Faster Subset Selection for Matrices and Applications
Haim Avron,Christos Boutsidis +1 more
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TLDR
It is shown that the combinatorial problem of finding a low-stretch spanning tree in an undirected graph corresponds to subset selection, and the various implications of this reduction are discussed.Abstract:
We study the following problem of subset selection for matrices: given a matrix $\mathbf{X} \in \mathbb{R}^{n \times m}$ ($m > n$) and a sampling parameter $k$ ($n \le k \le m$), select a subset of $k$ columns from $\mathbf{X}$ such that the pseudoinverse of the sampled matrix has as small a norm as possible. In this work, we focus on the Frobenius and the spectral matrix norms. We describe several novel (deterministic and randomized) approximation algorithms for this problem with approximation bounds that are optimal up to constant factors. Additionally, we show that the combinatorial problem of finding a low-stretch spanning tree in an undirected graph corresponds to subset selection, and discuss various implications of this reduction.read more
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Journal ArticleDOI
Discrete Signal Processing on Graphs: Sampling Theory
TL;DR: It is shown that perfect recovery is possible for graph signals bandlimited under the graph Fourier transform and the connection to the sampling theory of finite discrete-time signal processing and previous work on signal recovery on graphs is established.
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Signals on Graphs: Uncertainty Principle and Sampling
TL;DR: An uncertainty principle for graph signals is derived and the conditions for the recovery of band-limited signals from a subset of samples are illustrated and shown, showing an interesting link between uncertainty principle and sampling and proposing alternative signal recovery algorithms.
Proceedings ArticleDOI
Near Optimal Column-Based Matrix Reconstruction
TL;DR: In this article, the authors considered low-rank reconstruction of a matrix using a subset of its columns and presented asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction.
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Linear regression without correspondence
Daniel Hsu,Kevin Shi,Xiaorui Sun +2 more
TL;DR: This article considers algorithmic and statistical aspects of linear regression when the correspondence between the covariates and the responses is unknown, and a fully polynomial-time approximation scheme is given for the natural least squares optimization problem in any constant dimension.
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Provable deterministic leverage score sampling
TL;DR: In this paper, the authors provide a theoretical analysis of deterministic leverage score sampling and show that such sampling can be provably as accurate as its randomized counterparts, if the leverage scores follow a moderately steep power-law decay.
References
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Book
Matrix Analysis
Roger A. Horn,Charles R. Johnson +1 more
TL;DR: In this article, the authors present results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrate their importance in a variety of applications, such as linear algebra and matrix theory.
Journal ArticleDOI
Sensor Selection via Convex Optimization
Siddharth Joshi,Stephen Boyd +1 more
TL;DR: This paper describes a heuristic, based on convex optimization, that gives a subset selection as well as a bound on the best performance that can be achieved by any selection of k sensor measurements.
Proceedings ArticleDOI
Improved Approximation Algorithms for Large Matrices via Random Projections
TL;DR: In this paper, the authors present a (1 + ∆)-approximation algorithm for the singular value decomposition of an m? n matrix A with M non-zero entries that requires 2 passes over the data and runs in time O(n 2 ).