Related papers: Statistical Learning Theory for Neural Operators

This paper addresses inverse problems (in a broad sense) for two classes of multivariate neural network (NN) operators, with particular emphasis on saturation results, and both analytical and semi-analytical inverse theorems. One of the key…

Neural networks have emerged as effective tools for solving ill-posed inverse problems. In many scientific applications, however, observational training data are insufficient, and learned inverse operators must instead be trained on…

Blind inverse problems arise in many experimental settings where both the signal of interest and the forward operator are (partially) unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward…

We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of…

Graphical models have been popularly used for capturing conditional independence structure in multivariate data, which are often built upon independent and identically distributed observations, limiting their applicability to complex…

The functional linear model extends the notion of linear regression to the case where the response and covariates are iid elements of an infinite dimensional Hilbert space. The unknown to be estimated is a Hilbert-Schmidt operator, whose…

We consider ill-posed inverse problems where the forward operator $T$ is unknown, and instead we have access to training data consisting of functions $f_i$ and their noisy images $Tf_i$. This is a practically relevant and challenging…

We consider the problem of learning a one-hidden-layer neural network: we assume the input $x\in \mathbb{R}^d$ is from Gaussian distribution and the label $y = a^\top \sigma(Bx) + \xi$, where $a$ is a nonnegative vector in $\mathbb{R}^m$…

We develop a framework using Hilbert spaces as a proxy to analyze PAC learning problems with structural properties. We consider a joint Hilbert space incorporating the relation between the true label and the predictor under a joint…

Function approximation from input and output data pairs constitutes a fundamental problem in supervised learning. Deep neural networks are currently the most popular method for learning to mimic the input-output relationship of a general…

This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix…

Tree structured graphical models are powerful at expressing long range or hierarchical dependency among many variables, and have been widely applied in different areas of computer science and statistics. However, existing methods for…

Statistical divergences (SDs), which quantify the dissimilarity between probability distributions, are a basic constituent of statistical inference and machine learning. A modern method for estimating those divergences relies on…

Various iterative reconstruction algorithms for inverse problems can be unfolded as neural networks. Empirically, this approach has often led to improved results, but theoretical guarantees are still scarce. While some progress on…

Learning maps between function spaces with a strong inductive bias is a central challenge in soft computing, especially when training data are scarce and standard deep architectures overfit. We introduce a \emph{neural integral operator}…

We study random iterations of averaged operators in Hilbert spaces and prove that the associated residuals converge exponentially fast, both in expectation and almost surely. Our results provide quantitative bounds in terms of a single…

In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy…

This paper has two main goals: (a) establish several statistical properties---consistency, asymptotic distributions, and convergence rates---of stationary solutions and values of a class of coupled nonconvex and nonsmoothempirical risk…

We develop a general framework for estimating function-valued parameters under equality or inequality constraints in infinite-dimensional statistical models. Such constrained learning problems are common across many areas of statistics and…

While there are convergence guarantees for temporal difference (TD) learning when using linear function approximators, the situation for nonlinear models is far less understood, and divergent examples are known. Here we take a first step…