Related papers: An error bound for Lasso and Group Lasso in high d…

While Weighted Lasso sparse regression has appealing statistical guarantees that would entail a major real-world impact in finance, genomics, and brain imaging applications, it is typically scarcely adopted due to its complex…

We consider the theory for the high-dimensional generalized linear model with the Lasso. After a short review on theoretical results in literature, we present an extension of the oracle results to the case of quasi-likelihood loss. We prove…

A fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre $\beta$ ensemble, characterised by the Dyson parameter $\beta$, and the Laguerre…

We consider a sparse linear regression model Y=X\beta^{*}+W where X has a Gaussian entries, W is the noise vector with mean zero Gaussian entries, and \beta^{*} is a binary vector with support size (sparsity) k. Using a novel conditional…

We provide optimal lower bounds for two well-known parameter estimation (also known as statistical estimation) tasks in high dimensions with approximate differential privacy. First, we prove that for any $\alpha \le O(1)$, estimating the…

This paper establishes a statistical versus computational trade-off for solving a basic high-dimensional machine learning problem via a basic convex relaxation method. Specifically, we consider the {\em Sparse Principal Component Analysis}…

Exponential generalization bounds with near-tight rates have recently been established for uniformly stable learning algorithms. The notion of uniform stability, however, is stringent in the sense that it is invariant to the data-generating…

We give very precise bounds for the congruence subgroup growth of arithmetic groups. This allows us to determine the subgroup growth of irreducible lattices of semisimple Lie groups. In the most general case our results depend on the…

Let $G$ be a simple non-compact linear Lie group. Let $\pi$ be any irreducible unitary representation of $G$ with infinitesimal character $\Lambda$ whose continuous part is $\nu$. The beautiful Helgason-Jonson bound in 1969 says that the…

Let M_n denote the number of sites in the largest cluster in critical site percolation on the triangular lattice inside a box side length n. We give lower and upper bounds on the probability that M_n / E(M_n) > x of the form exp(- C…

The factorization norms of the lower-triangular all-ones $n \times n$ matrix, $\gamma_2(M_{count})$ and $\gamma_{F}(M_{count})$, play a central role in differential privacy as they are used to give theoretical justification of the accuracy…

We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form…

We consider estimation in a high-dimensional linear model with strongly correlated variables. We propose to cluster the variables first and do subsequent sparse estimation such as the Lasso for cluster-representatives or the group Lasso…

We show that computing the lattice programming gap of the group problems is NP-hard when the dimension is a part of input. We also obtain lower and upper bounds for the gap in terms of the cost vector and the determinant of the lattice.

We introduce the spike-and-slab group lasso (SSGL) for Bayesian estimation and variable selection in linear regression with grouped variables. We further extend the SSGL to sparse generalized additive models (GAMs), thereby introducing the…

We propose a residual randomization procedure designed for robust Lasso-based inference in the high-dimensional setting. Compared to earlier work that focuses on sub-Gaussian errors, the proposed procedure is designed to work robustly in…

The height gap theorem states that the finite subsets $F$ of matrices generating non-virtually solvable groups have normalized height $\widehat{h}(F)$ bounded below by a constant. It was first proved by Breuillard and another proof was…

We compute the second and third levels of the Lasserre hierarchy for the spherical finite distance problem. A connection is used between invariants in representations of the orthogonal group and representations of the general linear group,…

Large reasoning models (LRMs) exhibit diverse high-level reasoning patterns (e.g., direct solution, reflection-and-verification, and exploring multiple solutions), yet prevailing training recipes implicitly bias models toward a limited set…

In this article we investigate consistency of selection in regression models via the popular Lasso method. Here we depart from the traditional linear regression assumption and consider approximations of the regression function $f$ with…