Minimax and irreducible matrices

Irreducible Toeplitz and Hankel matrices

An infinite matrix is called irreducible if its directed graph is strongly connected. It is proved that an infinite Toeplitz matrix is irreducible if and only if almost every finite leading submatrix is irreducible. An infinite Hankel matrix may be irreducible even if all its finite leading submatrices are reducible. Irreducibility results are also obtained in the finite cases.

Minimax Estimation of Large Covariance Matrices 1369

Sharp minimax tests for large covariance matrices

We consider the detection problem of correlations in a p-dimensional Gaussian vector for p large, when we observe n independent, identically distributed random vectors. We assume that the covariance matrix vary in some ellipsoid with parameter α > 1/2 and total energy bounded by L > 0. We prove here both rate and sharp asymptotic results in the minimax setup. Our test procedure is a U-statistic...

Minimax Estimation of Bandable Precision Matrices

The inverse covariance matrix provides considerable insight for understanding statistical models in the multivariate setting. In particular, when the distribution over variables is assumed to be multivariate normal, the sparsity pattern in the inverse covariance matrix, commonly referred to as the precision matrix, corresponds to the adjacency matrix representation of the Gauss-Markov graph, wh...

Hoffman Polynomials of Nonnegative Irreducible Matrices and Strongly Connected Digraphs

For a nonnegative n× n matrix A, we find that there is a polynomial f(x) ∈ R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative...