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Subspace clustering aims to cluster unlabeled data that lies in a union of low-dimensional linear subspaces. Deep subspace clustering approaches based on auto-encoders have become very popular to solve subspace clustering problems. However,…

Consider a dataset of vector-valued observations that consists of noisy inliers, which are explained well by a low-dimensional subspace, along with some number of outliers. This work describes a convex optimization problem, called REAPER,…

This paper is concerned with the approximation of the solution of partial differential equations by means of artificial neural networks. Here a feedforward neural network is used to approximate the solution of the partial differential…

An encoder-decoder neural network has been used to examine the possibility for acceleration of a partial integro-differential equation, the Fokker-Planck-Landau collision operator. This is part of the governing equation in the massively…

Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…

Simplicia-simplicial regression concerns statistical modeling scenarios in which both the predictors and the responses are vectors constrained to lie on the simplex. \cite{fiksel2022} introduced a transformation-free linear regression…

Linear regression in $\ell_p$-norm is a canonical optimization problem that arises in several applications, including sparse recovery, semi-supervised learning, and signal processing. Generic convex optimization algorithms for solving…

Network pruning is a widely used technique to reduce computation cost and model size for deep neural networks. However, the typical three-stage pipeline, i.e., training, pruning and retraining (fine-tuning) significantly increases the…

Network representation learning seeks to embed networks into a low-dimensional space while preserving the structural and semantic properties, thereby facilitating downstream tasks such as classification, trait prediction, edge…

We develop an exact coordinate descent algorithm for high-dimensional regularized Huber regression. In contrast to composite gradient descent methods, our algorithm fully exploits the advantages of coordinate descent when the underlying…

In this article, we propose a general nonlinear sufficient dimension reduction (SDR) framework when both the predictor and response lie in some general metric spaces. We construct reproducing kernel Hilbert spaces whose kernels are fully…

Most state of the art deep neural networks are overparameterized and exhibit a high computational cost. A straightforward approach to this problem is to replace convolutional kernels with its low-rank tensor approximations, whereas the…

Dimensionality reduction (DR) is often used as a preprocessing step in classification, but usually one first fixes the DR mapping, possibly using label information, and then learns a classifier (a filter approach). Best performance would be…

This paper provides a least squares formulation for the training of a 2-layer convolutional neural network using quadratic activation functions, a 2-norm loss function, and no regularization term. Using this method, an analytic expression…

We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is…

This chapter offers a comprehensive introduction to the least-squares neural network (LSNN) method introduced in [14,16], for solving scalar first-order hyperbolic partial differential equations, specifically linear advection-reaction…

Deep neural networks have exhibited remarkable performance in image super-resolution (SR) tasks by learning a mapping from low-resolution (LR) images to high-resolution (HR) images. However, the SR problem is typically an ill-posed problem…

In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying…

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…

Higher-order tensor methods were recently proposed for minimizing smooth convex and nonconvex functions. Higher-order algorithms accelerate the convergence of the classical first-order methods thanks to the higher-order derivatives used in…