From Matrix ICA to Tensor ICA: Architectures and Decomposition Choices

Classical ICA treats data as vectors or matrices. But many EEG/neuroimaging datasets are naturally tensors (for example: channel x time x trial, sometimes with frequency or subject modes). Flattening loses mode structure.

Classical vs Tensor Formulation

Classical ICA:

Multilinear ICA uses mode-wise transforms:

The core idea is to preserve mode-specific information while still searching for statistically independent components.

Architecture I vs Architecture II (Practical Reading)

In my notes, a useful split is:

• Architecture I: apply ICA to transposed/unfolded data to recover spatially independent basis behavior.
• Architecture II: apply ICA to unfolded data to obtain independent coefficients with a different global basis interpretation.

Both are valid; they imply different interpretability priorities.

Decomposition Choices Matter

A compact design equation I use:

tensor ICA method = tensor decomposition + objective function + optimization algorithm

Examples of decomposition layer:

• Tucker/HOSVD (stable, but parameter growth can be heavy in high order).
• CP/PARAFAC (leaner parameterization, stronger uniqueness assumptions).
• Tensor Ring / block-term styles (more expressive with different complexity profiles).

Objective and Optimization Layers

Objective choices include:

• nongaussianity (kurtosis/negentropy),
• mutual information minimization,
• likelihood-based criteria.

Optimization choices include:

• fixed-point updates,
• gradient/natural-gradient,
• orthogonalization strategy.

Practical Takeaway

For tensor data, decomposition choice is not a preprocessing detail. It is part of the ICA definition itself and directly controls memory use, convergence behavior, and interpretability.