aNy-way Independent Component Analysis

Abstract

Multimodal data fusion is a topic of great interest. Several fusion methods have been proposed to investigate coherent patterns and corresponding linkages across modalities, such as joint independent component analysis (jICA), multiset canonical correlation analysis (mCCA), mCCA+jICA, disjoint subspace using ICA (DS-ICA) and parallel ICA. JICA exploits source independence but assumes shared loading parameters. MCCA maximizes correlation linkage across modalities directly but is limited to orthogonal features. While there is no theoretical limit to the number of modalities analyzed together by jICA, mCCA, or the two-step approach mCCA+jICA, these approaches can only extract common features and require the same number of sources/components for all modalities. On the other hand, DS-ICA and parallel ICA can identify both common and distinct features but are limited to two modalities. DS-ICA assumes shared loading parameters among common features, which works well when links are strong. Parallel ICA simultaneously maximizes correlation between modalities and independence of sources, while allowing different number of sources for each modality. However, only a very limited number of modalities and linkage pairs can be optimized. To overcome these limitations, we propose aNy-way ICA, a new model to simultaneously maximize the independence of sources and correlations across modalities. aNy-way ICA combines infomax ICA and Gaussian independent vector analysis (IVA-G) via a shared weight matrix model without orthogonality constraints. Simulation results demonstrate that aNy-way ICA not only accurately recovers sources and loadings, but also the true covariance/linkage patterns, whether different modalities have the same or different number of sources. Moreover, aNy-way ICA outperforms mCCA and mCCA+jICA in terms of source and loading recovery accuracy, especially under noisy conditions.Clinical Relevance-This establishes a model for N-way data fusion of any number of modalities and linkage pairs, allowing different number of non-orthogonal sources for different modalities.

Figure 1

Diagram of aNy-way ICA. Sources S_m are estimated by maximization of independence in each modality separately. Corresponding loadings A_m are then organized into SCVs ak, followed by minimization of their mutual information with IVA-G, which amounts to maximization of the correlation structure within the SCV without orthogonality constraints [8]. The model is optimized with stochastic gradient descent until convergence.

Figure 2

Scenario 1 sources for (a) sMRI, (b) fMRI, and (c) EEG data.

Figure 3

Scenario 1. Cross-correlation of SCVs from (a) groundtruth, (b) aNy-way ICA, (c) mCCA+jICA, and (d) mCCA. Correlation coefficients between (e) recovered sources and ground-truth, and between (f) recovered loadings and ground-truth for each independent component (IC): sMRI (downward triangle), fMRI (star), and EEG data (upward triangle) from aNy-way ICA (red), mCCA+jICA(green) and mCCA(blue). Note that shapes and colors are consistent throughout the paper.

Figure 4

Scenario 2(a), where the number of the SCVs varied from 3 to 11. Recovered source accuracies for (a) sMRI, (c) fMRI, and (e) EEG data, and loading accuracies for (b) sMRI, (d) fMRI, and (f) EEG data using aNy-way ICA (red), mCCA+jICA with maximum (solid green) or minimum component number (dashed green), mCCA with maximum (solid blue) or minimum component number (dashed blue). Values are averaged across matched components per modality.

Figure 5.

Scenario 2(b), where the number of subjects N varied from 100 to 1000. Recovered source accuracies for (a) sMRI, (c) fMRI, and (e) EEG data, and loading accuracies for (b) sMRI, (d) fMRI, and (f) EEG data. Values are averaged across runs and matched components per modality.

Figure 6.

Scenario 2(c), where SNR varied from 30db to 0db. Recovered source accuracies for (a) sMRI, (c) fMRI, and (e) EEG data, and loading accuracies for (b) sMRI, (d) fMRI, and (f) EEG data. Values are averaged across runs and matched components per modality.