Enhancing Blind Source Separation with Dissociative Principal Component Analysis

TL;DR

A sophisticated approach is proposed that preserves the interpretability advantages of sPCA while significantly enhancing its source extraction capabilities and outperformed traditional methods such as PCA+ICA, PPCA+ICA, SPCA+ICA, PMD, and GPower.

Abstract

Sparse principal component analysis (sPCA) enhances the interpretability of principal components (PCs) by imposing sparsity constraints on loading vectors (LVs). However, when used as a precursor to independent component analysis (ICA) for blind source separation (BSS), sPCA may underperform due to its focus on simplicity, potentially disregarding some statistical information essential for effective ICA. To overcome this limitation, a sophisticated approach is proposed that preserves the interpretability advantages of sPCA while significantly enhancing its source extraction capabilities. This consists of two tailored algorithms, dissociative PCA (DPCA1 and DPCA2), which employ adaptive and firm thresholding alongside gradient and coordinate descent approaches to optimize the proposed model dynamically. These algorithms integrate left and right singular vectors from singular value decomposition (SVD) through dissociation matrices (DMs) that replace traditional singular values, thus capturing latent interdependencies effectively to model complex source relationships. This leads to refined PCs and LVs that more accurately represent the underlying data structure. The proposed approach avoids focusing on individual eigenvectors, instead, it collaboratively combines multiple eigenvectors to disentangle interdependencies within each SVD variate. The superior performance of the proposed DPCA algorithms is demonstrated across four varied imaging applications including functional magnetic resonance imaging (fMRI) source retrieval, foreground-background separation, image reconstruction, and image inpainting. They outperformed traditional methods such as PCA+ICA, PPCA+ICA, SPCA+ICA, PMD, and GPower.

Figures (6)

• Figure 1: Row (A) displays the ground truth spatial maps with moderate (left, $\rho=6$) and significant (right, $\rho=12$) overlaps. Rows (B) through (I) show the maps recovered as loading vectors/sources by various algorithms including B) PCA+ICA, C) PPCA+ICA, D) SPCA+ICA, E) PMD, F) GPower, G) DPCA1a, H) DPCA1b, and I) DPCA2. Correlation values between the ground truth and the recovered sources are indicated for each map, with the mean correlation displayed on the left.
• Figure 2: a) Comparison of all algorithms on synthetic data to illustrate the variance explained as the number of principal components varied from $2$ to $8$, and b) mean F-score over $10$ trials calculated for various noise variances across all algorithms.
• Figure 3: a) Average computation time illustrating the time efficiency of various algorithms when processing synthetic data, and b) convergence rate displaying the convergence behavior of the DPCA variants, measured by the normalized difference between successive updates along with error bars indicating variability across runs.
• Figure 4: a) shows the original images from three distinct scenarios: highway traffic, people in a corridor, and a person crossing a street. b) displays the ground truth for foreground detection, (c) through (i) display the foregrounds recovered by various algorithms in the following order: SPCA, PMD, GPower, DPCA1a, DPCA1b, and DPCA2, each corresponding to the scenarios depicted.
• Figure 5: The first column display the original noisy images with an input PSNR of 18.58dB. Subsequent images illustrate the results of image reconstruction using eight different methods: PCA, ICA, SPCA, PMD, GPower, DPCA1a, DPCA1b, and DPCA2. Output PSNR values are noted above each image, demonstrating the effectiveness of each reconstruction technique across three distinct images: Barbara, Hill, and Man.