Abstract
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 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.
Highlights
This paper is concerned with approximating the trace of a symmetric matrix B ∈ Rn×n that is accessible only implicitly via matrix-vector products or, more precisely, quadratic forms
If X is a random vector of length n such that E[X ] = 0 and E[X X T ] = I, E[X T B X ] = tr(B)
Most existing non-asymptotic results for trace estimation are specific to an symmetric positive definite (SPD) matrix B; see [4,22,36] for examples
Summary
This paper is concerned with approximating the trace of a symmetric matrix B ∈ Rn×n that is accessible only implicitly via matrix-vector products or, more precisely,. Most existing non-asymptotic results for trace estimation are specific to an SPD matrix B; see [4,22,36] for examples They provide a bound on the estimated number N of probe vectors to ensure a small relative error with high probability:. Specialized to determinant computation, we combine our results with an improved analysis of the Lanczos method, to get a sharper total error bound for Rademacher vectors. We remark that some of our results are potentially of wider interest, beyond stochastic trace and determinant estimation, such as a tail bound for Rademacher chaos (Theorem 2) and an error bound (Corollary 3 combined with Corollary 5) on the polynomial approximation of the logarithm.
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