Noise-Augmented ℓ_0 Regularization of Tensor Regression with Tucker Decomposition
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Tensor data are multi-dimension arrays. Low-rank decomposition-based regression methods with tensor predictors exploit the structural information in tensor predictors while significantly reducing the number of parameters in tensor regression. We propose a method named NA_0CT^2 (Noise Augmentation for ℓ_0 regularization on Core Tensor in Tucker decomposition) to regularize the parameters in tensor regression (TR), coupled with Tucker decomposition. We establish theoretically that NA_0CT^2 achieves exact ℓ_0 regularization in linear TR and generalized linear TR on the core tensor from the Tucker decomposition. To our knowledge, NA_0CT^2 is the first Tucker decomposition-based regularization method in TR to achieve ℓ_0 in core tensor. NA_0CT^2 is implemented through an iterative procedure and involves two simple steps in each iteration – generating noisy data based on the core tensor from the Tucker decomposition of the updated parameter estimate and running a regular GLM on noise-augmented data on vectorized predictors. We demonstrate the implementation of NA_0CT^2 and its ℓ_0 regularization effect in both simulation studies and real data applications. The results suggest that NA_0CT^2 improves predictions compared to other decomposition-based TR approaches, with or without regularization and it also helps to identify important predictors though not designed for that purpose.
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