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In many applications, it is of interest to approximate data, given by mxn matrix A, by a matrix B of at most rank k, which is much smaller than m and n. The best approximation is given by singular value decomposition, which is too time…
Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature. Yet, the very nature of this quantity is…
We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or…
The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this…
We present a Hamiltonian Monte Carlo algorithm to sample from multivariate Gaussian distributions in which the target space is constrained by linear and quadratic inequalities or products thereof. The Hamiltonian equations of motion can be…
The power of randomized algorithms in numerical methods have led to fast solutions which use the Singular Value Decomposition (SVD) as a core routine. However, given the large data size of modern and the modest runtime of SVD, most…
LU and Cholesky matrix factorization algorithms are core subroutines used to solve systems of linear equations (SLEs) encountered while solving an optimization problem. Standard factorization algorithms are highly efficient but remain…
Solving symmetric positive definite linear problems is a fundamental computational task in machine learning. The exact solution, famously, is cubicly expensive in the size of the matrix. To alleviate this problem, several linear-time…
In the numerical linear algebra community, it was suggested that to obtain nearly optimal bounds for various problems such as rank computation, finding a maximal linearly independent subset of columns (a basis), regression, or low-rank…
We study the Sparse Plus Low-Rank decomposition problem (SLR), which is the problem of decomposing a corrupted data matrix into a sparse matrix of perturbations plus a low-rank matrix containing the ground truth. SLR is a fundamental…
Subsampling methods have been recently proposed to speed up least squares estimation in large scale settings. However, these algorithms are typically not robust to outliers or corruptions in the observed covariates. The concept of influence…
It has been known in potential theory that, for some kernels matrices corresponding to well-separated point sets, fast analytical low-rank approximation can be achieved via the use of proxy points. This proxy point method gives a…
A recently introduced Importance Sampling strategy based on a least squares optimization is applied to the Monte Carlo simulation of Libor Market Models. Such Least Squares Importance Sampling (LSIS) allows the automatic optimization of the…
This paper argues that randomized linear sketching is a natural tool for on-the-fly compression of data matrices that arise from large-scale scientific simulations and data collection. The technical contribution consists in a new algorithm…
A key goal in the design of probabilistic inference algorithms is identifying and exploiting properties of the distribution that make inference tractable. Lifted inference algorithms identify symmetry as a property that enables efficient…
This paper presents an algorithmic study of a class of covering mixed-integer linear programming problems which encompasses classic cover problems, including multidimensional knapsack, facility location and supplier selection problems. We…
Randomized algorithms for low-rank approximation of quaternion matrices have gained increasing attention in recent years. However, existing methods overlook pass efficiency, the ability to limit the number of passes over the input…
Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in machine learning, statistics, bioinformatics, computer vision, as well as signal and image processing. In theory, this problem can…
Many problems in data science can be treated as estimating a low-rank matrix from highly incomplete, sometimes even corrupted, observations. One popular approach is to resort to matrix factorization, where the low-rank matrix factors are…
We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…