Low-Rank Tensor Train Coefficient Array Estimation for Tensor-on-Tensor Regression.

Area of Science:

• Multilinear regression
• Tensor decomposition
• Machine learning

Background:

• Tensor-on-tensor regression generalizes existing multilinear methods but faces challenges with high-dimensional coefficient arrays.
• Current low CANDECOMP/PARAFAC (CP) rank approximation methods can be limited in stability and efficiency for such problems.

Purpose of the Study:

• To propose a novel tensor-on-tensor regression method utilizing low tensor train (TT) rank approximation for coefficient array estimation.
• To enhance the stability, efficiency, and prediction accuracy of tensor-on-tensor regression, particularly for high-dimensional data.

Main Methods:

• Employs a low TT rank approximation for the coefficient array.
• Utilizes a TT rounding procedure for adaptive rank selection.
• Incorporates an l2 constraint to prevent overfitting.
• Solves the optimization problem using hierarchical alternating least squares.

Main Results:

• The proposed method demonstrates superior prediction accuracy compared to state-of-the-art techniques on synthetic and real-life datasets.
• Achieves comparable computational complexity to existing methods.
• Shows improved computational efficiency for high-dimensional data with small mode sizes.

Conclusions:

• The low TT rank approximation offers a more stable and efficient approach for tensor-on-tensor regression.
• The adaptive rank selection and l2 regularization contribute to robust performance.
• This method advances the analysis of complex, high-dimensional tensor data.