Master Class: Lecture 1

SpeakerMartin Wainright
AffiliationUC Berkeley
DateMonday, 25 Feb 2013
Time13:00 - 14:00
LocationRoberts G06 Sir Ambrose Fleming LT
Event seriesMaster Class: Martin Wainwright (25 Feb - 1 Mar 2013)

High-dimensional statistics deals with problems in which the number of samples n is of the same order as, or substantially smaller than the ambient dimension p of the data. The study of such "large p, small n" problems dates back to work of Kolmogorov, and has been the subject of intensive study over the past several decades. Of course, these problems are ill-posed without further restrictions, and so a large body of research has focused on models that endowed with some form of low-dimensional structure. Examples include vectors that are sparse (with relatively few non-zero entries), matrices that are sparse and/or low-rank, and regression functions that are defined on manifolds.

In these lectures, we survey certain aspects of this rapidly evolving field, beginning with sparse vector estimation and its applications to graphical model selection, before moving onto more general high-dimensional M-estimators, and concluding with a look at high-dimensional non-parametrics.

Lecture 1, Sparse recovery and graphical models: We begin our exploration of high-dimensional statistics with the problem of sparse regression, and in particular, the behavior of "ell-one"-relaxations for estimating sparse vectors. We develop the restricted nullspace/eigenvalue conditions that are necessary and sufficient for low estimation error, and the irrepresentability conditions needed for correct variable selection. We illustrate some consequences of these results in application to high-dimensional graphical model selection.

Slides: Wainwright_Lecture1.pdf

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