相关论文: The Structure of Extremal Bad Science Matrices
We consider quadratic forms of deterministic matrices $A$ evaluated at the random eigenvectors of a large $N \times N$ GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove…
In this work we establish under certain hypotheses the $N \to +\infty$ asymptotic expansion of integrals of the form $$\mathcal{Z}_{N,\Gamma}[V] \, = \, \int_{\Gamma^N} \prod_{ a < b}^{N}(z_a - z_b)^\beta \, \prod_{k=1}^{N} \mathrm{e}^{ - N…
We study robust linear regression in high-dimension, when both the dimension $d$ and the number of data points $n$ diverge with a fixed ratio $\alpha=n/d$, and study a data model that includes outliers. We provide exact asymptotics for the…
We consider the sequential composite binary hypothesis testing problem in which one of the hypotheses is governed by a single distribution while the other is governed by a family of distributions whose parameters belong to a known set…
We consider the rate of convergence of the expected loss of empirically optimal vector quantizers. Earlier results show that the mean-squared expected distortion for any fixed distribution supported on a bounded set and satisfying some…
In this note, we study the asymptotics of the determinant $\det(I_N - \beta H_N)$ for $N$ large, where $H_N$ is the $N\times N$ restriction of a Hankel matrix $H$ with finitely many jump discontinuities in its symbol satisfying $\|H\|\leq…
We consider the lossless compression bound of any individual data sequence. If we fit the data by a parametric model, the entropy quantity $nH({\hat \theta}_n)$ obtained by plugging in the maximum likelihood estimate is an underestimate of…
In our previous works, we proved that the inverse of the stiffness matrix of an $h$-version finite element method (FEM) applied to scalar second order elliptic boundary value problems can be approximated at an exponential rate in the block…
We report new world records for the maximal determinant of an n-by-n matrix with entries +/-1. Using various techniques, we beat existing records for n=22, 23, 27, 29, 31, 33, 34, 35, 39, 45, 47, 53, 63, 69, 73, 77, 79, 93, and 95, and we…
Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these…
A matrix is called Bohemian if its entries are sampled from a finite set of integers. We determine the maximum absolute determinant of upper Hessenberg Bohemian Matrices for which the subdiagonal entries are fixed to be $1$ and upper…
We study the high-dimensional asymptotics of empirical risk minimization (ERM) in over-parametrized two-layer neural networks with quadratic activations trained on synthetic data. We derive sharp asymptotics for both training and test…
We provide optimal upper bounds on the growth of iterated sumsets $hA=A+\dots+A$ for finite subsets $A$ of abelian semigroups. More precisely, we show that the new upper bounds recently derived from Macaulay's theorem in commutative algebra…
By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…
We provide a statistical analysis of regularization-based continual learning on a sequence of linear regression tasks, with emphasis on how different regularization terms affect the model performance. We first derive the convergence rate…
A classical result of Steinitz from 1913 \cite{Ste13}, answering an earlier question of Riemann and L\'evy (e.g., \cite{Lev05}), states that for any norm $\|\cdot\|$ in $\mathbb{R}^d$ and any set of vectors $v_1, \cdots, v_n \in \R^d$…
Advanced science and technology provide a wealth of big data from different sources for extreme value analysis. Classical extreme value theory was extended to obtain an accelerated max-stable distribution family for modelling competing…
The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to…
We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix $A \in [0,1]^{m \times n}$, a vector $b \in [1,\infty)^m$, and a monotone…
We study maximal estimates for the wave equation with orthonormal initial data. In dimension $d=3$, we establish optimal results with the sharp regularity exponent up to the endpoint. In higher dimensions $d \ge 4$ and also in $d=2$, we…