Related papers: Randomized strong rank-revealing QR for column sub…
In this work, we analyze a sublinear-time algorithm for selecting a few rows and columns of a matrix for low-rank approximation purposes. The algorithm is based on an initial uniformly random selection of rows and columns, followed by a…
In this paper, we introduce an efficient algorithm for column subset selection that combines the column-pivoted QR factorization with sparse subspace embeddings. The proposed method, SE-QRSC, is particularly effective for wide matrices with…
The selection of most informative and discriminative features from high-dimensional data has been noticed as an important topic in machine learning and data engineering. Using matrix factorization-based techniques such as nonnegative matrix…
We introduce a Generalized Randomized QR-decomposition that may be applied to arbitrary products of matrices and their inverses, without needing to explicitly compute the products or inverses. This factorization is a critical part of a…
The classic rank-revealing QR factorization factorizes a matrix $A$ as $AP=QR$ where $P$ permutes the columns of $A$, $Q$ is an orthogonal matrix, and $R$ is upper triangular with non-increasing diagonal entries. This is called…
This manuscript describes a technique for computing partial rank-revealing factorizations, such as, e.g, a partial QR factorization or a partial singular value decomposition. The method takes as input a tolerance $\varepsilon$ and an…
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which…
The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed…
Given a matrix $A$ of size $m\times n$, the manuscript describes a algorithm for computing a QR factorization $AP=QR$ where $P$ is a permutation matrix, $Q$ is orthonormal, and $R$ is upper triangular. The algorithm is blocked, to allow it…
The dominant contribution to communication complexity in factorizing a matrix using QR with column pivoting is due to column-norm updates that are required to process pivot decisions. We use randomized sampling to approximate this process…
Rank-revealing matrix decompositions provide an essential tool in spectral analysis of matrices, including the Singular Value Decomposition (SVD) and related low-rank approximation techniques. QR with Column Pivoting (QRCP) is usually…
The problem of column subset selection asks for a subset of columns from an input matrix such that the matrix can be reconstructed as accurately as possible within the span of the selected columns. A natural extension is to consider a…
Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called…
We present Flip-Flop Spectrum-Revealing QR (Flip-Flop SRQR) factorization, a significantly faster and more reliable variant of the QLP factorization of Stewart, for low-rank matrix approximations. Flip-Flop SRQR uses SRQR factorization to…
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…
Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…
In this paper we introduce a new column selection strategy, named here ``Deviation Maximization", and apply it to compute rank-revealing QR factorizations as an alternative to the well known block version of the QR factorization with the…
A QR factorization of a tall and skinny matrix with n columns can be represented as a reduction. The operation used along the reduction tree has in input two n-by-n upper triangular matrices and in output an n-by-n upper triangular matrix…
Matrix factorization is a well-studied task in machine learning for compactly representing large, noisy data. In our approach, instead of using the traditional concept of matrix rank, we define a new notion of link-rank based on a…
Many applications in scientific computing and data science require the computation of a rank-revealing factorization of a large matrix. In many of these instances the classical algorithms for computing the singular value decomposition are…