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GPU parallelization of the sequential matrix diagonalization algorithm and its application to high-dimensional data

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Abstract

This paper presents the parallelization on a GPU of the sequential matrix diagonalization (SMD) algorithm, a method for diagonalizing polynomial covariance matrices, which is the most recent technique for polynomial eigenvalue decomposition. We first parallelize with CUDA the calculation of the polynomial covariance matrix. Then, following a formal transformation of the polynomial matrix multiplication code—extensively used by SMD—we insert in this code the cublasDgemm function of CUBLAS library. Furthermore, a specialized cache memory system is implemented within the GPU to greatly limit the PC-to-GPU transfers of slices of polynomial matrices. The resulting SMD code can be applied efficiently over high-dimensional data. The proposed method is verified using sequences of images of airplanes with varying spatial orientation. The performance of the parallel codes for polynomial covariance matrix generation and SMD is evaluated and reveals speedups of up to 161 and 67, respectively, relative to sequential execution on a PC.

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Abbreviations

PEVD:

Polynomial eigenvalue decomposition

SMD:

Sequential matrix diagonalization

MIMO:

Multi-input multi-output (convolution)

PCA:

Principal component analysis

n :

Dimensionality of the data

\(\mathbf{A}\) :

Polynomial matrix

\(\mathbf{R}\) :

Polynomial covariance matrix

\(\mathbf{E}\) :

Matrix of polynomial eigenvectors

W :

Lag window half-length used to calculate \(\mathbf{R}\)

\(m_r\) :

Lag window length used to calculate \(\mathbf{R}\)

\(\mathbf{[R]}_{\mathbf{p}}\) :

Slice of \(\mathbf{R}\) at lag index p

GPU:

Graphical processing unit

CUDA:

Compute unified device architecture

CUBLAS:

CUDA basic linear algebra subroutines

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Author information

Authors and Affiliations

  1. Via Mersin 10, Boǧaztepe, Turkey

    Manuel Carcenac

  2. Department of Electrical and Electronics Engineering, European University of Lefke, Via Mersin 10, Gemikonaǧı, Turkey

    Soydan Redif

  3. Department of Computer Engineering, College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait

    Server Kasap

Authors
  1. Manuel Carcenac
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  2. Soydan Redif
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  3. Server Kasap
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Correspondence to Manuel Carcenac.

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Carcenac, M., Redif, S. & Kasap, S. GPU parallelization of the sequential matrix diagonalization algorithm and its application to high-dimensional data. J Supercomput 73, 3603–3634 (2017). https://doi.org/10.1007/s11227-017-1961-6

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  • DOI: https://doi.org/10.1007/s11227-017-1961-6

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