Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On The Variance of Schatten $p$-Norm Estimation with Gaussian Sketching Matrices

Lior Horesh, Vasileios Kalantzis, Yingdong Lu, Tomasz Nowicki

Abstract

Monte Carlo matrix trace estimation is a popular randomized technique to estimate the trace of implicitly-defined matrices via averaging quadratic forms across several observations of a random vector. The most common approach to analyze the quality of such estimators is to consider the variance over the total number of observations. In this paper we present a procedure to compute the variance of the estimator proposed by Kong and Valiant [Ann. Statist. 45 (5), pp. 2218 - 2247] for the case of Gaussian random vectors and provide a sharper bound than previously available.

On The Variance of Schatten $p$-Norm Estimation with Gaussian Sketching Matrices

Abstract

Monte Carlo matrix trace estimation is a popular randomized technique to estimate the trace of implicitly-defined matrices via averaging quadratic forms across several observations of a random vector. The most common approach to analyze the quality of such estimators is to consider the variance over the total number of observations. In this paper we present a procedure to compute the variance of the estimator proposed by Kong and Valiant [Ann. Statist. 45 (5), pp. 2218 - 2247] for the case of Gaussian random vectors and provide a sharper bound than previously available.

Paper Structure

This paper contains 9 sections, 11 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Trace Lemma
  4. Recursion Structure of Trace Calculations
  5. Calculations of $N(q, \cdot, \cdot)$
  6. Calculations of $M(q, \cdot, \cdot)$
  7. Main Results
  8. Calculations of the Variance
  9. Bounds on the Variance

Key Result

Lemma 2.1

If $X$ is a Gaussian random vector of $d$ dimension, mean zero and covariance matrix $\Lambda$, and $A\in \mathbb R^{d\times d}$ is a fixed matrix, then,

Theorems & Definitions (14)

  • Definition 2.1
  • Example 1
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 5.1: Representation of the variance
  • Example 2
  • Lemma 5.1
  • ...and 4 more