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Register a new userOn the Distribution of an Arbitrary Subset of the Eigenvalues for some Finite Dimensional Random Matrices
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Added by
Marco Chiani Dr.
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.
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We consider the problem of learning a coefficient vector x_0in R^N from noisy linear observation y=Ax_0+w in R^n. In many contexts (ranging from model selection to image processing) it is desirable to construct a sparse estimator x. In this case, a popular approach consists in solving an L1-penalized least squares problem known as the LASSO or Basis Pursuit DeNoising (BPDN). For sequences of matrices A of increasing dimensions, with independent gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical models ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications.
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