Abstract
Phase-Locked Matrix Factorization (PLMF) is an algorithm to perform separation of synchronous sources. Such a problem cannot be addressed by orthodox methods such as Independent Component Analysis, because synchronous sources are highly mutually dependent. PLMF separates available data into the mixing matrix and the sources; the sources are then decomposed into amplitude and phase components. Previously, PLMF was applicable only if the oscillatory component, common to all synchronized sources, was known, which is clearly a restrictive assumption. The main goal of this paper is to present a version of PLMF where this assumption is no longer needed—the oscillatory component can be estimated alongside all the other variables, thus making PLMF much more applicable to real-world data. Furthermore, the optimization procedures in the original PLMF are improved. Results on simulated data illustrate that this new approach successfully estimates the oscillatory component, together with the remaining variables, showing that the general problem of separation of synchronous sources can now be tackled.
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Notes
- 1.
In EEG and MEG, the sources are not individual neurons, whose oscillations are too weak to be detected from outside the scalp even with no superposition. In this case, the sources are populations of closely located neurons oscillating together.
- 2.
“Real-valued” is used here to distinguish from other papers, where the absolute value operator is dropped, hence making the PLF a complex quantity[2011a].
- 3.
The vec(.) operator stacks the columns of a matrix into a column vector.
- 4.
- 5.
These numerical problems are the reason why no results for λ A = 0 are shown in this paper.
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Acknowledgements
This work was partially funded by the DECA-Bio project of the Institute of Telecommunications, and by the Academy of Finland through its Centres of Excellence Program 2006–2011.
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Almeida, M., Vigário, R., Bioucas-Dias, J. (2013). Phase-Locked Matrix Factorization with Estimation of the Common Oscillation. In: Latorre Carmona, P., Sánchez, J., Fred, A. (eds) Mathematical Methodologies in Pattern Recognition and Machine Learning. Springer Proceedings in Mathematics & Statistics, vol 30. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5076-4_4
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