Title :
The Sparse Principal Component of a Constant-Rank Matrix
Author :
Asteris, Megasthenis ; Papailiopoulos, Dimitris S. ; Karystinos, George N.
Author_Institution :
Dept. of Electr. & Comput. Eng., Univ. of Texas at Austin, Austin, TX, USA
Abstract :
The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem NP-hard. In this paper, we prove that, if the matrix is positive semidefinite and its rank is constant, then its sparse principal component is polynomially computable. Our proof utilizes the auxiliary unit vector technique that has been recently developed to identify problems that are polynomially solvable. In addition, we use this technique to design an algorithm which, for any sparsity value, computes the sparse principal component with complexity O(ND+1), where N and D are the matrix size and rank, respectively. Our algorithm is fully parallelizable and memory efficient.
Keywords :
computational complexity; eigenvalues and eigenfunctions; optimisation; principal component analysis; signal processing; NP-hard; auxiliary unit vector; constant-rank matrix; maximum eigenvalue; optimal submatrix; principal submatrix; sparse principal component; sparsity value; Complexity theory; Indexes; Optimization; Polynomials; Principal component analysis; Sparse matrices; Vectors; Eigenvalues and eigenfunctions; feature extraction; information processing; machine learning algorithms; principal component analysis; signal processing algorithms;
Journal_Title :
Information Theory, IEEE Transactions on
DOI :
10.1109/TIT.2014.2303975