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. 2018;27(3):638-647.
doi: 10.1080/10618600.2017.1401544. Epub 2018 Jun 6.

Tensor-on-tensor regression

Eric F Lock  1
Affiliations

Tensor-on-tensor regression

Eric F Lock. J Comput Graph Stat. 2018.

Abstract

We propose a framework for the linear prediction of a multi-way array (i.e., a tensor) from another multi-way array of arbitrary dimension, using the contracted tensor product. This framework generalizes several existing approaches, including methods to predict a scalar outcome from a tensor, a matrix from a matrix, or a tensor from a scalar. We describe an approach that exploits the multiway structure of both the predictors and the outcomes by restricting the coefficients to have reduced CP-rank. We propose a general and efficient algorithm for penalized least-squares estimation, which allows for a ridge (L 2) penalty on the coefficients. The objective is shown to give the mode of a Bayesian posterior, which motivates a Gibbs sampling algorithm for inference. We illustrate the approach with an application to facial image data. An R package is available at https://github.com/lockEF/MultiwayRegression.

Keywords: Multiway data; PARAFAC/CANDECOMP; reduced rank regression; ridge regression.

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Figures

Figure 1
Figure 1
Relative prediction error for characteristics of out-of-sample images for different parameter choices. The top row (full rank) gives the results under separate ridge regression models for each outcome without rank restriction.
Figure 2
Figure 2
Actual vs. predicted values for 1000 test images across 72 characteristics.
Figure 3
Figure 3
Example test image (left), and its posterior samples for 5 select characteristics (right).
See this image and copyright information in PMC

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