计算高斯过程上确界的一个多项式时间逼近格式
A polynomial time approximation scheme for computing the supremum of Gaussian processes
摘要：
给出了计算高斯过程上确界的一个多项式时间近似方案。也就是说，给定向量$V\subseteq\mathbb{R}^d$的有限集合，我们在时间上确定性地计算对$\mathop {\mathbb{E}}_{X\leftarrow\mathcal{N}^d}[\sup_{v\in V}|\langle v,X\rangle|]$的$(1+\varepsilon)$因子近似$\operatorname {poly}(d)\cdot|V|^{O_{\varepsilon}(1)}$。先前，由于Ding、Lee和Peres的工作，仅已知常数因子确定性多项式时间近似算法[Ann. of Math.（2）175（2012）1409-1471]。这回答了Lee（2010）和Ding [Ann. Probab. 42（2014）464-496]。高斯过程的上确界的研究在概率论中具有相当重要的意义，其应用于泛函分析、凸几何，并且鉴于Ding，Lee和Peres [Ann. of Math.（2）175（2012）1409-1471]最近对有限图上的随机游动的突破性工作。因此，我们的研究结果可以在其他地方使用。特别是，结合工作的丁[安。42（2014）464-496]，我们的结果产生用于计算有界度图的覆盖时间的PTAS。在此之前，这种算法只适用于树。同时，我们也给出了高斯空间中半范数的一个显式不经意估计，并使其具有最优的查询复杂度。我们的算法和分析是基本的性质，使用两个经典的比较不等式，Slepian的引理和坎特的引理。
英文摘要：
We give a polynomial time approximation scheme (PTAS) for computing the supremum of a Gaussian process. That is, given a finite set of vectors $V\subseteq\mathbb{R}^d$, we compute a $(1+\varepsilon)$-factor approximation to $\mathop {\mathbb{E}}_{X\leftarrow\mathcal{N}^d}[\sup_{v\in V}|\langle v,X\rangle|]$ 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.
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