An Improved Taylor Algorithm for Computing the Matrix Logarithm

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson–Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison.

On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

Randomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

COMPUTING THE MATRIX POWERS OF MATRIX

In this paper, considering fractional matrix power definition, we define matrix powers of matrixusing matrix exponential and matrix logarithm. Finally present a guide for computingthe matrix powers of matrix.

Sparsity induced by covariance transformation: some deterministic and probabilistic results

Motivated by statistical challenges arising in modern scientific fields, notably genomics, this paper seeks embeddings in which relevant covariance models are sparse. The work exploits a bijective mapping between a strictly positive definite matrix and its orthonormal eigen-decomposition, and between an orthonormal eigenvector matrix and its principle matrix logarithm. This leads to a representation of covariance matrices in terms of skew-symmetric matrices, for which there is a natural basis representation, and through which sparsity is conveniently explored. This theoretical work establishes the possibility of exploiting sparsity in the new parametrization and converting the conclusion back to the one of interest, a prospect of high relevance in statistics. The statistical aspects associated with this operation, while not a focus of the present work, are briefly discussed.

A polyconvex extension of the logarithmic Hencky strain energy

Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function [Formula: see text] which is equal to the classical Hencky strain energy [Formula: see text] in a neighborhood of the identity matrix 𝟙; here, [Formula: see text] denotes the set of [Formula: see text]-matrices with positive determinant, [Formula: see text] denotes the deformation gradient, [Formula: see text] is the corresponding stretch tensor, [Formula: see text] is the principal matrix logarithm of [Formula: see text], [Formula: see text] is the trace operator, [Formula: see text] is the Frobenius matrix norm and [Formula: see text] is the deviatoric part of [Formula: see text]. The extension can also be chosen to be coercive, in which case Ball’s classical theorems for the existence of energy minimizers under appropriate boundary conditions are immediately applicable. We also generalize the approach to energy functions [Formula: see text] in the so-called Valanis–Landel form [Formula: see text] with [Formula: see text], where [Formula: see text] denote the singular values of [Formula: see text].