Optimal Algorithms for Binary, Sparse, and L 1-Norm Principal Component Analysis

George N. Karystinos ()
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George N. Karystinos: Technical University of Crete, Department of Electronic and Computer Engineering

A chapter in Mathematics Without Boundaries, 2014, pp 339-382 from Springer

Abstract: Abstract The principal components of a data matrix based on the L 2 norm can be computed with polynomial complexity via the singular-value decomposition (SVD). If, however, the principal components are constrained to be finite-alphabet or sparse or the L 1 norm is used as an alternative of the L 2 norm, then the computation of them is NP-hard. In this work, we show that in all these problems, the optimal solution can be obtained in polynomial time if the rank of the data matrix is constant. Based on the auxiliary-unit-vector technique that we have developed over the past years, we present optimal algorithms and show that they are fully parallelizable and memory efficient, hence readily implementable. We analyze the properties of our algorithms, compare against the state of the art, and comment on communications and signal processing problems where they are directly applicable to. The efficiency of our auxiliary-unit-vector technique allows the development of a binary, sparse, or L 1 principal component analysis (PCA) line of research in parallel to the conventional L 2 PCA theory.

Keywords: 0-1 Variables; Algorithms; Binary sequences; Code-division multiplexing; Combinatorial optimization; Complexity; Dimensionality reduction; Eigenvalues and eigenfunctions; Feature extraction; Information processing; L 1 norm; L 2 norm; Machine learning; Maximization of quadratic forms; Maximum-likelihood detection; MIMO systems; Noncoherent communication; Outlier resistance; Polynomial algorithms; Principal component analysis; Quadratic programming; Subspace signal processing (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4939-1124-0_11

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DOI: 10.1007/978-1-4939-1124-0_11

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