|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
|| Affiliation: || Stanford University
|| Host: || Ron Kimmel
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 email@example.com.