Related papers: On $\ell_p$-Gaussian-Grothendieck problem

Motivated by the general matrix deviation inequality for i.i.d ensemble Gaussian matrix, we study its universality property. As a starting point for this problem, we show that this property holds for $\ell_{p}$-norm with $1\leq p< \infty$…

We study the free energy of a particle in (arbitrary) high-dimensional Gaussian random potentials with isotropic increments. We prove a computable saddle-point variational representation in terms of a Parisi-type functional for the free…

The subspace approximation problem Subspace($k$,$p$) asks for a $k$-dimensional linear subspace that fits a given set of points optimally, where the error for fitting is a generalization of the least squares fit and uses the $\ell_{p}$ norm…

In this paper, we consider the inverse Galois problem with described inertia behavior. For a finite group $G$, one of its subgroups $I$ and a prime integer $p$, we ask whether or not $G$ and $I$ can be realized as the Galois group and the…

While the Gibbs states of spin glass models have been noted to have an erratic dependence on temperature, one may expect the mean over the disorder to produce a continuously varying ``quenched state''. The assumption of such continuity in…

Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(\Delta_{i,n},\ 1\leq i\leq p)$, properly rescaled, and eventually included in any neighbourhood of the…

Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of size m x m, the positive semidefinite Grothendieck problem with rank-n-constraint (SDP_n) is maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1,…

We study the insulated conductivity problem with closely spaced insulators embedded in a homogeneous matrix where the current-electric field relation is the power law $J = |E|^{p-2}E$. The gradient of solutions may blow up as $\varepsilon$,…

We prove that for any $2<p<\infty$ and for every $n$-dimensional subspace $X$ of $L_p$, represented on $\mathbb R^n$, whose unit ball $B_X$ is in Lewis' position one has the following two-level Gaussian concentration inequality: \[ \mathbb…

The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from a variety of disciplines, ranging from genomics to social sciences. In this paper we provide a detailed asymptotic…

The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is…

We study the standard quadratic optimization problem over the simplex when the objective matrix is drawn from the Gaussian Orthogonal Ensemble (GOE). Let \(\kappa_n\) denote the support size of the almost surely unique global optimizer. We…

Let G be a compact group acting in a real vector space V. We obtain a number of inequalities relating the L^infinity norm of a matrix element of the representation of G with its L^p norm for p<infinity. We apply our results to obtain…

A formalism capable of handling the first step of hierarchical replica symmetry breaking in finite-connectivity models is introduced. The emerging order parameter is claimed to be a probability distribution over the space of field…

We consider a class of sparse random matrices which includes the adjacency matrix of the Erd\H{o}s-R\'enyi graph $\mathcal{G}(N,p)$. We show that if $N^{\varepsilon} \leq Np \leq N^{1/3-\varepsilon}$ then all nontrivial eigenvalues away…

Mean-field models of glasses that present a random first order transition exhibit highly non-trivial fluctuations. Building on previous studies that focused on the critical scaling regime, we here obtain a fully quantitative framework for…

Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in…

We consider the problem of computing a positive definite $p \times p$ inverse covariance matrix aka precision matrix $\theta=(\theta_{ij})$ which optimizes a regularized Gaussian maximum likelihood problem, with the elastic-net regularizer…

We consider eigenvalue problems for general elliptic operators of arbitrary order subject to homogeneous boundary conditions on open subsets of the euclidean N-dimensional space. We prove stability results for the dependence of the…

We consider maximum likelihood estimation for Gaussian Mixture Models (Gmms). This task is almost invariably solved (in theory and practice) via the Expectation Maximization (EM) algorithm. EM owes its success to various factors, of which…