Seminars and Colloquia by Series

Insights on gradient-based algorithms in high-dimensional non-convex learning

Series
School of Mathematics Colloquium
Time
Thursday, November 12, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/89107379948
Speaker
Lenka Zdeborová – EPFL

Gradient descent algorithms and their noisy variants, such as the Langevin dynamics or multi-pass SGD, are at the center of attention in machine learning. Yet their behaviour remains perplexing, in particular in the high-dimensional non-convex setting. In this talk, I will present several high-dimensional and non-convex statistical learning problems in which the performance of gradient-based algorithms can be analysed down to a constant. The common point of these settings is that the data come from a probabilistic generative model leading to problems for which, in the high-dimensional limit, statistical physics provides exact closed solutions for the performance of the gradient-based algorithms. The covered settings include the spiked mixed matrix-tensor model and the phase retrieval.

An Introduction to Gabor Analysis

Series
School of Mathematics Colloquium
Time
Thursday, October 29, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE at https://us02web.zoom.us/j/89107379948
Speaker
Kasso Okoudjou – Tufts University –

In 1946, Dennis Gabor claimed that any Lebesgue square-integrable function can be written as an infinite linear combination of time and frequency shifts of the standard Gaussian.  Since then, decomposition methods for larger classes of functions or distributions in terms of various elementary building blocks have lead to an impressive body of work in harmonic analysis. For example, Gabor analysis, which originated from Gabor's claim, is concerned with both the theory and the applications of the approximation properties of sets of time and frequency shifts of a given function. It re-emerged with the advent of wavelets at the end of the last century and is now at the intersection of many fields of mathematics, applied mathematics, engineering, and science. In this talk, I will introduce the fundamentals of the theory highlighting some applications and open problems.

Hypertrees

Series
School of Mathematics Colloquium
Time
Thursday, October 1, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/89107379948
Speaker
Nati Linial – Hebrew University of Jerusalem –

A finite connected acyclic graph is called a tree. Both properties - connectivity and being acyclic - make very good sense in higher dimensions as well. This has led Gil Kalai (1983) to define the notion of a $d$-dimensional hypertree for $d > 1$. The study of hypertrees is an exciting area of research, and I will try to give you a taste of the many open questions and what we know (and do not know) about them. No specific prior background is assumed.

The talk is based on several papers. The list of coauthors on these papers includes Roy Meshulam, Mishael Rosenthal, Yuval Peled, Lior Aronshtam, Tomsz Luczak, Amir Dahari, Ilan Newman and Yuri Rabinovich.

Symmetrization for functions of bounded mean oscillation

Series
School of Mathematics Colloquium
Time
Thursday, September 24, 2020 - 11:00 for
Location
https://us02web.zoom.us/j/89107379948
Speaker
Almut Burchard – University of Toronto –

Spaces of bounded mean oscillation (BMO) are relatively
large function spaces that are often used in place
of L^\infinity to do basic Fourier analysis.
It is not well-understood how geometric properties
of the underlying point space enters into the functional
analysis of BMO.  I will describe recent work with
Galia Dafni and Ryan Gibara, where we take some
steps towards geometric inequalities.
Specifically, we show that the symmetric decreasing
rearrangement in n-dimensions is bounded, but not
continuous in BMO. The question of sharp bounds
remains open. 

Recording: https://us02web.zoom.us/rec/share/pjIM7jMdtcDAl70hT8e7V_MBqUzPwnl1scdcQUsE6WDuKGLev6hz468_v1F_mwc1.t31L3k8qvvmXiexP

Geometry of nodal sets of Laplace eigenfunctions

Series
School of Mathematics Colloquium
Time
Thursday, September 17, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/89107379948
Speaker
Alexander Logunov – Princeton University –

We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interestingrelation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. Zero sets of Laplace eigenfunctions on surfaces are unions of smooth curves with equiangular intersections. The topology of the zero set could be quite complicated, but Yau conjectured that the total length of the zero set is comparable to the square root of the eigenvalue for all eigenfunctions. We will start with open questions about spherical harmonics and explain some methods to study nodal sets, which are zero sets of solutions of elliptic PDE. 

Zoom: https://us02web.zoom.us/j/89107379948

A tale of two polytopes: The bipermutahedron and the harmonic polytope

Series
School of Mathematics Colloquium
Time
Thursday, September 3, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/89107379948
Speaker
Federico Ardila – San Francisco State University –

This talk's recording is available here.

The harmonic polytope and the bipermutahedron are two related polytopes which arose in our work with Graham Denham and June Huh on the Lagrangian geometry of matroids. This talk will explain their geometric origin and discuss their algebraic and geometric combinatorics.

The bipermutahedron is a (2n−2)-dimensional polytope with (2n!)/2^n vertices and 3^n−3 facets. Its f-polynomial, which counts the faces of each dimension, is given by a simple evaluation of the three variable Rogers-Ramanujan function. Its h-polynomial, which gives the dimensions of the intersection cohomology of the associated topic variety, is real-rooted, so its coefficients are log-concave.

The harmonic polytope is a (2n−2)-dimensional polytope with (n!)^2(1+1/2+...+1/n) vertices and 3^n−3 facets. Its volume is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra.

These two polytopes are related by a surprising fact: in any dimension, the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.

The talk will be as self-contained as possible, and will feature joint work with Graham Denham, Laura Escobar, and June Huh.

A dynamic view on the probabilistic method: random graph processes

Series
School of Mathematics Colloquium
Time
Thursday, May 7, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/344615810
Speaker
Lutz Warnke – Georgia Tech

 Random graphs are the basic mathematical models for large-scale disordered networks in many different fields (e.g., physics, biology, sociology).
Since many real world networks evolve over time, it is natural to study various random graph processes which arise by adding edges (or vertices) step-by-step in some random way.

The analysis of such random processes typically brings together tools and techniques from seemingly different areas (combinatorial enumeration, differential equations, discrete martingales, branching processes, etc), with connections to the analysis of randomized algorithms.
Furthermore, such processes provide a systematic way to construct graphs with "surprising" properties, leading to some of the best known bounds in extremal combinatorics (Ramsey and Turan Theory).

In this talk I shall survey several random graph processes of interest (in the context of the probabilistic method), and give a glimpse of their analysis.
If time permits, we shall also illustrate one of the main proof techniques (the "differential equation method") using a simple toy example.

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