Related papers: A Framework for Computational Lower Bounds in Nont…

We study extensions of compressive sensing and low rank matrix recovery to the recovery of low rank tensors from incomplete linear information. While the reconstruction of low rank matrices via nuclear norm minimization is rather…

We show the sup-norm convergence of deep neural network estimators with a novel adversarial training scheme. For the nonparametric regression problem, it has been shown that an estimator using deep neural networks can achieve better…

Deep generative modeling has led to new and state of the art approaches for enforcing structural priors in a variety of inverse problems. In contrast to priors given by sparsity, deep models can provide direct low-dimensional…

This work considers low-rank canonical polyadic decomposition (CPD) under a class of non-Euclidean loss functions that frequently arise in statistical machine learning and signal processing. These loss functions are often used for certain…

Conventional coded computing frameworks are predominantly tailored for structured computations, such as matrix multiplication and polynomial evaluation. Such tasks allow the reuse of tools and techniques from algebraic coding theory to…

We focus on \emph{row sampling} based approximations for matrix algorithms, in particular matrix multipication, sparse matrix reconstruction, and \math{\ell_2} regression. For \math{\matA\in\R^{m\times d}} (\math{m} points in \math{d\ll m}…

We investigate a generalized framework to estimate a latent low-rank plus sparse tensor, where the low-rank tensor often captures the multi-way principal components and the sparse tensor accounts for potential model mis-specifications or…

We propose a unified framework for estimating low-rank matrices through nonconvex optimization based on gradient descent algorithm. Our framework is quite general and can be applied to both noisy and noiseless observations. In the general…

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…

Deep neural networks (DNNs) have shown great success in many machine learning tasks. Their training is challenging since the loss surface of the network architecture is generally non-convex, or even non-smooth. How and under what…

Noisy $k$-XOR is a basic average-case inference problem in which one observes random noisy $k$-ary parity constraints and seeks to recover, or more weakly, detect, a hidden Boolean assignment. A central question is to characterize the…

This paper is devoted to the problem of determining the concentration bounds that are achievable in non-parametric regression. We consider the setting where features are supported on a bounded subset of $\mathbb{R}^d$, the regression…

We assume the direct sum <A> o <B> for the signal subspace. As a result of post- measurement, a number of operational contexts presuppose the a priori knowledge of the LB -dimensional "interfering" subspace <B> and the goal is to estimate…

Decoding algorithms based on approximate tensor network contraction have proven tremendously successful in decoding 2D local quantum codes such as surface/toric codes and color codes, effectively achieving optimal decoding accuracy. In this…

Modern large-scale statistical models require to estimate thousands to millions of parameters. This is often accomplished by iterative algorithms such as gradient descent, projected gradient descent or their accelerated versions. What are…

We define a reduction mechanism for LP and SDP formulations that degrades approximation factors in a controlled fashion. Our reduction mechanism is a minor restriction of classical reductions establishing inapproximability in the context of…

Due to the multi-linearity of tensors, most algorithms for tensor optimization problems are designed based on the block coordinate descent method. Such algorithms are widely employed by practitioners for their implementability and…

Unlike matrix completion, tensor completion does not have an algorithm that is known to achieve the information-theoretic sample complexity rate. This paper develops a new algorithm for the special case of completion for nonnegative…

We propose rigorous lower and upper error bounds for neural network (NN) approximations to PDEs by efficiently computing the Riesz representations of suitable extension and restrictions of the NN residual towards geometrically simpler…

We show that computing even very coarse approximations of critical points is intractable for simple classes of nonconvex functions. More concretely, we prove that if there exists a polynomial-time algorithm that takes as input a polynomial…