Generalized Multi-Linear Models for Sufficient Dimension Reduction on Tensor Valued Predictors

TL;DR

This paper introduces a novel multi-linear sufficient dimension reduction method for tensor-valued predictors in supervised learning, providing efficient estimation procedures with proven statistical properties and demonstrated superior performance.

Contribution

The paper develops a new tensor-based dimension reduction technique with theoretical guarantees and practical algorithms applicable to high-dimensional and binary data.

Findings

Efficient estimation algorithms for tensor predictors.

Proven consistency and asymptotic normality of estimators.

Superior performance in simulations and real data.

Abstract

We consider supervised learning (regression/classification) problems with tensor-valued input. We derive multi-linear sufficient reductions for the regression or classification problem by modeling the conditional distribution of the predictors given the response as a member of the quadratic exponential family. We develop estimation procedures of sufficient reductions for both continuous and binary tensor-valued predictors. We prove the consistency and asymptotic normality of the estimated sufficient reduction using manifold theory. For continuous predictors, the estimation algorithm is highly computationally efficient and is also applicable to situations where the dimension of the reduction exceeds the sample size. We demonstrate the superior performance of our approach in simulations and real-world data examples for both continuous and binary tensor-valued predictors.