Time+Place: Sunday 16/02/2014 14:30 Room 337-8 Taub Bld.
Title: Optimal Shrinkage of Singular Values for Matrix Denoising
Speaker: Matan Gavish - CS-Lecture http://www.stanford.edu/~gavish/
Affiliation: Stanford University
Host: Ron Kimmel

Abstract:

It is common practice in multivariate and matrix-valued data analysis to 
reduce dimensionality by performing a Singular Value Decomposition or 
Principal Component Analysis, and keeping only r singular values or 
principal components, the rest being presumably associated with noise.
However, the literature does not propose a disciplined criterion to 
determine r; most practitioners still look for the ``elbow in the Scree 
Plot'', a 48-years-old heuristic performed by eye.  Formally, this is a 
matrix denoising problem, in which one recovers an unknown matrix X from 
a noisy observation Y=3DX+Z.  We show that, for white noise and 
appropriate asymptotic frameworks, random matrix theory successfully 
describes the random behavior of the singular values and vectors of Y. 
It delivers simple, convincing answers to a range of fundamental 
questions, such as the location of the optimal singular value threshold 
(2.309) and the shape of the optimal singular value shrinker (reflected 
Quarter Circle density).  Our framework has been used to discover 
optimal eigenvalue shrinkers for high-dimensional covariance estimation, 
and seems to apply to various other estimators that rely on 
eigendecomposition of signal or data matrices. Moreover, several methods 
for low-rank matrix recovery from incomplete observations rely on 
iterative matrix denoising; we discuss evidence that improved matrix 
denoising can lead to improved matrix compressed sensing.

Short Bio:

Matan Gavish is a doctoral student in statistics at Stanford University, 
in collaboration with the Yale University program in applied 
mathematics. His research interests include statistics, harmonic 
analysis, applied mathematics and computing. His doctoral thesis focuses 
on theoretical aspects of matrix inference: denoising, recovery and 
organization. Matan has an MSc in mathematics from the Hebrew University 
of Jerusalem. Contact him as gavish@stanford.edu.