Empirical Approximation of $L_p$ Norms

Abstract

We study empirical $L_p$ moments of a random vector $\pmb\varphi$ based on its i.i.d.\ copies $\pmb\varphi^1,\ldots,\pmb\varphi^m$, that is, $\frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p$. Our main result is a new estimate for the expected uniform deviation \[ \mathbb{E}\sup_{y\in D}\biggl| \frac1m\sum_{j=1}^m |\langle \pmb\varphi^j,y\rangle|^p -\mathbb{E}|\langle \pmb\varphi,y\rangle|^p \biggr| \] over an arbitrary index set $D$. The proof is based on a new bound for Talagrand's $\gamma$-functional, sharper than the standard Dudley-type entropy estimate. We then apply this estimate to the following two problems. First, for $p>2$, we study Marcinkiewicz-type discretization of $L_p$ norms on an $N$-dimensional subspace $X_N\subset B(\Omega)$ of bounded functions on a probability space $(\Omega,\mu)$. We obtain bounds in terms of the norm of the embedding $ (X_N,\|\cdot\|_{L_p(\mu)})\hookrightarrow B(\Omega). $ In particular, we prove that when this norm is of order $N^{1/p}$ and \[ m \ge C(p)\, N\log N\,(\log\log N)^{p-1}, \] then $m$ random samples suffice to approximate the $L_p(\mu)$ norm uniformly on $X_N$ by the sampled discrete $L_p$ norm. This substantially improves the previously known bound in this setting $ m \ge C(p)\, N(\log N)^{\min\{p,3\}}, $ and is optimal up to the factor $(\log\log N)^{p-1}$ in the random-sampling setting. Second, for $1\le p<2$, we obtain an $L_p$ analogue of the restricted isometry property via random sampling for bounded orthogonal systems and, more generally, for $N$-element systems $\mathcal D_N$ satisfying a Riesz-type condition. We prove that when \[ m \ge C(p)\, s\log N\,(\log s)^2\,\log\log s, \] then $m$ random samples suffice to guarantee an $L_p$ restricted isometry-type property uniformly over the class of all $s$-sparse functions generated by $\mathcal D_N$.