統計数学セミナー
Seminar on Probability and Statistics |
Home : Archive [ 2003 to 04 ] [ 2004 to 05 ] [ 2005 to 06 ] [ 2006 to 07 ] [ 2007 to 08 ] [ 2008 to 09 ] [ 2009 to 10 ] [ 2010 to 11 ] [ 2011 to 12 ] [ 2012 to 13 ] [ 2013 to 14 ] [ 2014 to 15 ] |
Previous Seminar : Next Seminar |
Seminar on Probability and Statistics Wednesday July 25 2007 Tokyo 122 4:20-5:30 pm
大規模ランダム行列のスペクトル理論とデータ解析への応用(Review)
小林 景 / KOBAYASHI, Kei 統計数理研究所, 学振特別研究員 / Institute of Statistical Mathematics, JSPS postdoctoral fellow Abstract The empirical spectral distribution of random matrices have been studied
since Wigner's pioneering work on the semicircular law in the 1950's.
The result says that the empirical spectral distribution of a symmetric
matrix with i.i.d. random elements converges to the semicircular law as
the size of the matrix becomes large. Though this result is beautiful in
theory, its application has been limited to a few problems in nuclear
physics and coding theory. The next breakthrough was the Marcenko-Pastur
(M-P) law for the asymptotic spectral distribution of sample covariance
matrices. The M-P law has found more applications, in particular high
dimensional statistical data analysis, than the semicircular law.
In this talk I will first review these two significant results. Each of them has three completely different proofs. Then I will explain several other theoretical results that have mostly been studied this decade. Finally, I will present some of the applications of these results. This review is partly based on lectures on random matrices given by P. Bickel, N. El-Karoui and A. Guionnet, and also some seminars at UC Berkeley. (# This talk is almost the same as the talk I gave at ISM on June 1.) |
Previous Seminar : Next Seminar | Seminar on Probability and Statistics |