相关论文: Narrow-width approximation limitations
In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main tools for estimating the metric entropy, approximation rates,…
The literature on stochastic programming typically restricts attention to problems that fulfill constraint qualifications. The literature on estimation and inference under partial identification frequently restricts the geometry of…
We consider spline estimates which preserve prescribed piecewise convex properties of the unknown function. A robust version of the penalized likelihood is given and shown to correspond to a variable halfwidth kernel smoother where the…
We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results…
Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large…
Certified defenses against small-norm adversarial examples have received growing attention in recent years; though certified accuracies of state-of-the-art methods remain far below their non-robust counterparts, despite the fact that…
A major problem in numerical weather prediction (NWP) is the estimation of high-dimensional covariance matrices from a small number of samples. Maximum likelihood estimators cannot provide reliable estimates when the overall dimension is…
Logistic regression involving high-dimensional covariates is a practically important problem. Often the goal is variable selection, i.e., determining which few of the many covariates are associated with the binary response. Unfortunately,…
In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector- or even Hilbert space-valued. Our main objective is to study how the…
Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of…
We consider the problem of approximating a function in a general nonlinear subset of $L^2$, when only a weighted Monte Carlo estimate of the $L^2$-norm can be computed. Of particular interest in this setting is the concept of sample…
Convex sample approximations of chance-constrained optimization problems are considered, in which chance constraints are replaced by sets of sampled constraints. We propose a randomized sample selection strategy that allows tight bounds to…
We survey most of the different types of approximation algorithms which minimize the number of output vertices. We present their main qualities and their inherent drawbacks.
The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of…
This paper discusses a methodology for determining a functional representation of a random process from a collection of scattered pointwise samples. The present work specifically focuses onto random quantities lying in a high dimensional…
Various natural language processing tasks are structured prediction problems where outputs are constructed with multiple interdependent decisions. Past work has shown that domain knowledge, framed as constraints over the output space, can…
The finite sensitivity of instruments or detection methods means that data sets in many areas of astronomy, for example cosmological or exoplanet surveys, are necessarily systematically incomplete. Such data sets, where the population being…
Conformal prediction is a generic methodology for finite-sample valid distribution-free prediction. This technique has garnered a lot of attention in the literature partly because it can be applied with any machine learning algorithm that…
Nonparametric estimation of the gap time distribution in a simple renewal process may be considered a problem in survival analysis under particular sampling frames corresponding to how the renewal process is observed. This note describes…
Large Language Models work quite well with general-purpose data and many tasks in Natural Language Processing. However, they show several limitations when used for a task such as domain-specific abstractive text summarization. This paper…