Direct ICA on data tensor via random matrix modeling

Independent Component Analysis (ICA) is a fundamental method for Blind Source Separation (BSS). Classical ICA takes data matrix input formed by vector data. This paper focuses on ICA for BSS with third-order data tensor input formed by matrix data, such as 2D images. Two approaches exist for this problem. The first approach reshapes each matrix into a vector to apply classical ICA, with structural information lost. The second approach unfolds a data tensor into a data matrix along different modes to perform classical ICA mode-wise, which partially preserves structures but has strong or ill BSS assumptions. This paper proposes a third approach via RAndom Matrix ICA (RAMICA) modeling. RAMICA works on data tensor directly, without vectorization or unfolding, and preserves row or column structures under more general BSS assumptions. We develop the RAMICA model, algorithm, and related theories via defining new statistics for random matrices and new procedures for whitening and independent component estimation. We study the identifiability, higher-order extension, and relationships with existing methods. Experiments on both synthetic and real data show superior BSS performance of RAMICA over competing methods and offer insights on the trade-offs between different factors.