Abstract

Tucker decomposition can provide an intuitive summary to understand brain function by decomposing multi-subject fMRI data into a core tensor and multiple factor matrices, and was mostly used to extract functional connectivity patterns across time/subjects using orthogonality constraints. However, these algorithms are unsuitable for extracting common spatial and temporal patterns across subjects due to distinct characteristics such as high-level noise. Motivated by a successful application of Tucker decomposition to image denoising and the intrinsic sparsity of spatial activations in fMRI, we propose a low-rank Tucker-2 model with spatial sparsity constraint to analyze multi-subject fMRI data. More precisely, we propose to impose a sparsity constraint on spatial maps by using an ℓp norm (0 < p ≤ 1), in addition to adding low-rank constraints on factor matrices via the Frobenius norm. We solve the constrained Tucker-2 model using alternating direction method of multipliers, and propose to update both sparsity and low-rank constrained spatial maps using half quadratic splitting. Moreover, we extract new spatial and temporal features in addition to subject-specific intensities from the core tensor, and use these features to classify multiple subjects. The results from both simulated and experimental fMRI data verify the improvement of the proposed method, compared with four related algorithms including robust Kronecker component analysis, Tucker decomposition with orthogonality constraints, canonical polyadic decomposition, and block term decomposition in extracting common spatial and temporal components across subjects. The spatial and temporal features extracted from the core tensor show promise for characterizing subjects within the same group of patients or healthy controls as well.

Highlights

  • TENSOR decompositions have attracted increasing attention in blind source separation (BSS) of multi-subject functional magnetic resonance imaging data, since it can be readily represented by a three-way/mode tensor

  • Taking S1 as example, RKCA performs much better than Canonical polyadic decomposition (CPD) when estimating spatial maps (SMs) under higher signal-to-noise ratio (SNR), but it may become worse than CPD when estimating time courses (TCs) under lower SNRs

  • We impose the sparsity constraint on SMs to incorporate the intrinsic characteristics of functional magnetic resonance imaging (fMRI) data, in addition to the low-rank constraint on factor matrices and the sparsity constraint on the core tensor

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Summary

Introduction

TENSOR decompositions have attracted increasing attention in blind source separation (BSS) of multi-subject functional magnetic resonance imaging (fMRI) data, since it can be readily represented by a three-way/mode tensor (voxel× time ×subject). Canonical polyadic decomposition (CPD), Tucker decomposition (TKD), and block term decomposition (BTD) are commonly adopted forms for tensor decomposition. TKD decomposes a three-way tensor into a core tensor and three factor matrices. It can be seen as a complete form of CPD with polyadic but not necessarily canonical expansion in rank-1 terms [3], and CPD is a special case of TKD model with a diagonal core tensor [4][5]. As a complete form of CPD [5], Tucker-2 model can obtain shared SMs and TCs from the two factor matrices and subject-specific intensities from the core tensor. Some researchers studied the dynamic functional connectivity networks of fMRI using the core tensor of TKD [10][11][18][19] and the core tensor was proven to carry features of original data in image processing [20] and analyses of electroencephalography (EEG) data [3][17]

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