Sampling from large matrices: an approach through geometric functional analysis

We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix of the smallest possible size O(r log r) with a small error in the spectral norm, where r = ||A||_F^2 / ||A||_2^2 is the numerical rank of A. The numerical rank is always bounded by, and is a stable relaxation of, the rank of A. This yields an asymptotically optimal guarantee in an algorithm for computing low-rank approximations of A. We also prove asymptotically optimal estimates on the spectral norm and the cut-norm of random submatrices of A. The result for the cut-norm yields a slight improvement on the best known sample complexity for an approximation algorithm for MAX-2CSP problems. We use methods of Probability in Banach spaces, in particular the law of large numbers for operator-valued random variables.Comment: Our initial claim about Max-2-CSP problems is corrected. We put an exponential failure probability for the algorithm for low-rank approximations. Proofs are a little more explaine

Fast computation of the matrix exponential for a Toeplitz matrix

The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured $n\times n$ matrix it can be computed in $\mathcal{O}(n^3)$ operations. An interesting problem arises if the input matrix is a Toeplitz matrix, for example as the result of discretizing integral equations with a time invariant kernel. In this case it is not obvious how to take advantage of the Toeplitz structure, as the exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself. The main contribution of this work are fast algorithms for the computation of the Toeplitz matrix exponential. The algorithms have provable quadratic complexity if the spectrum is real, or sectorial, or more generally, if the imaginary parts of the rightmost eigenvalues do not vary too much. They may be efficient even outside these spectral constraints. They are based on the scaling and squaring framework, and their analysis connects classical results from rational approximation theory to matrices of low displacement rank. As an example, the developed methods are applied to Merton's jump-diffusion model for option pricing

Computing approximate PSD factorizations

We give an algorithm for computing approximate PSD factorizations of nonnegative matrices. The running time of the algorithm is polynomial in the dimensions of the input matrix, but exponential in the PSD rank and the approximation error. The main ingredient is an exact factorization algorithm when the rows and columns of the factors are constrained to lie in a general polyhedron. This strictly generalizes nonnegative matrix factorizations which can be captured by letting this polyhedron to be the nonnegative orthant.Comment: 10 page

Wishart Mechanism for Differentially Private Principal Components Analysis

We propose a new input perturbation mechanism for publishing a covariance matrix to achieve $(\epsilon,0)$-differential privacy. Our mechanism uses a Wishart distribution to generate matrix noise. In particular, We apply this mechanism to principal component analysis. Our mechanism is able to keep the positive semi-definiteness of the published covariance matrix. Thus, our approach gives rise to a general publishing framework for input perturbation of a symmetric positive semidefinite matrix. Moreover, compared with the classic Laplace mechanism, our method has better utility guarantee. To the best of our knowledge, Wishart mechanism is the best input perturbation approach for $(\epsilon,0)$-differentially private PCA. We also compare our work with previous exponential mechanism algorithms in the literature and provide near optimal bound while having more flexibility and less computational intractability.Comment: A full version with technical proofs. Accepted to AAAI-1