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

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Mathematics Without Boundaries

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.

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Notes

  1. 1.

    We ignore the other semicircle because any pair of angles ϕ 1 and ϕ 2 with difference π results in opposite vectors \(\mathbf{c}(\phi _{1}) = -\mathbf{c}(\phi _{2})\) and, hence, opposite binary vectors \(\mathbf{x}(\mathbf{c}(\phi _{1})) = -\mathbf{x}(\mathbf{c}(\phi _{2}))\) in (16) which, however, are equivalent with respect to the optimization metric in (9).

  2. 2.

    As in the rank-2 case, we ignore the other semihypersphere because any pair of vectors \(\boldsymbol{\phi }\) and \(\tilde{\boldsymbol{\phi }}\) whose first elements ϕ 1 and \(\tilde{\phi }_{1}\), respectively, have difference π results in opposite vectors \(\mathbf{c}(\boldsymbol{\phi }) = -\mathbf{c}(\tilde{\boldsymbol{\phi }})\) and, hence, opposite binary vectors \(\mathbf{x}(\mathbf{c}(\boldsymbol{\phi })) = -\mathbf{x}(\mathbf{c}(\tilde{\boldsymbol{\phi }}))\) in (16) which, however, are equivalent with respect to the optimization metric in (9).

  3. 3.

    If V 1: D−1, :  is full-rank, then its null space has rank 1 and \(\mathbf{c}(\boldsymbol{\phi })\) is uniquely determined (within a sign ambiguity which is resolved by c D  ≥ 0). If, instead, V 1: D−1, :  is rank-deficient, then the intersection of the D − 1 hypersurfaces (i.e., the solution of (28)) is a p-manifold (with p ≥ 1) in the (D − 1)-dimensional space and does not generate a new cell. Hence, linearly dependent combinations of D − 1 rows of V are ignored.

  4. 4.

    An alternative way of resolving the sign ambiguities at the intersections of hypersurfaces was developed in [34] and led to the direct construction of a set S of size \(\sum _{i=0}^{D-1}\binom{N - 1}{i} = O(N^{D-1})\) with complexity O(N D).

  5. 5.

    We again allow c(ϕ) to lie on the unit-radius semicircle and ignore the other semicircle because any pair of angles ϕ 1 and ϕ 2 with difference π results in opposite vectors \(\mathbf{c}(\phi _{1}) = -\mathbf{c}(\phi _{2})\) which, however, are equivalent with respect to the optimization metric in (40) and produce the same support \(I(\mathbf{c}(\phi _{1})) = I(\mathbf{c}(\phi _{2}))\) in (42).

  6. 6.

    The exact detailed steps that are taken to define, with complexity O(N), the support I for each interval that is adjacent to an intersection point are described in [4].

  7. 7.

    As in the rank-2 case, we allow \(\mathbf{c}(\boldsymbol{\phi })\) to lie on the unit-radius semihypersphere, since we can ignore the other semihypersphere because any pair of vectors \(\boldsymbol{\phi }\) and \(\tilde{\boldsymbol{\phi }}\) whose first elements ϕ 1 and \(\tilde{\phi }_{1}\), respectively, have difference π results in opposite vectors \(\mathbf{c}(\boldsymbol{\phi }) = -\mathbf{c}(\tilde{\boldsymbol{\phi }})\) which, however, are equivalent with respect to the optimization metric in (40) and produce the same support \(I(\mathbf{c}(\boldsymbol{\phi })) = I(\mathbf{c}(\tilde{\boldsymbol{\phi }}))\) in (42).

  8. 8.

    If the (D − 1) × D matrix is full-rank, then its null space has rank 1 and \(\mathbf{c}(\boldsymbol{\phi })\) is uniquely determined (within a sign ambiguity which, however, does not affect the final decision on the index-set). If, instead, the (D − 1) × D matrix is rank-deficient, then the intersection of the D hypersurfaces (i.e., the solution of (51)) is a p-manifold (with p ≥ 1) on the D-dimensional space and does not generate a new cell. Hence, combinations of D rows of V that result in linearly dependent rows of the (D − 1) × D matrix in (51) can be simply ignored.

  9. 9.

    Absolute-value errors put significantly less emphasis on extreme errors than squared-error expressions.

  10. 10.

    A combined L 1/L 2-norm approach has been followed in [16, 46].

  11. 11.

    We note that without the presented algorithm, computation of the L 1 principal component of A 2×50 would have required complexity proportional to 250, which is of course infeasible.

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  1. Department of Electronic and Computer Engineering, Technical University of Crete, Chania, 73100, Greece

    George N. Karystinos

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  1. Dept. Industrial & Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Karystinos, G.N. (2014). Optimal Algorithms for Binary, Sparse, and L 1-Norm Principal Component Analysis. In: Pardalos, P., Rassias, T. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1124-0_11

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