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1 Maarten De Vos SISTA – SCD - BIOMED K.U.Leuven On the combination of ICA and CPA Maarten De Vos Dimitri Nion Sabine Van Huffel Lieven De Lathauwer

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2 Maarten De Vos What is ICA? What is CPA? Why combining ICA and CPA? Our algorithm Results Conclusion Roadmap

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3 Maarten De Vos What is ICA? What is CPA? Why combining ICA and CPA? Our algorithm Results Conclusion Roadmap

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4 Maarten De Vos EEG 1 EEG 2 EEG 3 Independent Component Analysis (ICA) ICA: Estimate statistically independent sources s 1, s 2 and s 3 ; and mixing coefficients m i i Decomposing a measurement (EEG) into contributing sources. EEG 3 = m 31 s 1 + m 32 s 2 + m 33 s 3 EEG 1 = m 11 s 1 + m 12 s 2 + m 13 s 3 EEG 2 = m 21 s 1 + m 22 s 2 + m 23 s 3

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5 Maarten De Vos Decomposition of a measured signal S1S1 SRSR M1M1 MRMR =++ …. + E Y PCA estimates orthogonal sources (basis) = MQQ * S S1S1 SRSR M1M1 MRMR =++ …. + E Y ICA estimates statistically independent sources = MS Matrix decompositions (e.g. PCA) are often not unique. ICA imposes statistical independence to sources.

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6 Maarten De Vos Different implementations of ‘independence’ Jade: Joint Approximate Diagonalization of Eigenmatrices All the higher-order cross-cumulants are zero Fourth order tensor cumulant is diagonal Mixing matrix is the matrix that approximately diagonalizes the eigenmatrices of cumulant Computation of ICA

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7 Maarten De Vos = * * * * * Approximate diagonalization Sobi: Second Order Blind Identification Assumption that sources are autocorrelated Mixing matrix also diagonalizes set of matrices Matrices are correlation matrices at different time lags * * * * * *

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8 Maarten De Vos Decomposition of a measured signal S1S1 SRSR M1M1 MRMR =++ …. + E Y B A = Y S C PCA Tucker / HOSVD : estimates subspace If a signal is multi-dimensional (=higher order tensor), multilinear algebra tools can be used that better exploit the multi-dimensional nature of the data. = B A S C QQ*PP* OO*

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9 Maarten De Vos Decomposition of a measured signal S1S1 SRSR M1M1 MRMR =++ …. + E Y S1S1 SRSR A1A1 ARAR =+ Y E BRBR B1B1 PCA CPA: Canonical / Parallel Factor Analysis If a signal is multi-dimensional (=higher order tensor), multilinear algebra tools can be used that better exploit the multi-dimensional nature of the data.

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10 Maarten De Vos CPA components are not orthogonal The best Rank – R approximation may not exist The R components are not ordered But the decomposition is unique and no rotation is possible without changing the model part … Something about CPA S1S1 SRSR A1A1 ARAR =++ …. Y E BRBR B1B1

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11 Maarten De Vos CPA computed often by ALS: –Minimization of the (Frobenius -) norm of residuals –Minimize 1)Initialize A,S,B 2)Update A, given S and B : 3)Update S, given A and B : 4)Update B, given A and S : 5)Iterate (2-3-4) until convergence Computation of CPA S1S1 SRSR A1A1 ARAR =++ …. Y E BRBR B1B1

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12 Maarten De Vos Sometimes long swamps, meaning that the costfunction converges very slowly. Computation of CPA (2)

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13 Maarten De Vos In order to reduce swamps, interpolate A, B and S from the estimates of 2 previous iterations and use the interpolated matrices at the current iteration 1.Line Search: 2.Then ALS update Choice of crucial =1 annihilates LS step (i.e. we get standard ALS) Search directions Improvement of ALS: Line search

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14 Maarten De Vos [Harshman, 1970] « LSH » [Bro, 1997] « LSB » [Rajih, Comon, 2005] « Enhanced Line Search (ELS) » [Nion, De Lathauwer, 2006] «Enhanced Line Search with Complex Step (ELSCS) » Improvement of ALS : Line search

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15 Maarten De Vos

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16 Maarten De Vos What is ICA? What is CPA? Why combining ICA and CPA? Our algorithm Simulation results Conclusion Overview

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17 Maarten De Vos These activations have different ratios in different subjects Gives rise to a trilinear CPA structure [Beckmann et al., 2005]

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18 Maarten De Vos Beckmann et al (2005) Combination of ICA and CPA : tensor pICA Tensor pICA outperforms CPA due to low Signal-to-Noise Ratio Algo from paper: –One iteration step to optimize ICA costfunction –One iteration step to optimize trilinear structure –Optimize ‘until convergence’ Algo implemented in paper: –Compute ICA on matricized tensor –Decompose afterwards mixing vector to obtain trilinear decomposition Tensor pICA

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19 Maarten De Vos Does it make sense to add constraints? –- : uniqueness –+: robustness –+: more identifiable if constraints make sense –+: see results

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20 Maarten De Vos What is ICA? What is CPA? Why combining ICA and CPA? Our algorithm Results Conclusion Overview

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21 Maarten De Vos We developed a new algorithm that simultaneously imposed the independence and the trilinear constraint A1A1 ARAR B1B1 BRBR =+ …. + Y SRSR S1S1

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22 Maarten De Vos Compute fourth-order cumulant tensor Compute the ‘eigenmatrices’ of this tensor -> 3 rd order tensor Add slice with covariance matrix to this tensor This tensor has a 3 rd order CPA structure ICA - CPA * * * * * * * * * * *

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23 Maarten De Vos Compute fourth-order cumulant tensor Compute the ‘eigenmatrices’ of this tensor -> 3 rd order tensor This tensor has a 3 rd order CPA structure When the mixing matrix has a bilinear structure (the mixing vector has a Khatri-Rao structure) and this tensor can be rewritten as a 5 th order tensor with CPA structure: ICA - CPA

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24 Maarten De Vos How to compute the 5 th order CPA? ALS breaks symmetry, simulations showed bad performance Taking into account the partial symmetry naturally preserved in a line-search scheme –Search directions: between current estimate and ALS update –Step size: rooting real polynomial of degree 10

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25 Maarten De Vos What is ICA? What is CPA? Why combining ICA and CPA? Our algorithm Results Conclusion Overview

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26 Maarten De Vos Application in fMRI? [Stegeman, 2007]: CPA on fMRI comparable to tensor pICA if correct number of components is chosen [Daubechies, 2009]: ICA on fMRI works rather because of sparsity than because of independence. Infomax

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27 Maarten De Vos We consider narrow-band sources received by a uniform circular array (UCA) of I identical sensors of radius P. We assume free-space propagation. The entries of A represent the gain between a transmitter and an antenna We generated BPSK user signals: all source distributions are binary (1 or -1), with an equal probability of both values. B contains the chips of the spreading codes for the different users. Application in telecommunications

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28 Maarten De Vos Well-conditioned mixtureMixture (5,2,1000) Rank overestimated Colored noise

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29 Maarten De Vos Outperforms orthogonality constraint

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30 Maarten De Vos We developed a new algorithm to impose both independence and trilinear constraints simultaneously: ICA-CPA We showed that the method outperforms both standard ICA and CPA in certain situations It should only be used when assumptions are validated … Conclusion

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31 Maarten De Vos

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