## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 25, Number 2 (2015), 465-476.

### A polynomial time approximation scheme for computing the supremum of Gaussian processes

#### Abstract

We give a polynomial time approximation scheme (PTAS) for computing the supremum of a Gaussian process. That is, given a finite set of vectors $V\subseteq\mathbb{R}^{d}$, we compute a $(1+\varepsilon)$-factor approximation to $\mathbb{E}_{X\leftarrow\mathcal{N}^{d}}[\sup_{v\in V}|\langle v,X\rangle|]$ deterministically in time $\operatorname{poly} (d)\cdot|V|^{O_{\varepsilon}(1)}$. Previously, only a constant factor deterministic polynomial time approximation algorithm was known due to the work of Ding, Lee and Peres [*Ann. of Math.* (*2*) **175** (2012) 1409–1471]. This answers an open question of Lee (2010) and Ding [*Ann. Probab.* **42** (2014) 464–496].

The study of supremum of Gaussian processes is of considerable importance in probability with applications in functional analysis, convex geometry, and in light of the recent breakthrough work of Ding, Lee and Peres [*Ann. of Math.* (*2*) **175** (2012) 1409–1471], to random walks on finite graphs. As such our result could be of use elsewhere. In particular, combining with the work of Ding [*Ann. Probab.* **42** (2014) 464–496], our result yields a PTAS for computing the cover time of bounded-degree graphs. Previously, such algorithms were known only for trees.

Along the way, we also give an explicit oblivious estimator for semi-norms in Gaussian space with optimal query complexity. Our algorithm and its analysis are elementary in nature, using two classical *comparison inequalities*, Slepian’s lemma and Kanter’s lemma.

#### Article information

**Source**

Ann. Appl. Probab., Volume 25, Number 2 (2015), 465-476.

**Dates**

First available in Project Euclid: 19 February 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1424355120

**Digital Object Identifier**

doi:10.1214/13-AAP997

**Mathematical Reviews number (MathSciNet)**

MR3313745

**Zentralblatt MATH identifier**

1323.60056

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40]

**Keywords**

Gaussian processes derandomization cover time random walks $\varepsilon$-nets

#### Citation

Meka, Raghu. A polynomial time approximation scheme for computing the supremum of Gaussian processes. Ann. Appl. Probab. 25 (2015), no. 2, 465--476. doi:10.1214/13-AAP997. https://projecteuclid.org/euclid.aoap/1424355120