The paper investigates restricted eigenvalue (RE) bounds for norm‑regularized estimators under heavy‑tailed designs. It shows that the previously conjectured sample‑size law based on Gaussian width fails for heavy‑tailed measurements, due to a phenomenon called simultaneous threshold occupancy. The authors provide explicit counterexamples, derive worst‑case sample‑complexity bounds, and compare the behavior of heavy‑tailed versus Gaussian designs on constant‑width polyhedral descent cones.
By Shi Fu, Huibo Xu, Qixin Zhang, Dacheng Tao
arXiv:2004. 05813v3 Announce Type: replace-cross Abstract: Suppose that we are given independent, identically distributed random samples $x_1,\cdots,x_n$ from a mixture at most $k$ many $d$-dimensional spherical Gaussian distributions $\mu_1,\cdots,\mu_{k_0}$ of identical and known variance $\sigma^2$ in each coordinate, such that the minimum $\ell^2$ distance between two distinct centers $y_l$ and $y_j$ is greater than $2\Delta\sigma \min\{\sqrt{d},\sqrt k\}$, where $\Delta>C_0$, and $C_0$ is a sufficiently large universal constant.
By Somnath Chakraborty, Hariharan Narayanan
arXiv:2609.27836v1 Announce Type: cross
Abstract: Let $P$ be an irreducible reversible Markov kernel on a $m$-state space $\Omega$, and denote its right spectral gap $\gamma=1-\lambda_2(P)$. From a s...
By Yanjin Xiang, Zhihua Zhang
arXiv:2609.06327v2 Announce Type: replace-cross
Abstract: A query-oblivious coreset for a softmax-attention head is a subset of the key-value pairs whose attention output is within $\varepsilon$ of t...
By Ofek I. Cohen
arXiv:2609.15179v1 Announce Type: cross
Abstract: The Gaussian kernel is a widely used similarity measure underlying kernel methods such as kernel PCA and spectral clustering, but computing Gaussian...
By Soumik Dutta, Kunal Dutta
arXiv:2609. 12594v1 Announce Type: new Abstract: We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for $\pi(\,\mathrm{d} x)\propto e^{-f(x)-g(x)}\,\mathrm{d} x$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g:\mathbb{R}^d\to\mathbb{R}$ is convex and globally $G$-Lipschitz.
By Yuchen Xin, Zhihua Zhang