Randomized algorithms have propelled advances in artificial intelligence and represent a foundational research area in advancing AI for Science.

The key finding of the manuscript is that the convergence order can be greatly improved by modifying only a very small number of elements in the coefficient matrix.

Numerical Analysis Numerical Analysis 65R20 (primary) 65D32, 45B05 (secondary)

This paper describes efficient algorithms for computing rank-revealing factorizations of matrices that are too large to fit in RAM, and must instead be stored on slow external memory devices such as solid-state or spinning disk hard drives (out-of-core or out-of-memory).

The purpose of this text is to provide an accessible introduction to a set of recently developed algorithms for factorizing matrices.

Numerical Analysis

A fundamental problem when adding column pivoting to the Householder QR factorization is that only about half of the computation can be cast in terms of high performing matrix-matrix multiplications, which greatly limits the benefits that can be derived from so-called blocking of algorithms.

Numerical Analysis Numerical Analysis

The method takes as input a tolerance $\varepsilon$ and an $m\times n$ matrix $A$, and returns an approximate low rank factorization of $A$ that is accurate to within precision $\varepsilon$ in the Frobenius norm (or some other easily computed norm).

Numerical Analysis

The ID and CUR factorizations pick subsets of the rows/columns of a matrix to use as bases for its row/column space.

Numerical Analysis Mathematical Software

The manuscript describes efficient algorithms for the computation of the CUR and ID decompositions.

Numerical Analysis Numerical Analysis

These methods use random sampling to identify a subspace that captures most of the action of a matrix.

Numerical Analysis Probability

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