Tensor-on-tensor regression.

Area of Science:

• Multivariate statistics
• Tensor analysis
• Machine learning

Background:

• Traditional regression models often struggle with high-dimensional, multi-way data structures.
• Existing tensor prediction methods may lack generality or computational efficiency.
• The need for flexible frameworks to handle complex array data is growing.

Purpose of the Study:

• To propose a generalized linear prediction framework for multi-way arrays (tensors).
• To develop an efficient algorithm for penalized estimation in tensor regression.
• To demonstrate the framework's applicability to real-world data, such as facial images.

Main Methods:

• Utilizing the contracted tensor product for linear prediction between multi-way arrays.
• Employing reduced CP-rank constraints on coefficients to exploit multi-way structure.
• Implementing penalized least-squares estimation with L2 regularization.
• Deriving a Gibbs sampling algorithm for Bayesian inference.

Main Results:

• The proposed framework unifies and extends various existing tensor prediction approaches.
• An efficient penalized estimation algorithm is developed, incorporating ridge penalties.
• The objective function corresponds to the mode of a Bayesian posterior, enabling Gibbs sampling.
• Successful application to facial image data demonstrates practical utility.

Conclusions:

• The contracted tensor product provides a powerful and generalizable framework for tensor prediction.
• Reduced CP-rank coefficients effectively leverage the inherent multi-way structure of data.
• The developed penalized estimation and Bayesian inference algorithms are efficient and applicable.
• This approach offers a robust method for analyzing complex, high-dimensional array data.