On sparse representations of linear operators and the approximation of matrix products

Thus far, sparse representations have been exploited largely in the context of robustly estimating functions in a noisy environment from a few measurements. In this context, the existence of a basis in which the signal class under consideration is sparse is used to decrease the number of necessary measurements while controlling the approximation error. In this paper, we instead focus on applications in numerical analysis, by way of sparse representations of linear operators with the objective of minimizing the number of operations needed to perform basic operations (here, multiplication) on these operators. We represent a linear operator by a sum of rank-one operators, and show how a sparse representation that guarantees a low approximation error for the product can be obtained from analyzing an induced quadratic form. This construction in turn yields new algorithms for computing approximate matrix products.Comment: 6 pages, 3 figures; presented at the 42nd Annual Conference on Information Sciences and Systems (CISS 2008

An Efficient Parallel Solver for SDD Linear Systems

We present the first parallel algorithm for solving systems of linear equations in symmetric, diagonally dominant (SDD) matrices that runs in polylogarithmic time and nearly-linear work. The heart of our algorithm is a construction of a sparse approximate inverse chain for the input matrix: a sequence of sparse matrices whose product approximates its inverse. Whereas other fast algorithms for solving systems of equations in SDD matrices exploit low-stretch spanning trees, our algorithm only requires spectral graph sparsifiers

Geometrical inverse preconditioning for symmetric positive definite matrices

We focus on inverse preconditioners based on minimizing $F(X) = 1-\cos(XA,I)$, where $XA$ is the preconditioned matrix and $A$ is symmetric and positive definite. We present and analyze gradient-type methods to minimize $F(X)$ on a suitable compact set. For that we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of $F(X)$ on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included

Practical Gauss-Newton Optimisation for Deep Learning

We present an efficient block-diagonal ap- proximation to the Gauss-Newton matrix for feedforward neural networks. Our result- ing algorithm is competitive against state- of-the-art first order optimisation methods, with sometimes significant improvement in optimisation performance. Unlike first-order methods, for which hyperparameter tuning of the optimisation parameters is often a labo- rious process, our approach can provide good performance even when used with default set- tings. A side result of our work is that for piecewise linear transfer functions, the net- work objective function can have no differ- entiable local maxima, which may partially explain why such transfer functions facilitate effective optimisation.Comment: ICML 201

Computationally efficient approximations of the joint spectral radius

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper a procedure for approximating the joint spectral radius of a finite set of matrices with arbitrary high accuracy. Our approximation procedure is polynomial in the size of the matrices once the number of matrices and the desired accuracy are fixed

Covariance Estimation in High Dimensions via Kronecker Product Expansions

This paper presents a new method for estimating high dimensional covariance matrices. The method, permuted rank-penalized least-squares (PRLS), is based on a Kronecker product series expansion of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. For covariance matrices of low separation rank, our results establish that PRLS has significantly faster convergence than the standard sample covariance matrix (SCM) estimator. The convergence rate captures a fundamental tradeoff between estimation error and approximation error, thus providing a scalable covariance estimation framework in terms of separation rank, similar to low rank approximation of covariance matrices. The MSE convergence rates generalize the high dimensional rates recently obtained for the ML Flip-flop algorithm for Kronecker product covariance estimation. We show that a class of block Toeplitz covariance matrices is approximatable by low separation rank and give bounds on the minimal separation rank $r$ that ensures a given level of bias. Simulations are presented to validate the theoretical bounds. As a real world application, we illustrate the utility of the proposed Kronecker covariance estimator for spatio-temporal linear least squares prediction of multivariate wind speed measurements.Comment: 47 pages, accepted to IEEE Transactions on Signal Processin