相关论文: On Single Index Models beyond Gaussian Data

The problem of learning single index and multi index models has gained significant interest as a fundamental task in high-dimensional statistics. Many recent works have analysed gradient-based methods, particularly in the setting of…

We consider the problem of jointly learning a one-dimensional projection and a univariate function in high-dimensional Gaussian models. Specifically, we study predictors of the form $f(x)=\varphi^\star(\langle w^\star, x \rangle)$, where…

Single-Index Models are high-dimensional regression problems with planted structure, whereby labels depend on an unknown one-dimensional projection of the input via a generic, non-linear, and potentially non-deterministic transformation. As…

Single-index models are a class of functions given by an unknown univariate ``link'' function applied to an unknown one-dimensional projection of the input. These models are particularly relevant in high dimension, when the data might…

We study gradient flow on the multi-index regression problem for high-dimensional Gaussian data. Multi-index functions consist of a composition of an unknown low-rank linear projection and an arbitrary unknown, low-dimensional link…

Significant advances have been made recently on training neural networks, where the main challenge is in solving an optimization problem with abundant critical points. However, existing approaches to address this issue crucially rely on a…

Few neural architectures lend themselves to provable learning with gradient based methods. One popular model is the single-index model, in which labels are produced by composing an unknown linear projection with a possibly unknown scalar…

We study the problem of learning single-index models, where the label $y \in \mathbb{R}$ depends on the input $\boldsymbol{x} \in \mathbb{R}^d$ only through an unknown one-dimensional projection $\langle…

Gaussian processes (GPs) are a powerful tool for probabilistic inference over functions. They have been applied to both regression and non-linear dimensionality reduction, and offer desirable properties such as uncertainty estimates,…

The problem of statistical inference for regression coefficients in a high-dimensional single-index model is considered. Under elliptical symmetry, the single index model can be reformulated as a proxy linear model whose regression…

Gaussian processes with derivative information are useful in many settings where derivative information is available, including numerous Bayesian optimization and regression tasks that arise in the natural sciences. Incorporating derivative…

Single index models provide an effective dimension reduction tool in regression, especially for high dimensional data, by projecting a general multivariate predictor onto a direction vector. We propose a novel single-index model for…

Neural networks can identify low-dimensional relevant structures within high-dimensional noisy data, yet our mathematical understanding of how they do so remains scarce. Here, we investigate the training dynamics of two-layer shallow neural…

Recent works have demonstrated that the sample complexity of gradient-based learning of single index models, i.e. functions that depend on a 1-dimensional projection of the input data, is governed by their information exponent. However,…

Models with dimension more than the available sample size are now commonly used in various applications. A sensible inference is possible using a lower-dimensional structure. In regression problems with a large number of predictors, the…

In deep learning, a central issue is to understand how neural networks efficiently learn high-dimensional features. To this end, we explore the gradient descent learning of a general Gaussian Multi-index model…

We study the problem of gradient descent learning of a single-index target function $f_*(\boldsymbol{x}) = \textstyle\sigma_*\left(\langle\boldsymbol{x},\boldsymbol{\theta}\rangle\right)$ under isotropic Gaussian data in $\mathbb{R}^d$,…

This paper addresses the problem of learning a sparse structure Bayesian network from high-dimensional discrete data. Compared to continuous Bayesian networks, learning a discrete Bayesian network is a challenging problem due to the large…

To learn (statistical) dependencies among random variables requires exponentially large sample size in the number of observed random variables if any arbitrary joint probability distribution can occur. We consider the case that sparse data…

Understanding the advantages of deep neural networks trained by gradient descent (GD) compared to shallow models remains an open theoretical challenge. In this paper, we introduce a class of target functions (single and multi-index Gaussian…