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Improved bounds on sample size for implicit matrix trace

estimators

Farbod Roosta-Khorasani and Uri Ascher ∗

July 23, 2014

Abstract

This article is concerned with Monte-Carlo methods for the estimation of the trace of an implicitly given matrix A whose information is only available through matrix-vector products. Such a method approximates the trace by an average of N expressions of the form wt(Aw), with random vectors w drawn from an appropriate distribution. We prove, discuss and experiment with bounds on the number of realizations N required in order to guarantee a probabilistic bound on the relative error of the trace estimation upon employing Rademacher (Hutchinson), Gaussian and uniform unit vector (with and without replacement) probability distributions.

In total, one necessary bound and six sufficient bounds are proved, improving upon and extending similar estimates obtained in the seminal work of Avron and Toledo (2011) in several dimensions. We first improve their bound on N for the Hutchinson method, dropping a term that relates to rank(A) and making the bound comparable with that for the Gaussian estimator.

We further prove new sufficient bounds for the Hutchinson, Gaussian and the unit vector estimators, as well as a necessary bound for the Gaussian estimator, which depend more specifically on properties of the matrix A. As such they may suggest for what type of matrices one distribution or another provides a particularly effective or relatively ineffective stochastic estimation method.

Keywords: randomized algorithms, trace estimation, Monte-Carlo methods, implicit linear operators

Mathematics Subject Classification (2010): 65C20, 65C05, 68W20

Communicated by Emmanuel Candès.

1 Introduction

The need to estimate the trace of an implicit square matrix is of fundamental importance [15] and arises in many applications; see for instance [10, 5, 4, 9, 7, 18, 13, 11, 8, 3] and references

∗Dept. of Computer Science, University of British Columbia, Vancouver, Canada farbod/ascher@cs.ubc.ca . This work was supported in part by NSERC Discovery Grant 84306.

1

2

therein. By “implicit” we mean that the matrix of interest is not available explicitly: only probes in the form of matrix-vector products for any appropriate vector are available. The standard approach for estimating the trace of such a matrix A is based on a Monte-Carlo method, where one generates N random vector realizations wi from a suitable probability distribution D and computes

trND (A) := 1

N

N∑ i=1

wtiAwi. (1)

For the popular case where A is symmetric positive semi-definite (SPSD), the original method for estimating its trace, tr(A), is due to Hutchinson [10] and uses the Rademacher distribution for D.

Until the work by Avron and Toledo [4], the main analysis and comparison of such meth- ods was based on the variance of one sample. It is known that compared to other methods the Hutchinson method has the smallest variance, and as such it has been extensively used in many applications. In [4] so-called (ε, δ) bounds are derived in which, using Chernoff-like analysis, a lower bound is obtained on the number of samples required to achieve a prob- abilistically guaranteed relative error of the estimated trace. More specifically, for a given pair (ε, δ) of small (say, < 1) positive values and an appropriate probability distribution D, a lower bound on N is provided such that

Pr ( |trND (A)− tr(A)| ≤ ε tr(A)

) ≥ 1− δ. (2)

These authors further suggest that minimum-variance estimators may not be practically best, and conclude based on their analysis that the method with the best bound is the one using the Gaussian distribution. Let us denote

c = c(ε, δ) := ε−2 ln(2/δ), (3a)

r = rank(A). (3b)

Then [4] showed that, provided A is real SPSD, (2) holds for the Hutchinson method if N ≥ 6(c+ ε−2 ln r) and for the Gaussian distribution if N ≥ 20c.

In the present paper we continue to consider the same objective as in [4], and our first task is to improve on these bounds. Specifically, in Theorems 1 and 3 we show that (2) holds for the Hutchinson method if

N ≥ 6c(ε, δ), (4)

and for the Gaussian distribution if

N ≥ 8c(ε, δ). (5)

The bound (4) removes a previous factor involving the rank of the matrix A, conjectured in [4] to be indeed redundant. Note that these two bounds are astoundingly simple and general: they hold for any SPSD matrix, regardless of size or any other matrix property. Thus, we cannot expect them to be tight in practice for many specific instances of A that arise in applications.

3

Although practically useful, the bounds on N given in (4) and (5) do not provide insight into how different types of matrices are handled with each probability distribution. Our next contribution is to provide different bounds for the Gaussian and Hutchinson trace estimators which, though generally not computable for implicit matrices, do shed light on this question.

Furthermore, for the Gaussian estimator we prove a practically useful necessary lower bound on N , for a given pair (ε, δ).

A third probability distribution we consider was called the unit vector distribution in [4]. Here, the vectors wi in (1) are uniformly drawn from the columns of a scaled identity matrix,√ nI, and A need not be SPSD. We slightly generalize the bound in [4], obtained for the case

where the sampling is done with replacement. Our bound, although not as simply computed as (4) or (5), can be useful in determining which types of matrices this distribution works best on. We then give a tighter bound for the case where the sampling is done without replacement, suggesting that when the difference between the bounds is significant (which happens when N is large), a uniform random sampling of unit vectors without replacement may be a more advisable distribution to estimate the trace with.

This paper is organized as follows. Section 2 gives two bounds for the Hutchinson method as advertised above, namely the improved bound (4) and a more involved but potentially more informative bound. Section 3 deals likewise with the Gaussian method and adds also a necessary lower bound, while Section 4 is devoted to the unit vector sampling methods.

In Section 5 we give some numerical examples verifying that the trends predicted by the theory are indeed realized. Conclusions and further thoughts are gathered in Section 6.

In what follows we use the notation trNH (A), tr N G (A), tr

N U1

(A), and trNU2(A) to refer, respectively, to the trace estimators using Hutchinson, Gaussian, and uniform unit vector with and without replacement, in lieu of the generic notation trND (A) in (1) and (2). We also denote for any given random vector of size n, wi = (wi1, wi2, . . . , win)

t. We restrict attention to real-valued matrices, although extensions to complex-valued ones are possible, and employ the 2-norm by default.

2 Hutchinson estimator bounds

In this section we consider the Hutchinson trace estimator, trNH (A), obtained by setting D = H in (1), where the components of the random vectors wi are i.i.d Rademacher random variables (i.e., Pr(wij = 1) = Pr(wij = −1) = 12).

2.1 Improving the bound in [4]

Theorem 1 Let A be an n× n SPSD matrix. Given a pair (ε, δ), the inequality (2) holds with D = H if N satisfies (4).

Proof Since A is SPSD, it can be diagonalized by a unitary similarity transformation as A = U tΛU . Consider N random vectors wi, i = 1, . . . , N , whose components are i.i.d and drawn from the Rademacher distribution, and define zi = Uwi for each. We have

4

Pr ( trNH (A) ≤ (1− ε)tr(A)

) = Pr

( 1

N

N∑ i=1

wtiAwi ≤ (1− ε)tr(A)

)

= Pr

( 1

N

N∑ i=1

ztiΛzi ≤ (1− ε)tr(A)

)

= Pr

N∑ i=1

r∑ j=1

λjz 2 ij ≤ N(1− ε)tr(A)

= Pr

r∑ j=1

λj tr(A)

N∑ i=1

z2ij ≤ N(1− ε)

≤ exp{tN(1− ε)}E

exp{ r∑ j=1

λj tr(A)

N∑ i=1

−tz2ij}

, where the last inequality holds for any t > 0 by Markov’s inequality.

Next, using the convexity of the exp function and the linearity of expectation, we obtain

E

exp{ r∑ j=1

λj tr(A)

N∑ i=1

−tz2ij}

≤ r∑ j=1

λj tr(A)

E

( exp{

N∑ i=1

−tz2ij}

)

= r∑ j=1

λj tr(A)

E

( N∏ i=1

exp{−tz2ij}

)

=

r∑ j=1

λj tr(A)

N∏ i=1

E ( exp{−tz2ij}

) ,

where the last equality holds since, for a given j, zij ’s are independent with respect to i.

Now, we want to have that exp{tN(1−ε)} ∏N i=1 E

( exp{−tz2ij}

) ≤ δ/2. For this we make

use of the inequalities in the end of the proof of Lemma 5.1 of [2]. Following inequalities (15)–(19) in [2] and letting t = ε/(2(1 + ε)), we get

exp{tN(1− ε)} N∏ i=1

E ( exp{−tz2ij}

) < exp{−N

2 ( ε2

2 − ε

3

3 )}.

Next, if N satisfies (4) then exp{−N2 ( ε2

2 − ε3

3 )} < δ/2, and thus it follows that

Pr ( trNH (A) ≤ (1− ε)tr(A)

) < δ/2.

By a similar argument, making use of inequalities (11)–(14) in [2] with the same t as above, we also o