SIAM Journal on Scientific and Statistical Computing
Some applications of the rank revealing QR factorization
SIAM Journal on Scientific and Statistical Computing
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
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
Free Bits, PCPs, and Nonapproximability---Towards Tight Results
SIAM Journal on Computing
Clustering in large graphs and matrices
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Some optimal inapproximability results
Journal of the ACM (JACM)
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Fast monte-carlo algorithms for finding low-rank approximations
Journal of the ACM (JACM)
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Near-optimal algorithms for unique games
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Algorithm 853: An efficient algorithm for solving rank-deficient least squares problems
ACM Transactions on Mathematical Software (TOMS)
Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix
SIAM Journal on Computing
SIAM Journal on Computing
ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT
Computational Complexity
Information Processing Letters
Improved Approximation Algorithms for Large Matrices via Random Projections
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
On the Nyström Method for Approximating a Gram Matrix for Improved Kernel-Based Learning
The Journal of Machine Learning Research
Fast computation of low-rank matrix approximations
Journal of the ACM (JACM)
Sampling from large matrices: An approach through geometric functional analysis
Journal of the ACM (JACM)
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Relative-Error $CUR$ Matrix Decompositions
SIAM Journal on Matrix Analysis and Applications
An improved approximation algorithm for the column subset selection problem
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
On selecting a maximum volume sub-matrix of a matrix and related problems
Theoretical Computer Science
Sensor selection via convex optimization
IEEE Transactions on Signal Processing
Rank-Deficient Nonlinear Least Squares Problems and Subset Selection
SIAM Journal on Numerical Analysis
Near Optimal Column-Based Matrix Reconstruction
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Bypassing UGC from some optimal geometric inapproximability results
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Optimal column-based low-rank matrix reconstruction
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Column subset selection via sparse approximation of SVD
Theoretical Computer Science
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
Subspace sampling and relative-error matrix approximation: column-based methods
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
Randomized Algorithms for Matrices and Data
Foundations and Trends® in Machine Learning
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We address two problems related to selecting an optimal subset of columns from a matrix. In one of these problems, we are given a matrix A@?R^m^x^n and a positive integer k, and we want to select a sub-matrix C of k columns to minimize @?A-@P"CA@?"F, where @P"C=CC^+ denotes the matrix of projection onto the space spanned by C. In the other problem, we are given A@?R^m^x^n, positive integers c and r, and we want to select sub-matrices C and R of c columns and r rows of A, respectively, to minimize @?A-CUR@?"F, where U@?R^c^x^r is the pseudo-inverse of the intersection between C and R. Although there is a plethora of algorithmic results, the complexity of these problems has not been investigated thus far. We show that these two problems are NP-hard assuming UGC.