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2008 | OriginalPaper | Buchkapitel
Decoupling and Partial Independence
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The initial motivation of this note was the question: How many samples are needed to approximate the inertia matrix (variance-covariance matrix) of a density on
R
n
? It first arose in a joint paper with L. Lovász and M. Simonovits on an algorithm for computing volumes of convex sets. Rudelson proved a very interesting result (answering the question) based on a classical theorem from Functional Analysis (see Square Form Theorem below) due to Lust-Piquard, which is proved using the beautiful technique of Decoupling. This note gives a self-contained proof of the theorem and its application to this problem as well as a different question dealing with extending the basic result of Random Matrix Theory to partially random matrices (see Theorem 3) below.