Norm and trace estimation with random rank-one vectors

04/14/2020
by   Zvonimir Bujanović, et al.
0

A few matrix-vector multiplications with random vectors are often sufficient to obtain reasonably good estimates for the norm of a general matrix or the trace of a symmetric positive semi-definite matrix. Several such probabilistic estimators have been proposed and analyzed for standard Gaussian and Rademacher random vectors. In this work, we consider the use of rank-one random vectors, that is, Kronecker products of (smaller) Gaussian or Rademacher vectors. It is not only cheaper to sample such vectors but it can sometimes also be much cheaper to multiply a matrix with a rank-one vector instead of a general vector. In this work, theoretical and numerical evidence is given that the use of rank-one instead of unstructured random vectors still leads to good estimates. In particular, it is shown that our rank-one estimators multiplied with a modest constant constitute, with high probability, upper bounds of the quantity of interest. Partial results are provided for the case of lower bounds. The application of our techniques to condition number estimation for matrix functions is illustrated.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/20/2020

On randomized trace estimates for indefinite matrices with an application to determinants

Randomized trace estimation is a popular and well studied technique that...
research
06/16/2016

Estimation of matrix trace using machine learning

We present a new trace estimator of the matrix whose explicit form is no...
research
01/26/2022

Stochastic diagonal estimation: probabilistic bounds and an improved algorithm

We study the problem of estimating the diagonal of an implicitly given m...
research
05/24/2022

Approximate Euclidean lengths and distances beyond Johnson-Lindenstrauss

A classical result of Johnson and Lindenstrauss states that a set of n h...
research
05/12/2018

Randomization Approaches for Reducing PAPR with Partial Transmit Sequences and Semidefinite Relaxation

To reduce peak-to-average power ratio, we propose a method to choose a s...
research
09/30/2022

Optimal Query Complexities for Dynamic Trace Estimation

We consider the problem of minimizing the number of matrix-vector querie...
research
10/09/2018

Matrix-free construction of HSS representation using adaptive randomized sampling

We present new algorithms for the randomized construction of hierarchica...

Please sign up or login with your details

Forgot password? Click here to reset