Decomposable Sparse Tensor on Tensor Regression

TL;DR

This paper introduces a novel method for sparse low-rank tensor-on-tensor regression that transforms the problem into simpler steps, leading to improved accuracy and predictor selection by leveraging tensor structure.

Contribution

It proposes a new decomposition-based approach for high-dimensional tensor regression, enabling efficient and accurate modeling of tensor predictors and responses.

Findings

Outperforms existing methods in accuracy

More effective predictor selection

Efficient stagewise search algorithm

Abstract

Most regularized tensor regression research focuses on tensors predictors with scalars responses or vectors predictors to tensors responses. We consider the sparse low rank tensor on tensor regression where predictors $\mathcal{X}$ and responses $\mathcal{Y}$ are both high-dimensional tensors. By demonstrating that the general inner product or the contracted product on a unit rank tensor can be decomposed into standard inner products and outer products, the problem can be simply transformed into a tensor to scalar regression followed by a tensor decomposition. So we propose a fast solution based on stagewise search composed by contraction part and generation part which are optimized alternatively. We successfully demonstrate our method can out perform current methods in terms of accuracy and predictors selection by effectively incorporating the structural information.