Related papers: Randomized Matrix Decompositions using R
In this paper, we study the problem of decomposing a superposition of a low-rank matrix and a sparse matrix when a relatively few linear measurements are available. This problem arises in many data processing tasks such as aligning multiple…
We present a new restricted SVD-based CUR (RSVD-CUR) factorization for matrix triplets $(A, B, G)$ that aims to extract meaningful information by providing a low-rank approximation of the three matrices using a subset of their rows and…
Principal component analysis (PCA) requires the computation of a low-rank approximation to a matrix containing the data being analyzed. In many applications of PCA, the best possible accuracy of any rank-deficient approximation is at most a…
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…
Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that exploit low-rank structure. However, estimating the rank has been done largely in an…
The manuscript describes efficient algorithms for the computation of the CUR and ID decompositions. The methods used are based on simple modifications to the classical truncated pivoted QR decomposition, which means that highly optimized…
Matrix decompositions are ubiquitous in machine learning, including applications in dimensionality reduction, data compression and deep learning algorithms. Typical solutions for matrix decompositions have polynomial complexity which…
The robust principal component analysis (RPCA) decomposes a data matrix into a low-rank part and a sparse part. There are mainly two types of algorithms for RPCA. The first type of algorithm applies regularization terms on the singular…
Constrained low-rank matrix approximations have been known for decades as powerful linear dimensionality reduction techniques to be able to extract the information contained in large data sets in a relevant way. However, such low-rank…
The Hadamard decomposition is a powerful technique for data analysis and matrix compression, which decomposes a given matrix into the element-wise product of two or more low-rank matrices. In this paper, we develop an efficient algorithm to…
In this paper, we tackle two important problems in low-rank learning, which are partial singular value decomposition and numerical rank estimation of huge matrices. By using the concepts of Krylov subspaces such as Golub-Kahan…
In 1954, Alston S. Householder published \textit{Principles of Numerical Analysis}, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower…
This paper considers the use of Robust PCA in a CUR decomposition framework and applications thereof. Our main algorithms produce a robust version of column-row factorizations of matrices $\mathbf{D}=\mathbf{L}+\mathbf{S}$ where…
The Singular Value Decomposition is a matrix decomposition technique widely used in the analysis of multivariate data, such as complex space-time images obtained in both physical and biological systems. In this paper, we examine the…
The randomized singular value decomposition (SVD) is a popular and effective algorithm for computing a near-best rank $k$ approximation of a matrix $A$ using matrix-vector products with standard Gaussian vectors. Here, we generalize the…
Traditional low-rank approximation is a powerful tool to compress the huge data matrices that arise in simulations of partial differential equations (PDE), but suffers from high computational cost and requires several passes over the PDE…
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…
The singular value decomposition (SVD) is commonly used in applications requiring a low rank matrix approximation. However, the singular vectors cannot be interpreted in terms of the original data. For applications requiring this type of…
This article studies how to form CUR decompositions of low-rank matrices via primarily random sampling, though deterministic methods due to previous works are illustrated as well. The primary problem is to determine when a column submatrix…
This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions,…