Sparsifying Suprema of Gaussian Processes

We give a dimension-independent sparsification result for suprema of centered Gaussian processes: Let $T$ be any (possibly infinite) bounded set of vectors in $\mathbb{R}^n$, and let $\{{\boldsymbol{X}}_t\}_{t\in T}$ be the canonical Gaussian process on $T$. We show that there is an $O_\varepsilon(1)$-size subset $S \subseteq T$ and a set of real values $\{c_s\}_{s \in S}$ such that $\sup_{s \in S} \{{\boldsymbol{X}}_s + c_s\}$ is an $\varepsilon$-approximator of $\sup_{t \in T} {\boldsymbol{X}}_t$. Notably, the size of $S$ is completely independent of both the size of $T$ and of the ambient dimension $n$. We use this to show that every norm is essentially a junta when viewed as a function over Gaussian space: Given any norm $\nu(x)$ on $\mathbb{R}^n$, there is another norm $\psi(x)$ which depends only on the projection of $x$ along $O_\varepsilon(1)$ directions, for which $\psi({\boldsymbol{g}})$ is a multiplicative $(1 \pm \varepsilon)$-approximation of $\nu({\boldsymbol{g}})$ with probability $1-\varepsilon$ for ${\boldsymbol{g}} \sim N(0,I_n)$. We also use our sparsification result for suprema of centered Gaussian processes to give a sparsification lemma for convex sets of bounded geometric width: Any intersection of (possibly infinitely many) halfspaces in $\mathbb{R}^n$ that are at distance $O(1)$ from the origin is $\varepsilon$-close, under $N(0,I_n)$, to an intersection of only $O_\varepsilon(1)$ many halfspaces. We describe applications to agnostic learning and tolerant property testing.