Structure-Preserving Nonlinear Sufficient Dimension Reduction for Tensor Regression

Music City Center 

We present a novel approach to nonlinear sufficient dimension reduction for scalar-on-tensor regression and classification problems. Our method defines a Tensor Product Space within several RKHS and introduces two kinds of the dimension-folding subspace alongside the conventional SDR subspace within the Tensor Product Space. We demonstrate that, under mild conditions, the range of the regression operator in the Tensor Product Space resides within the conventional SDR subspace. Furthermore, we propose the Tucker and CP Tensor Envelope frameworks, designed to preserve the intrinsic multidimensional structure of tensor-valued predictors while achieving effective dimension reduction. This framework bridges the subspaces, enabling us to establish that the tensor envelope of regression operator is also contained within the Dimension Folding Subspace. By leveraging Population-level and Sample-level Estimation, and drawing inspiration from this two common tensor decomposition methods, we develop two optimization algorithms to enhance the operator's objective function. We evaluate the performance of our proposed estimators through comprehensive simulations and real-world applications.

Nonlinear Sufficient Dimension Reduction

Dimension Folding

Tensor Decomposition

Reproducing Kernel Hilbert Space

Tensor Envelope and Tensor Product Space

Coordinate Mapping