Abstract
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.
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© 2008 János Bolyai Mathematical Society and Springer-Verlag
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Kannan, R. (2008). Decoupling and Partial Independence. In: Grötschel, M., Katona, G.O.H., Sági, G. (eds) Building Bridges. Bolyai Society Mathematical Studies, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85221-6_11
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