types of the operators aX+bY, for a and b element of the base field. At its most basic level, the work is concerned with linear algebra. However, it has implications for group representation theory and algebraic geometry.

We discuss variational problems arising in machine learning and their limits as the number of data points goes to infinity. Consider point clouds obtained as random samples of an underlying "ground-truth" measure. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points. Many machine learning tasks, such as clustering and classification, can be posed as minimizing functionals on such graphs. We consider functionals involving graph cuts and their limits as the number of data points goes to infinity. In particular we establish under what conditions the minimizers of discrete problems have a well defined continuum limit, and characterize the limit. The question is considered using the Gamma convergence. The Gamma limit, and associated compactness property, are considered with respect to a topology which uses optimal transportation to suitably compare functions defined on graphs with functions defined with respect to the continuum ground-truth measure. The talk is primarily based on joint works with Nicolas Garcia Trillos, as well as on works with Xavier Bresson, Thomas Laurent, and James von Brecht.

A recent trend in signal processing and machine learning research is that exact reconstruction is achievable from highly subsampled data by passing to nonlinear, sparsity-inducing, reconstruction methods such as l1 minimization. Such guarantees often require strong structural conditions on the data in addition to sparsity, such as incoherence, which render the theory unusable on problems of practical importance. Here, we show that many of these strong assumptions are tied to i.i.d uniform sampling, and can be dropped by allowing weighted, or importance sampling. First, we explain why importance sampling works in this context: it aims to make the inverse problem as well-conditioned as possible given a fixed sample budget. We then discuss several problem domains where importance sampling strategies can be derived explicitly, and outperform state-of-the-art sampling strategies used in practice: medical imaging, collaborative filtering, uncertainty quantification, and stochastic gradient methods. Along the way, we derive results at the intersection of applied harmonic analysis and random matrix theory that are of independent interest.

An impedance-match surface has the property that an incident wave generates no reflection. Here we demonstrate that by using simple construct, an acoustically reflecting surface can acquire hybrid resonances and becomes impedance-matched to airborne sound at tunable frequencies, so that no reflection is generated. Each resonant cell of the metasurface is deep-subwavelength in all its spatial dimensions, with its thickness less than the peak absorption wavelength by two orders of magnitude. As there can be no transmission, the impedance-matched acoustic wave is hence either completely absorbed at one or multiple frequencies, or converted into other form(s) of energy such as electrical current. A high acoustic-electrical energy conversion efficiency of 23% is achieved. We use the geometric perspective to derive the hybrid resonance condition, and show that the hybrid resonance is a by-product of the anti-resonance condition for the membrane-type acoustic metamaterials. This is work in collaboration with Guancong MA, Min YANG, Songwen XIAO, and Zhiyu YANG. Spring 2016