A POLYNOMIAL TIME APPROXIMATION SCHEME FOR COMPUTING THE SUPREMUM OF GAUSSIAN PROCESSES

• 成果类型: Article
• 署名作者: Meka, Raghu
• 署名单位: Microsoft
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP997
• 发表日期: 2015
• 页码: 465-476
• 关键词: cover times

摘要

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 subset of R-d, we compute a (1 + epsilon)-factor approximation to E-X <- N-d [sup(v) (is an element of V) vertical bar < v, X >vertical bar] deterministically in time poly(d) . vertical bar V vertical bar(O epsilon(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 seminorms 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.