Abstract
This paper introduces an adaptation of the sequential matrix diagonalization (SMD) method to high-dimensional functional magnetic resonance imaging (fMRI) data. SMD is currently the most efficient statistical method to perform polynomial eigenvalue decomposition. Unfortunately, with current implementations based on dense polynomial matrices, the algorithmic complexity of SMD is intractable and it cannot be applied as such to high-dimensional problems. However, for certain applications, these polynomial matrices are mostly filled with null values and we have consequently developed an efficient implementation of SMD based on a GPU-parallel representation of sparse polynomial matrices. We then apply our “sparse SMD” to fMRI data, i.e. the temporal evolution of a large grid of voxels (as many as 29,328 voxels). Because of the energy compaction property of SMD, practically all the information is concentrated by SMD on the first polynomial principal component. Brain regions that are activated over time are thus reconstructed with great fidelity through analysis-synthesis based on the first principal component only, itself in the form of a sequence of polynomial parameters.
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References
Alrmah MA, Weiss S, Redif S, Lambotharan S, McWhirter JG (2013) Angle of arrival estimation for broadband signals: a comparison. In: Proceedings on intelligent signal processing conference, IET, London, UK
Bai Z, Demmel J, Dongarra J, Ruhe A, Van der Vorst H (2000) Templates for the solution of algebraic eigenvalue problems: a practical guide. SIAM, Philadelphia
Baumgartner R, Ryner L, Richter W, Summers R, Jarmasz M, Somorjai R (2000) Comparison of two exploratory data analysis methods for fMRI: fuzzy clustering vs. principal component analysis. Magn Reson Imaging 18:89–94
Beckmann C, Smith S (2005) Tensorial extensions of independent component analysis for multisubject fMRI analysis. NeuroImage 25(1):294–311
Bell AJ, Sejnowski TJ (1995) An information maximisation approach to blind separation and blind deconvolution. Neural Comput 7(6):1129–1159
Carcenac M, Redif S (2016) A highly scalable modular bottleneck neural network for image dimensionality reduction and image transformation. Appl Intell 44(3):557–610
Carcenac M, Redif S, Kasap S (2017) GPU parallelization of the sequential matrix diagonalization algorithm and its application to high-dimensional data. J Supercomput 73(8):3603–3634
Comon P (1994) Independent component analysis—a new concept? Signal Process 36(3):287–314
Corr J, Thomson K, Weiss S, McWhirter JG, Redif S, Proudler IK (2014) Multiple shift maximum element sequential matrix diagonalisation for parahermitian matrices. In: IEEE workshop on statistical signal processing, Gold Coast, Australia, pp 312–315
Fisher M (2014) Marching cubes. https://graphics.stanford.edu/~mdfisher/MarchingCubes.html. Accessed 20 July 2018
Friston K, Josephs O, Rees G, Tuner R (1998) Nonlinear event-related responses in fMRI. Magn Reson Med 39:41–52
Glover G (2011) Overview of functional magnetic resonance imaging. Neurosurg Clin N Am 22(2):133–139
Golay X, Kollias S, Stoll G, Meier D, Valavanis A, Boesiger PA (1998) New correlation-based fuzzy logic clustering algorithm for fMRI. Magn Reson Imaging 40:249–260
Golub GH, Van Loan CF (2013) Matrix computations, 4th edn. The Johns Hopkins Univ. Press, Baltimore, MD
Hanke M, Dinga R, Hausler C, Guntupalli JS, Casey M, Kaule FR, Stadler J (2015) High-resolution 7-Tesla fMRI data on the perception of musical genres—an extension to the studyforrest dataset. https://f1000research.com/articles/4-174/v1. Accessed 20 July 2018
Intel Corporation (2018) Developer reference for Intel Math Kernel Library 2018 - C. https://software.intel.com/en-us/mkl-reference-manual-for-c. Accessed 20 July 2018
Kailath T (1980) Linear systems. Prentice-Hall, Englewood Cliffs, NJ
Kasap S, Redif S (2014) Novel field-programmable gate array architecture for computing the eigenvalue decomposition of para-Hermitian polynomial matrices. IEEE Trans VLSI Syst 22(3):522–536
Kim SG, Ugurbil K (1997) Functional magnetic resonance imaging of the human brain. J Neurosci Methods 74:229–243
Krishnan A, Williams LJ, McIntosh AR, Abdi H (2011) Partial least squares methods for neuroimaging: a tutorial and review. NeuroImage 56:455–475
Lazar NA, Luna B, Sweeney JA, Eddy WF (2002) Combining brains: a survey of methods for statistical pooling of information. NeuroImage 16:538–550
Mandelkow H, De Zwart JA, Duyn JH (2016) Linear discriminant analysis achieves high classification accuracy for the BOLD fMRI response to naturalistic movie stimuli. Front Hum Neurosci 10:128
McLachlan GJ (1992) Discriminant analysis and statistical pattern recognition. Wiley, Hoboken
McWhirter JG, Baxter PD, Cooper T, Redif S, Foster J (2007) An EVD algorithm for para-Hermitian polynomial matrices. IEEE Trans Signal Process 55(5):2158–2169
Miller K, Luh W, Lie T, Martinez A, Obata T, Wong E, Frank L, Buxton R (2001) Nonlinear temporal dynamics the cerebral blood flow response. Hum Brain Mapp 13:1–12
Moret N, Tonello A, Weiss S (2011) MIMO precoding for filter bank modulation systems based on PSVD. In: Proceedings on IEEE 73rd vehicle technology conference, pp 1–95
Nvidia Corporation (2018) CUDA Toolkit Documentation v9.1.85. http://docs.nvidia.com/cuda. Accessed 20 July 2018
Poldrack Lab and Center for Reproducible Neuroscience at Stanford University (2018) OpenfMRI. https://openfmri.org. Accessed 20 July 2018 (recently superseded by https://openneuro.org)
Polizzi E (2009) Density-matrix-based algorithm for solving eigenvalue problems. Phys Rev B 79:115112
Redif S (2015) Fetal electrocardiogram estimation using polynomial eigenvalue decomposition. Turk J Electr Eng Comput Sci. https://doi.org/10.3906/elk-1401-19
Redif S, Kasap S (2015) Novel reconfigurable hardware architecture for polynomial matrix multiplications. IEEE Trans VLSI Syst 23(3):454–465
Redif S, McWhirter JG, Baxter P, Cooper T (2006) Robust broadband adaptive beamforming via polynomial eigenvalues. In: Proceeding on IEEE OCEAN conference, pp 1–6
Redif S, Weiss S, McWhirter JG (2011) An approximate polynomial matrix eigenvalue decomposition algorithm for para-Hermitian matrices. In: Proceedings on 11th IEEE international symposium on signal processing and information technology, Bilbao, Spain, pp 421–425
Redif S, Weiss S, McWhirter JG (2015) Sequential matrix diagonalisation algorithms for polynomial EVD of parahermitian matrices. IEEE Trans Signal Process 63(1):81–89
Redif S, Weiss S, McWhirter JG (2017) Relevance of polynomial matrix decompositions to broadband blind signal separation. Signal Process 134:76–86
Ros BP, Bijma F, De Gunst MC, De Munck JC (2015) A three domain covariance framework for EEG/MEG data. NeuroImage 119:305–315
Shen H, Wang LB, Liu YD, Hu DW (2014) Discriminative analysis of resting-state functional connectivity patterns of schizophrenia using low dimensional embedding of fMRI. NeuroImage 49:3110–3121
Tkacenko A (2010) Approximate eigenvalue decomposition of para-Hermitian systems through successive FIR paraunitary transformations. In: Proceedings on IEEE international conference on acoustics, speech and signal processing, Dallas, TX, USA, pp 4074–4077
Tohidian M, Amindavar H, Reza AM (2013) A DFT-based approximate eigenvalue and singular value decomposition of polynomial matrices. EURASIP J Adv Signal Process 1:1–16
Turner B, Paul E, Miller M, Barbey A (2018) Small sample sizes reduce the replicability of task-based fMRI studies. Commun Biol. https://doi.org/10.1038/s42003-018-0073-z
Vaidyanathan PP (1993) Multirate systems and filter banks. Prentice-Hall, Englewood Cliffs, NJ
Virta J, Li B, Nordhausen K, Oja H (2016) Independent component analysis for tensor-valued data. Preprint available as arXiv:1602.00879
Weiss S, Pestana J, Proudler IK (2018) On the existence and uniqueness of the eigenvalue decomposition of a parahermitian matrix. Trans Signal Process 66(10):2659–2672
Wellcome Trust Centre for Neuroimaging (2018) Statistical parametric mapping (SPM). http://www.fil.ion.ucl.ac.uk/spm. Accessed 20 July 2018
Zarahn E, Aquirre GK, D’Esposito M (1997) Empirical analysis of BOLD fMRI statistics. NeuroImage 5:179–197
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Carcenac, M., Redif, S. Application of the sequential matrix diagonalization algorithm to high-dimensional functional MRI data. Comput Stat 35, 579–605 (2020). https://doi.org/10.1007/s00180-019-00925-8
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DOI: https://doi.org/10.1007/s00180-019-00925-8