# Sturm: Sparse Tubal-Regularized Multilinear Regression for fMRI

## TL;DR

This paper introduces Sturm, a novel sparse multilinear regression method utilizing t-SVD and tubal tensor nuclear norm for improved classification of high-dimensional fMRI data with limited samples.

## Contribution

The paper proposes a new tensor regression approach based on t-SVD that enhances sparsity and classification accuracy in fMRI analysis, overcoming limitations of previous tensor methods.

## Key findings

- Sturm outperforms existing methods in fMRI classification tasks.
- Effective with a small number of voxels.
- Demonstrates superior accuracy on multiple datasets.

## Abstract

While functional magnetic resonance imaging (fMRI) is important for healthcare/neuroscience applications, it is challenging to classify or interpret due to its multi-dimensional structure, high dimensionality, and small number of samples available. Recent sparse multilinear regression methods based on tensor are emerging as promising solutions for fMRI, yet existing works rely on unfolding/folding operations and a tensor rank relaxation with limited tightness. The newly proposed tensor singular value decomposition (t-SVD) sheds light on new directions. In this work, we study t-SVD for sparse multilinear regression and propose a Sparse tubal-regularized multilinear regression (Sturm) method for fMRI. Specifically, the Sturm model performs multilinear regression with two regularization terms: a tubal tensor nuclear norm based on t-SVD and a standard L1 norm. We further derive the algorithm under the alternating direction method of multipliers framework. We perform experiments on four classification problems, including both resting-state fMRI for disease diagnosis and task-based fMRI for neural decoding. The results show the superior performance of Sturm in classifying fMRI using just a small number of voxels.