L1-Regularized ICA: A Novel Method for Analysis of Task-related fMRI Data

TL;DR

In this method, the ℓ1-regularization term is added to the cost function of ICA, and minimization of the cost function is performed by a difference of convex functions algorithm.

Abstract

We propose a new method of independent component analysis (ICA) in order to extract appropriate features from high-dimensional data. In general, matrix factorization methods including ICA have a problem regarding the interpretability of extracted features. For the improvement of interpretability, it is considered that sparse constraint on a factorized matrix is helpful. With this background, we construct a new ICA method with sparsity. In our method, the L1-regularization term is added to the cost function of ICA, and minimization of the cost function is performed by difference of convex functions algorithm. For the validity of our proposed method, we apply it to synthetic data and real functional magnetic resonance imaging data.

Figures (9)

• Figure 1: The behaviors of Sparsity($\bm{A}$), $\mathrm{MAK}(\bm{S})$, and $\mathrm{RMSE}_{\bm{X}}$ versus $\alpha$ (left column) or $\kappa$ (right column) for $\bm X^{(1)}$ (top, mixture of Laplace/uniform) and $\bm X^{(2)}$ (bottom, mixture of log-normal/uniform): In both cases of $\bm X^{(1)}$ and $\bm X^{(2)}$, the red ($\kappa=0$) and the black lines (FastICA) are overlapped on the horizontal axis in the bottom left figure, and all lines by our method ($\alpha=10^{-5}, 10^{-3}, 10^{-1}$) are overlapped in the bottom right figure.
• Figure 2: Frequency and SR versus ${\rm Sparsity}(\bm{A})$: The blue bar and red line indicate the frequency of ${\rm Sparsity} (\bm A)$ and SR at each bin, respectively. 10 samples with ${\rm Sparsity}(\bm A) < 0.05$ are not displayed in this figure.
• Figure 3: The behaviors of Sparsity($\bm{A}$), $\mathrm{MAK}(\bm{S})$, and $\mathrm{RMSE}_{\bm{X}}$ versus $\alpha$ (left column) or $\kappa$ (right column) for large size data $\bm X^{(3)}$ (top) and $\bm X^{(4)}$ (bottom, noisy): In both cases of $\bm X^{(3)}$ and $\bm X^{(4)}$, the red ($\kappa=0$) and the black lines (FastICA) are overlapped on the horizontal axis in the bottom left figure, and all lines by our method ($\alpha=10^{-5}, 10^{-3}, 10^{-1}$) are overlapped in the bottom right figure.
• Figure 4: Correlation by our method: The values of Correlation between the respective temporal feature vector in $\bm{S}$ and the ground-truth timing vector $\bm{d}^{\mathrm{REST}}$ under various $\alpha, \kappa$ are shown.
• Figure 5: (A) The 18th row vector in $\bm S$ by our proposed method, $\bm s_{18}^{\rm (ours)}$: The orange background indicates the timing of showing image to a test subject. (B) Spatial map depicted on the cross-section of the brain: Each point has one-to-one correspondence with the element in the 18th spatial feature vector in $\bm A$, $\bm a_{18}^{\rm (ours)}$. The points are displayed according to the mapping between voxel position and each element in the spatial feature vector. The color represents the value of the element.

Theorems & Definitions (2)

• Theorem 1
• proof