Seminars and Colloquia by Series

Robustly Clustering a Mixture of Gaussians

Series
High Dimensional Seminar
Time
Wednesday, March 11, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Santosh Vempala Georgia Tech

Please Note: We give an efficient algorithm for robustly clustering of a mixture of two arbitrary Gaussians, a central open problem in the theory of computationally efficient robust estimation, assuming only that the the means of the component Gaussians are well-separated or their covariances are well-separated. Our algorithm and analysis extend naturally to robustly clustering mixtures of well-separated logconcave distributions. The mean separation required is close to the smallest possible to guarantee that most of the measure of the component Gaussians can be separated by some hyperplane (for covariances, it is the same condition in the second degree polynomial kernel). Our main tools are a new identifiability criterion based on isotropic position, and a corresponding Sum-of-Squares convex programming relaxation. This is joint work with He Jia.

The A. D. Aleksandrov problem of existence of convex hypersurfaces in Space with given Integral Gaussian curvature and optimal transport on the sphere

Series
High Dimensional Seminar
Time
Wednesday, March 4, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Vladimir OlikerEmory University

In his book Convex Polyhedra, ch. 7 (end of subsection 2) A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of convex polytopes with given geometric data. As an example of a geometric problem in which variational approach was successfully applied, Aleksandrov quotes the Minkowski problem. He also mentions the Weyl problem of isometric embedding for which a variational approach was proposed (but not fully developed and not completed) by W. Blashke and G. Herglotz. The first goal of this talk is to give a variational formulation and solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gaussian curvature (also posed by Aleksandrov who solved it using topological methods). The second goal of this talk is to show that in variational form the Aleksandrov problem is closely connected to the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.

Gaussian methods in randomized Dvoretzky theorem

Series
High Dimensional Seminar
Time
Wednesday, February 26, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Petros ValettasUniversity of Missouri, Columbia

The cornerstone in local theory of Banach spaces is Dvoretzky’s theorem, which asserts that almost euclidean structure is locally present in any high-dimensional normed space. The random version of this remarkable phenomenon was put forth by V. Milman in 70’s, who employed the concentration of measure on the sphere. Purpose of the talk is to present how Gaussian tools from high-dimensional probability (e.g., Gaussian convexity, hypercontractivity, superconcentration) can be exploited for obtaining optimal results in random forms of Dvoretzky’s theorem. Based on joint work(s) with Grigoris Paouris and Konstantin Tikhomirov.

Small Ball Probability for the Smallest Singular Value of a Complex Random Matrix

Series
High Dimensional Seminar
Time
Wednesday, February 19, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michail SarantisGeorgiaTech

Let $N_n$ be an $n\times n$ matrix whose entries are i.i.d. copies of a random variable $\zeta=\xi+i\xi'$, where $\xi,\xi'$ are i.i.d., mean zero, variance one, subgaussian random variables. We will present a result of Luh, according to which the probability that $N_n$ has a real eigenvalue is exponentially small in $n$. An interesting part of the proof is a small ball probability estimate for the smallest singular value of a complex perturbation $M_n=M+N_n$ of the original matrix.

Improved bounds for Hadwiger covering problem via the thin shell estimates

Series
High Dimensional Seminar
Time
Wednesday, January 29, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Han HuangGeorgia Tech

Let $K$ be a n dimensional convex body with of volume $1$. and barycenter of $K$ is the origin.  It is known that $|K \cap -K|>2^{-n}$.  Via thin shell estimate by Lee-Vempala (earlier versions were done by Guedon-Milman, Fleury, Klartag), we improve the bound by a sub-exponential factor.  Furthermore, we can improve  the Hadwiger’s Conjecture in the non-symmetric case by a sub-exponential factor.  This is a joint work with Boaz A. Slomka, Tomasz Tkocz, and Beatrice-Helen Vritsiou. 

On the L_p-Brunn-Minkowski inequality for measures

Series
High Dimensional Seminar
Time
Wednesday, January 22, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Galyna LivshytsGeorgia Tech

The first part of this pair of talks will be given at the Analysis seminar right before; attending it is not necessary, as all the background will be given in this lecture as well, and the talks will be sufficiently independent of each other.

I will discuss the L_p-Brunn-Minkowski inequality for log-concave measures, explain ‘’Bochner’s method’’ approach to this problem and state and prove several new results. This falls into a general framework of isoperimetric type inequalities in high-dimensional euclidean spaces. Joint with Hosle and Kolesnikov.

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