Seminars and Colloquia by Series

Heat semigroup approach to isoperimetric inequalities in metric measure spaces

Series
Stochastics Seminar
Time
Thursday, January 30, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Patricia Alonso-RuizTexas A&M University

The classical isoperimetric problem consists in finding among all sets with the same volume (measure) the one that minimizes the surface area (perimeter measure). In the Euclidean case, balls are known to solve this problem. To formulate the isoperimetric problem, or an isoperimetric inequality, in more general settings, requires in particular a good notion of perimeter measure.

The starting point of this talk will be a characterization of sets of finite perimeter original to Ledoux that involves the heat semigroup associated to a given stochastic process in the space. This approach put in connection isoperimetric problems and functions of bounded variation (BV) via heat semigroups, and we will extend these ideas to develop a natural definition of BV functions and sets of finite perimeter on metric measure spaces. In particular, we will obtain corresponding isoperimetric inequalies in this setting.

The main assumption on the underlying space will be a non-negative curvature type condition that we call weak Bakry-Émery and is satisfied in many examples of interest, also in fractals such as (infinite) Sierpinski gaskets and carpets. The results are part of joint work with F. Baudoin, L. Chen, L. Rogers, N. Shanmugalingam and A. Teplyaev.

Probabilistic approach to Bourgain's hyperplane conjecture

Series
Stochastics Seminar
Time
Thursday, January 16, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Arnaud MarsigliettiUniversity of Florida

The hyperplane conjecture, raised by Bourgain in 1986, is a major unsolved problem in high-dimensional geometry. It states that every convex set of volume 1 in the Euclidean space has a section that is lower bounded away from 0 uniformly over the dimension. We will present a probabilistic approach to the conjecture. 

Iterated Jackknives, Two-Sided Variance Inequalities and $\Phi$-Entropy

Series
Stochastics Seminar
Time
Thursday, December 5, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian HoudréGeorgia Institute of Technology

In analogy with the Gaussian setting, we provide, at first, inequalities on the variance of a function of $n$ independent random variables generalizing results obtained for i.i.d.~ones. In particular, we obtain various upper and lower bounds on this variance, via the iterated Jackknife statistics, which can be considered as generalizations of the Efron-Stein inequality.  Relations with the Hoeffding decomposition are then presented.  Finally, the case of the $\Phi$-entropy is also considered.   

Joint work with O. Bousquet from Google Brain Team. 

Zero-free regions and central limit theorems

Series
Stochastics Seminar
Time
Thursday, November 14, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Marcus MichelenUniversity of Illinois, Chicago

Let X be a random variable taking values in {0,...,n} and f(z) be its probability generating function.  Pemantle conjectured that if the variance of X is large and f has no roots close to 1 in the complex plane, then X must be approximately normal. We will discuss a complete resolution of this conjecture in a strong quantitative form, thereby giving the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. Additionally, if f has no roots with small argument, then X must be approximately normal, again in a sharp quantitative form. These results also imply a multivariate central limit theorem that answers a conjecture and completes a program of Ghosh, Liggett and Pemantle.  This talk is based on joint work with Julian Sahasrabudhe.

The minimal distance of random linear codes

Series
Stochastics Seminar
Time
Thursday, November 7, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Han HuangGeorgiaTech

When Alice wants to send a k-bits message v to Bob over a noisy channel, she encodes it as a longer n-bits message Mv, where M is a n times k matrix over F_2. The minimal distance d_M of the linear code M is defined as the minimum Hamming distance between Mw and Mu over all distinct points w,u in F_2^k. In this way, if there are less than d_M/2 corrupted bits in the message, Bob can recover the original message via a nearest neighbor search algorithm.

The classical Gilbert-Varshamov Bound provides a lower bound for d_M if the columns of M are independent copies of X, where X is the random vector uniformly distributed on F_2^n. Under the same assumption on M, we show that the distribution of d_M is essentially the same as the minimum of Hamming weight (Hamming distance to origin) of 2^k-1 i.i.d copies of X.

The result is surprising since M is only generated by k independent copies of X. Furthermore, our results also work for arbitrary finite fields.

This is joint work with Jing Hao, Galyna Livshyts, Konstantin Tikhomirov.

Finite time dynamics of chaotic and random systems

Series
Stochastics Seminar
Time
Thursday, October 24, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Leonid BunimovichGeorgia Institute of Technology

Everybody are convinced that everything is known about the simplest random process of coin tossing. I will show that it is not the case. Particularly not everything was known for the simplest chaotic dynamical systems like the tent map (which is equivalent to coin tossing). This new area of finite time predictions emerged from a natural new question in the theory of open dynamical systems.

Understanding statistical-vs-computational tradeoffs via the low-degree likelihood ratio

Series
Stochastics Seminar
Time
Thursday, October 17, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex WeinNew York University

High-dimensional inference problems such as sparse PCA and planted clique often exhibit statistical-vs-computational tradeoffs whereby there is no known polynomial-time algorithm matching the performance of the optimal estimator. I will discuss an emerging framework -- based on the so-called low-degree likelihood ratio -- for precisely predicting these tradeoffs and giving rigorous evidence for computational hardness in the conjectured hard regime. This method was originally proposed in a sequence of works on the sum-of-squares hierarchy, and the key idea is to study whether or not there exists a low-degree polynomial that succeeds at a given statistical task.

In the second part of the talk, I will give an application to the algorithmic problem of finding an approximate ground state of the SK (Sherrington-Kirkpatrick) spin glass model. I will explain two variants of this problem: "optimization" and "certification." While optimization can be solved in polynomial time [Montanari'18], we give rigorous evidence (in the low-degree framework) that certification cannot be. This result reveals a fundamental discrepancy between two classes of algorithms: local search succeeds while convex relaxations fail.

Based on joint work with Afonso Bandeira and Tim Kunisky (https://arxiv.org/abs/1902.07324 and https://arxiv.org/abs/1907.11636).

Maximum height of low-temperature 3D Ising interfaces

Series
Stochastics Seminar
Time
Thursday, October 10, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Reza GheissariUniversity of California, Berkeley

Consider the random surface given by the interface separating the plus and minus phases in a low-temperature Ising model in dimensions $d\geq 3$. Dobrushin (1972) famously showed that in cubes of side-length $n$ the horizontal interface is rigid, exhibiting order one height fluctuations above a fixed point. 

We study the large deviations of this interface and obtain a shape theorem for its pillar, conditionally on it reaching an atypically large height. We use this to analyze the law of the maximum height $M_n$ of the interface: we prove that for every $\beta$ large, $M_n/\log n \to c_\beta$, and $(M_n - \mathbb E[M_n])_n$ forms a tight sequence. Moreover, even though this centered sequence does not converge, all its sub-sequential limits satisfy uniform Gumbel tail bounds. Based on joint work with Eyal Lubetzky. 

Pages