TBA
Please Note: The preceding talk will be given on Tuesday September 15 at 10:30 am via https://technion.zoom.us/j/99202255210. More info here: http://people.math.gatech.edu/~glivshyts6/AGAonline.html
In this follow-up talk to the talk at the AGA seminar, we will discuss some aspects of a new algorithm for rounding and volume computation, including its proof, an efficient implementation for polytopes and open questions. We will begin the talk with a recap of the algorithm.
Joint work with He Jia, Aditi Laddha and Yin Tat Lee.
Link to attend: https://us02web.zoom.us/j/88203571169
It has been conjectured that for a sufficiently large $n$, and $p = p_n \in [\log(n)/n, 1/2)$, the probability that a $n\times n$ Bernoulli($p$) matrix $A$ is singular equals to the probability that $A$ contains of a zero row or zero column up to a negligible error.
This conjecture has been recently proved by Litvak-Tikhomirov in the regime $ C\log(n)/ n < p < 1/C$ for some universal constant $C>1$ with their new tool. While for $p = (1+o(1)) \log(n) /n$, it also holds due to a result of Basak-Rudelson. In this talk, we will discuss how to extend their results to fill the gap between these two regions. ( $1\le pn/\log(n) <\infty$ )
Tba
TBA
TBA
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.
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.
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.
