On the smallest singular value of unstructured heavy-tailed matrices

Speaker: 

Galyna Livshyts

Institution: 

Georgia Tech

Time: 

Tuesday, March 5, 2019 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

In this talk we discuss questions related to invertibility of random matrices, and the estimates for the smallest singular value. We outline the main results: an optimal small-ball behavior for the smallest singular value of square matrices under mild assumptions, and an essentially optimal small ball behavior for heavy-tailed rectangular random matrices. We make several remarks and outline some examples. We then explain the relation between such estimates and net constructions, and outline our main result in regards to existence of a net around the sphere with good properties. If time permits, we outline two more implications of this result.

The diffusion analogue to a tree-valued Markov chain.

Speaker: 

Noah Forman

Institution: 

University of Washington

Time: 

Friday, November 16, 2018 - 1:00pm to 2:00pm

Host: 

Location: 

340P

 

 

In '99, David Aldous conjectured that a certain natural "random walk" on the space of binary combinatorial trees should have a continuum analogue, which would be a diffusion on the Gromov-Hausdorff space of continuum trees. This talk discusses ongoing work by F-Pal-Rizzolo-Winkel that has recently verified this conjecture with a path-wise construction of the diffusion. This construction combines our work on dynamics of certain projections of the combinatorial tree-valued random walk with our previous construction of interval-partition-valued diffusions.

Minimal Gaussian partitions, clustering hardness and voting

Speaker: 

Steven Heilman

Institution: 

USC

Time: 

Tuesday, January 15, 2019 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

A single soap bubble has a spherical shape since it minimizes its surface area subject to a fixed enclosed volume of air.  When two soap bubbles collide, they form a “double-bubble” composed of three spherical caps.  The double-bubble minimizes total surface area among all sets enclosing two fixed volumes.  This was proven mathematically in a landmark result by Hutchings-Morgan-Ritore-Ros and Reichardt using the calculus of variations in the early 2000s.  The analogous case of three or more Euclidean sets is considered difficult if not impossible.  However, if we replace Lebesgue measure in these problems with the Gaussian measure, then recent work of myself (for 3 sets) and of Milman-Neeman (for any number of sets) can actually solve these problems.  We also use the calculus of variations.  Time permitting, we will discuss an improvement to the Milman-Neeman result and applications to optimal clustering of data and to designing elections that are resilient to hacking.  http://arxiv.org/abs/1901.03934

Edge universality of separable covariance matrices

Speaker: 

Fan Yang

Institution: 

UCLA

Time: 

Tuesday, November 20, 2018 - 11:00am to 12:00pm

 

 In this talk, we consider the largest singular value of the so-called separable covariance matrix Y=A^{1/2}XB^{1/2}, where X is a random matrix with i.i.d. entries and A, B are deterministic covariance matrices (which are non-negative definite symmetric). The separable covariance matrix is commonly used in e.g. environmental study, wireless communications and financial study to model sampling data with spatio-temporal correlations. However, the spectral properties of separable covariance matrices are much less known compared with sample covariance matrices.  

 

Recently, we prove that the distribution of the largest singular value of Y converges to the Tracy-Widom law under the minimal moment assumption on the entries of X. This is the first edge universality result for separable covariance matrices. As a corollary, if B=I, we obtain the edge universality for sample covariance matrices with correlated data and heavy tails. This improves the previous results for sample covariance matrices, which usually assume diagonal A or high moments of the X entries. The core parts of the proof are two comparison arguments: the Lindeberg replacement method, and a continuous self-consistent comparison argument.

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