On Rank-Revealing Factorisations
SIAM Journal on Matrix Analysis and Applications
Efficient algorithms for computing a strong rank-revealing QR factorization
SIAM Journal on Scientific Computing
Fast monte-carlo algorithms for finding low-rank approximations
Journal of the ACM (JACM)
Matrix approximation and projective clustering via volume sampling
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix
SIAM Journal on Computing
SIAM Journal on Computing
Adaptive sampling and fast low-rank matrix approximation
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra
SIAM Journal on Numerical Analysis
Compression and direct manipulation of complex blendshape models
Proceedings of the 2011 SIGGRAPH Asia Conference
Randomized Algorithms for Matrices and Data
Foundations and Trends® in Machine Learning
CageR: Cage-Based Reverse Engineering of Animated 3D Shapes
Computer Graphics Forum
Matrix factorization as search
ECML PKDD'12 Proceedings of the 2012 European conference on Machine Learning and Knowledge Discovery in Databases - Volume Part II
Column Subset Selection Problem is UG-hard
Journal of Computer and System Sciences
| Hi-index | 5.23 |
Given a matrix A@?R^m^x^n (n vectors in m dimensions), we consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible. This notion turned out to be important in low-rank approximations to matrices and rank revealing QR factorizations which have been investigated in the linear algebra community and can be quantified in a few different ways. In this paper, from a complexity theoretic point of view, we propose four related problems in which we try to find a sub-matrix C@?R^m^x^k of a given matrix A@?R^m^x^n such that (i) @s"m"a"x(C) (the largest singular value of C) is minimum, (ii) @s"m"i"n(C) (the smallest singular value of C) is maximum, (iii) @k(C)=@s"m"a"x(C)/@s"m"i"n(C) (the condition number of C) is minimum, and (iv) the volume of the parallelepiped defined by the column vectors of C is maximum. We establish the NP-hardness of these problems and further show that they do not admit PTAS. We then study a natural Greedy heuristic for the maximum volume problem and show that it has approximation ratio 2^-^O^(^k^l^o^g^k^). Our analysis of the Greedy heuristic is tight to within a logarithmic factor in the exponent, which we show by explicitly constructing an instance for which the Greedy heuristic is 2^-^@W^(^k^) from optimal. When A has unit norm columns, a related problem is to select the maximum number of vectors with a given volume. We show that if the optimal solution selects k columns, then Greedy will select @W(k/logk) columns, providing a logk approximation.