A main goal of regression is to derive statistical conclusions on the conditional distribution of the output variable Y given the input values x. Two of the most important characteristics of a single distribution are location and scale. Support vector machines (SVMs) are well established to estimate location functions like the conditional median or the conditional mean. We investigate the estimation of scale functions by SVMs when the conditional median is unknown, too. Estimation of scale functions is important e.g. to estimate the volatility in finance. We consider the median absolute deviation (MAD) and the interquantile range (IQR) as measures of scale. Our main result shows the consistency of MAD-type SVMs.
We study the distribution and uncertainty of nonconvex optimization for noisy tensor completion -- the problem of estimating a low-rank tensor given incomplete and corrupted observations of its entries. Focusing on a two-stage estimation algorithm proposed by Cai et al. (2019), we characterize the distribution of this nonconvex estimator down to fine scales. This distributional theory in turn allows one to construct valid and short confidence intervals for both the unseen tensor entries and the unknown tensor factors. The proposed inferential procedure enjoys several important features: (1) it…
Benchmarking Tabular Foundation Models for Conditional Density Estimation in Regression
Department of Statistics, Federal University of São Carlos (UFSCar)（圣卡洛斯联邦大学统计系）；Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK（格勒诺布尔阿尔卑斯大学、法国国家信息与自动化研究所、法国国家科学研究中心、格勒诺布尔国立理工学院、让·库尔坦数学实验室）
UIUC（伊利诺伊大学厄巴纳-香槟分校）；Arizona State University（亚利桑那州立大学）
The problem of estimating the support of a distribution is of great importance in many areas of machine learning, computer science, physics and biology. Most of the existing work in this domain has focused on settings that assume perfectly accurate sampling approaches, which is seldom true in practical data science. Here we introduce the first known approach to support estimation in the presence of sampling artifacts and errors where each sample is assumed to arise from a Poisson repeat channel which simultaneously captures repetitions and deletions of samples. The proposed estimator is based …
Huazhong University of Science and Technology（华中科技大学）；Peking University Health Science Center（北京大学医学部）；Beijing International Center for Mathematical Research, Peking University（北京国际数学研究中心，北京大学）；Colorado State University（科罗拉多州立大学）
We analyze the $\ell_1$ and $\ell_\infty$ convergence rates of k nearest neighbor density estimation method. Our analysis includes two different cases depending on whether the support set is bounded or not. In the first case, the probability density function has a bounded support and is bounded away from zero. We show that kNN density estimation is minimax optimal under both $\ell_1$ and $\ell_\infty$ criteria, if the support set is known. If the support set is unknown, then the convergence rate of $\ell_1$ error is not affected, while $\ell_\infty$ error does not converge. In the second case,…
In this paper, we study meta learning for support (i.e., the set of non-zero entries) recovery in high-dimensional precision matrix estimation where we reduce the sufficient sample complexity in a novel task with the information learned from other auxiliary tasks. In our setup, each task has a different random true precision matrix, each with a possibly different support. We assume that the union of the supports of all the true precision matrices (i.e., the true support union) is small in size. We propose to pool all the samples from different tasks, and \emph{improperly} estimate a single pre…
In high dimensional sparse regression, pivotal estimators are estimators for which the optimal regularization parameter is independent of the noise level. The canonical pivotal estimator is the square-root Lasso, formulated along with its derivatives as a "non-smooth + non-smooth" optimization problem. Modern techniques to solve these include smoothing the datafitting term, to benefit from fast efficient proximal algorithms. In this work we show minimax sup-norm convergence rates for non smoothed and smoothed, single task and multitask square-root Lasso-type estimators. Thanks to our theoretic…
