Related papers: Solving underdetermined systems with error-correct…
Integer least-squares problems, concerned with solving a system of equations where the components of the unknown vector are integer-valued, arise in a wide range of applications. In many scenarios the unknown vector is sparse, i.e., a large…
We present a new decoding algorithm based on error locating pairs and correcting an amount of errors exceeding half the minimum distance. When applied to Reed--Solomon or algebraic geometry codes, the algorithm is a reformulation of the…
Deconvolution is a statistical inverse problem to estimate the distribution of a random variable based on its noisy observations. Despite the extensive studies on the topic, deconvolution with unknown noise distribution remains as a…
Quantum error correction is widely believed to be essential for large-scale quantum computation, but the required qubit overhead remains a central challenge. Quantum low-density parity-check codes can substantially reduce this overhead…
In solving a system of $n$ linear equations in $d$ variables $Ax=b$, the condition number of the $n,d$ matrix $A$ measures how much errors in the data $b$ affect the solution $x$. Estimates of this type are important in many inverse…
We study symmetric tensor decompositions, i.e., decompositions of the form $T = \sum_{i=1}^r u_i^{\otimes 3}$ where $T$ is a symmetric tensor of order 3 and $u_i \in \mathbb{C}^n$.In order to obtain efficient decomposition algorithms, it is…
This paper discusses a stylized communications problem where one wishes to transmit a real-valued signal x in R^n (a block of n pieces of information) to a remote receiver. We ask whether it is possible to transmit this information reliably…
This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candes and Recht, Candes and Tao, and Keshavan, Montanari, and…
In this paper, we prove a new identity for the least-square solution of an over-determined set of linear equation $Ax=b$, where $A$ is an $m\times n$ full-rank matrix, $b$ is a column-vector of dimension $m$, and $m$ (the number of…
In this paper we analyze the blind deconvolution of an image and an unknown blur in a coded imaging system. The measurements consist of subsampled convolution of an unknown blurring kernel with multiple random binary modulations (coded…
Many numerical problems with input $x$ and output $y$ can be formulated as a system of equations $F(x, y) = 0$ where the goal is to solve for $y$. The condition number measures the change of $y$ for small perturbations to $x$. From this…
Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the…
Machine learning algorithms are typically run on large scale, distributed compute infrastructure that routinely face a number of unavailabilities such as failures and temporary slowdowns. Adding redundant computations using coding-theoretic…
Quantum technologies have the potential to solve certain computationally hard problems with polynomial or super-polynomial speedups when compared to classical methods. Unfortunately, the unstable nature of quantum information makes it prone…
Reed--Solomon error-correcting codes are ubiquitous across computer science and information theory, with applications in cryptography, computational complexity, communication and storage systems, and more. Most works on efficient error…
Consider a d*n matrix A, with d<n. The problem of solving for x in y=Ax is underdetermined, and has infinitely many solutions (if there are any). Given y, the minimum Kolmogorov complexity solution (MKCS) of the input x is defined to be an…
Many applications, including rank aggregation, crowd-labeling, and graphon estimation, can be modeled in terms of a bivariate isotonic matrix with unknown permutations acting on its rows and/or columns. We consider the problem of estimating…
An algorithm is proposed, analyzed, and tested experimentally for solving stochastic optimization problems in which the decision variables are constrained to satisfy equations defined by deterministic, smooth, and nonlinear functions. It is…
Given the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish deterministic conditions under which exact recovery of the low-rank and sparse…
This work introduces a decoding strategy for binary self-dual codes possessing an automorphism of a specific type. The proposed algorithm is a hard decision iterative decoding scheme. The enclosed experiments show that the new decoding…