Seminars and Colloquia by Series

Smoothed analysis of the least singular value without inverse Littlewood--Offord theory

Series
High Dimensional Seminar
Time
Wednesday, November 6, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Vishesh JainMIT

We will discuss a novel approach to obtaining non-asymptotic estimates on the lower tail of the least singular value of an $n \times n$ random matrix $M_{n} := M + N_{n}$, where $M$ is a fixed matrix with operator norm at most $O(\exp(n^{c}))$ and $N_n$ is a random matrix, each of whose entries is an independent copy of a random variable with mean 0 and variance 1. This has been previously considered in a series of works by Tao and Vu, and our results improve upon theirs in two ways: 

(i) We are able to deal with $\|M\| = O(\exp(n^{c}))$ whereas previous work was applicable for $\|M\| = O(\poly(n))$. 

(ii) Even for $\|M\| = O(poly(n))$, we are able to extract more refined information – for instance, our results show that for such $M$, the probability that $M_n$ is singular is $O(exp(-n^{c}))$, whereas even in the case when $N_n$ is an i.i.d. Bernoulli matrix, the results of Tao and Vu only give inverse polynomial singularity probability.  

 

Singular Brascamp-Lieb inequalities

Series
Analysis Seminar
Time
Wednesday, November 6, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Polona DurcikCaltech

Brascamp-Lieb inequalities are estimates for certain multilinear forms on functions on Euclidean spaces. They generalize several classical inequalities, such as Hoelder's inequality or Young's convolution inequality. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions in the Brascamp-Lieb inequality is replaced by a singular integral kernel. Examples include multilinear singular integral forms such as paraproducts or the multilinear Hilbert transform. We survey some results in the area. 

 

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