Midterm colloquium Pierre-Antoine Denarié

20 May 2022 14:00 till 14:30 - Location: lecture room E (Robert Hooke), 3mE - By: DCSC | Add to my calendar

"Tensor decomposition for algebraic Independent Component Analysis methods through efficient computation, storing and diagonalization of the cumulant tensor"

In signal analysis there is often a need for separating source signals that are measured as mixtures as is the case with analysis of functional Magnetic Resonance Imaging (fMRI) data.
Defined as Blind Source Seperation (BSS), the problem is inherently ill-posed in nature. Through the central limit theorem, Independent Component Analysis (ICA) manages to provide solutions to the problem by assuming that the source components are statistically independent. Common practice is to use the fourth-order statistic kurtosis, identical to the fourth-order cumulant, as it provides a measure of independence. ICA methods are categorized as optimization methods or as algebraic methods. Analogous to Principal ComponentAnalysis (PCA), algebraic methods diagonalize the fourth-order cumulant tensor containing the cross-kurtosis values which consequently represents statistically independent components. When a low Signal to Noise Ratio (SNR) is present, which often is the case in fMRI data, it has been shown that this produces results superior to the current go-to optimization method fastICA. However, the computation, storing and diagonalization of the tensor suffers from the curse of dimensionality which hinders algebraic methods from being practical for the estimation of many components. No research exists which directly addresses the scalability issue of algebraic ICA. Even tensor decomposition, which is often used as a means of lifting the curse of dimensionality, has no mention of addressing the issue at hand.

The primary objective of this literature study is to identify any gaps in the knowledge on how to address the scalability issues related to diagonalizing the cumulant tensor. By clearly
defining the root causes of the scalability issue it is shown how tensor decomposition by itself fails as a solution due to not being able to address the high cumulant tensor forming cost and intermediate storage cost. However, the forming and storage cost can be addressed through efficient computation methods which exploit the symmetry of the tensor. Of these methods, the implicit Canonical Polyadic Decomposition (CPD) computation of the cumulant tensor bypasses the initial forming cost and drastically reduces the storage needed of the tensor while simultaneously providing a solution to the BSS problem due to its diagonalization property. Besides the optimization based methods for computing CPD there exists a fast algebraic method based on General EigenValue Decomposition (GEVD) for which it is not studied yet whether implicit computation of the cumulant tensor can be used. The second identified method called Block Compact Symmetric Storage (BCSS), allows for more efficient computation and storing of the cumulant tensor in comparison to the regular case which can potentially speed up already existing algebraic ICA methods. No research exists on the abovementioned methods for algebraic ICA and for the fMRI case which further emphasizes the presence of a knowledge gap and the direction for future research.

Dr. K. Batselier and B. Hunyadi

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