Related papers: Improving CUR Matrix Decomposition and the Nystr\"…
We study superfast algorithms that computes low rank approximation of a matrix (hereafter referred to as LRA) that use much fewer memory cells and arithmetic operations than the input matrix has entries. We first specify a family of 2mn…
We derive a new adaptive leverage score sampling strategy for solving the Column Subset Selection Problem (CSSP). The resulting algorithm, called Adaptive Randomized Pivoting, can be viewed as a randomization of Osinsky's recently proposed…
We study algorithms for approximating pairwise similarity matrices that arise in natural language processing. Generally, computing a similarity matrix for $n$ data points requires $\Omega(n^2)$ similarity computations. This quadratic…
In the setting of nonparametric regression, we propose and study a combination of stochastic gradient methods with Nystr\"om subsampling, allowing multiple passes over the data and mini-batches. Generalization error bounds for the studied…
Randomized algorithms in numerical linear algebra have proven to be effective in ameliorating issues of scalability when working with large matrices, efficiently producing accurate low-rank approximations. A key remaining challenge,…
Spectral clustering has shown a superior performance in analyzing the cluster structure. However, its computational complexity limits its application in analyzing large-scale data. To address this problem, many low-rank matrix approximating…
Kernel matrices (e.g. Gram or similarity matrices) are essential for many state-of-the-art approaches to classification, clustering, and dimensionality reduction. For large datasets, the cost of forming and factoring such kernel matrices…
A CUR factorization is often utilized as a substitute for the singular value decomposition (SVD), especially when a concrete interpretation of the singular vectors is challenging. Moreover, if the original data matrix possesses properties…
Spectral clustering techniques are valuable tools in signal processing and machine learning for partitioning complex data sets. The effectiveness of spectral clustering stems from constructing a non-linear embedding based on creating a…
To achieve greater accuracy, hypergraph matching algorithms require exponential increases in computational resources. Recent kd-tree-based approximate nearest neighbor (ANN) methods, despite the sparsity of their compatibility tensor, still…
Kernel $k$-means clustering can correctly identify and extract a far more varied collection of cluster structures than the linear $k$-means clustering algorithm. However, kernel $k$-means clustering is computationally expensive when the…
We propose a generalized CUR (GCUR) decomposition for matrix pairs $(A, B)$. Given matrices $A$ and $B$ with the same number of columns, such a decomposition provides low-rank approximations of both matrices simultaneously, in terms of some…
We propose a new technique for constructing low-rank approximations of matrices that arise in kernel methods for machine learning. Our approach pairs a novel automatically constructed analytic expansion of the underlying kernel function…
Recent work in the matrix completion literature has shown that prior knowledge of a matrix's row and column spaces can be successfully incorporated into reconstruction programs to substantially benefit matrix recovery. This paper proposes a…
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…
This work proposes and analyzes a new class of numerical integrators for computing low-rank approximations to solutions of matrix differential equation. We combine an explicit Runge-Kutta method with repeated randomized low-rank…
We consider the problem of matrix column subset selection, which selects a subset of columns from an input matrix such that the input can be well approximated by the span of the selected columns. Column subset selection has been applied to…
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…
CUR decompositions approximate a matrix using selected columns, rows, and their intersection. Classical CUR theory provides exactness results for low-rank matrices and perturbation bounds controlled by the size of the noise. In this work we…
Interpolative and CUR decompositions involve "natural bases" of row and column subsets, or skeletons, of a given matrix that approximately span its row and column spaces. These low-rank decompositions preserve properties such as sparsity or…