This work develops an approach to the spectral norm of the matrix product $\Pi U_V$, based on entropy estimates for level sets of vectors $x\in V$, and shows that matching results hold for other random models with negatively associated entries.
Abstract
Let $\Pi$ be a $k\times n$ sparse random matrix. For a fixed $r$-dimensional subspace $V\subset{\mathbb R}^n$, let $U_V:{\mathbb R}^r\to{\mathbb R}^n$ denote an isometry from ${\mathbb R}^r$ onto $V$. The product $\Pi U_V$ is a central model in randomized dimension reduction and has been studied primarily through trace and Gaussian comparison inequalities. In this work, we develop an approach to the spectral norm of the matrix product $\Pi U_V$, based on entropy estimates for level sets of vectors $x\in V$. Combining the method with existing estimates, we show the following. Assume that \[ k\ge C\,r(\log\log r)^2,\qquad p\ge (\log k)/k. \] Let $\Pi$ be a $k\times n$ matrix with i.i.d. entries equidistributed with the product $b\,\xi$, where $b$ is a Bernoulli($p$) random variable and $\xi$ is mean-zero, independent of $b$, and satisfies $|\xi|\le1$ almost surely. Then with high probability \[ \|\Pi U_V\|\le C\sqrt{kp}. \] Matching results hold for other random models with negatively associated entries.
Given a subset $A \subseteq \mathbb F_2^n$, we can consider the distribution of the intersection size of $A$ with a uniformly random $d$-flat $F$. Motivated by the edge statistics problem and the hypercube statistics problem, the affine subspace statistics problem concerns the maximum of $\mathbb{P}[|F\cap A|=s]$ among $A \subseteq \mathbb F_2^n$ for any fixed $s\in\{1,\dots,2^d\}$ over a uniformly random $d$-flat $F$. We use $\lambda^*(d,s)$ to denote the limit of the maximum when $n$ goes to infinity. In this note, we prove tight bounds for $\lambda^*(d,s)$ in two different regimes. For $s=j2^k$ where $j$ is a positive odd integer, the best known lower bound construction achieving $\lambda^*(d,s)\ge 1-2^{-k}$ is due to taking $A$ as the union of $j$ parallel $(n-d+k)$-flats in $\mathbb F_2^n$. Our main result is a matching upper bound with an additive error term of $O(2^{-3k/2})$. We also study the case $s=1$, where we determine $\lambda^*(d,1)$ exactly. We show that the random construction where each point is included with probability $2^{-d}$ is optimal.
Ting-Wei Chao, Zixuan Xu, D. Zakharov· 0 citations
Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In particular, this resolves a question of E.M. Stein from the mid 1990s. The principal ingredient in our proof is that, when $t\lesssim d$ with $t$ sufficiently large, the associated multiplier $\mathfrak{m}_{\sqrt{\lfloor t^2\rfloor}}(\xi)$ admits an asymptotic expansion of arbitrary prescribed order, uniform in $\xi$, whose resulting maximal operators can be controlled by the discrete normalized Gaussian maximal function studied by Mirek--Szarek--Wr\'obel \cite{MSW25}.
Let $V$ be an $n$-dimensional vector space over a finite field of order $q$. Let $r\geq 3$, $(r-1)n\geq rk$ and let $\mathcal F_1,\ldots,\mathcal F_r\subset \genfrac{[}{]}{0pt}{}{V}{k}$, where $\genfrac{[}{]}{0pt}{}{V}{k}$ denotes the set of $k$-dimensional subspaces of $V$. Suppose that $F_1\cap\cdots\cap F_r\neq\{0\}$ holds for all $F_i\in\mathcal F_i$, $1\leq i\leq r$. Then we show that $\prod_{i=1}^r|\mathcal F_i|\leq\genfrac{[}{]}{0pt}{}{n-1}{k-1}$, provided $n-k$ is sufficiently large for fixed $q$ and $r$. Moreover, equality holds if and only if there is a common line $L$ such that every family $\mathcal F_i$ consists of all $k$-dimensional subspaces containing the line $L$. One of the main tools of the proof is a junta theorem concerning intersecting linear maps obtained by Ellis, Kindler, and Lifshitz.
We prove an improved restricted isometry bound for Gaussian partial circulant matrices with arbitrary prescribed sampling sets. There is a universal constant $C>0$ such that the following holds. Let $1\leq K\leq m\leq N$ be positive integers, let $\Omega\subset\mathbb Z_N$ be any fixed set with $|\Omega|=m$, and let $g\sim\mathcal N(0,I_N)$. For every $\delta,\eta\in(0,1)$, the normalized partial circulant matrix generated by $g$ has the RIP of order $K$ with constant at most $\delta$, with probability at least $1-\eta$ over the draw of $g$, provided \[ m\geq C\delta^{-2}K \max\{\log^2(eK)\log(2N)\log(em),\log(2/\eta)\}. \] The proof refines the Maurey entropy step in the chaos-process argument by combining a noncommutative Khintchine inequality with a Schatten moment estimate controlled by $m$, replacing one factor $\log(2N)$ in the Krahmer--Mendelson--Rauhut bound by $\log(em)$.
Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\textsf{E}f(X^{\otimes d})\right\rvert>t \right\} \le C\exp\left[-c_K\mathcal I_{n,d}\left( \frac{t}{L n^{(d-1)/2}} \right)\right], \] where \[ \mathcal I_{n,d}(s)= \min\left\{ \frac{s^2}{d^2}, \frac{s^2}{d\log(e+nd/s^2)} \right\},\qquad s>0, \qquad \mathcal I_{n,d}(0)=0. \] The first rate is forced by changes in $\|X\|$. The second comes from changes of $X$ when its norm is nearly fixed. The proof constructs one coupling that controls both the coordinatewise conditional displacement and the mean squared Euclidean distance, and combines these bounds with a second-order estimate for $x\mapsto x^{\otimes d}$. The rate is minimax sharp, scale by scale, even when the subgaussian norms are bounded by an absolute constant. For bounded coordinates the logarithm in the second rate disappears.
In this paper, we study the sample complexity of the empirical plug-in estimator for the $2$-Gromov-Wasserstein distance $D_2$ between compactly supported probability measures on Euclidean spaces. Let $\mu$ and $\nu$ be supported on compact subsets of $\mathbb{R}^{d_x}$ and $\mathbb{R}^{d_y}$, respectively, and let $\widehat\mu_n$ and $\widehat\nu_n$ be their empirical measures based on independent samples of size $n$. We prove that \[ \mathbb{E}\left|D_2^2(\widehat\mu_n,\widehat\nu_n)-D_2^2(\mu,\nu)\right| \lesssim n^{-2/((d_x\wedge d_y)\vee 4)} (\log n)^{\mathbf 1_{\{d_x\wedge d_y=4\}}}. \] This rate is sharp up to the logarithmic factor in the critical dimension. The proof is based on a geometric representation of the Euclidean distance as a squared $L^2$-distance between half-space feature maps. This yields a variational dual formulation of the Gromov-Wasserstein functional in terms of a family of classical optimal transport problems indexed by an infinite-dimensional auxiliary parameter. Although the resulting cost functions need not be semiconcave in either argument, we introduce a marginal recentering of the costs that restores the concavity structure needed for sharp metric-entropy bounds. Combining this representation with empirical-process estimates gives a rate governed by the smaller of the two ambient dimensions.
Pui-Kuen Leung, Riku Okada, Samuel Lok-Hei Wong· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
