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Some random matrix problems in high-dimensional Statistics

Mathematical Physics & Probability

Speaker: Noureddine El Karoui, UC Berkeley Statistics Department
Location: 1147 MSB
Start time: Wed, Nov 28 2007, 4:10PM

In the first part of the talk, I will talk about ``kernel" random matrices, random matrices that arise in Statistics and computer science. Their spectral properties have been investigated for low (or fixed) dimensional data vectors, but not in the high-dimensional setting that is now sometimes of interest in Statistics. I will describe the limiting spectra of a class of such kernel random matrices used in practice, for data vectors sampled from models classically studied in RMT. Interestingly, the results may be interpreted as indicating that certain heuristics occasionally advocated in practice do not give good insights for high-dimensional problems. The analysis also highlights some potential statistical limitations of the ``standard" RMT models. In the second part of the talk, I will discuss the theoretical aspects of an algorithm I proposed fairly recently to estimate the population spectral distribution of a covariance matrix from the observed spectral distribution of a high-dimensional sample covariance matrix (by means of a classic result of Marchenko-Pastur and convex optimization). The proof relies on various properties of Stieltjes transforms.