Soft Tensor Regression.

Area of Science:

• Statistics
• Machine Learning
• Neuroscience

Background:

• Traditional tensor regression methods often vectorize predictors or use low-rank approximations, which can be limiting.
• Classic Parallel Factors (PARAFAC) approximations may struggle when the true tensor rank is high.

Purpose of the Study:

• To propose a novel tensor regression framework, Soft Tensor Regression (Softer), based on a softened PARAFAC approximation.
• To enhance model flexibility, estimation accuracy, and predictive precision in tensor regression.
• To theoretically establish the consistency of the Softer posterior distribution.

Main Methods:

• Developed a Bayesian inference approach for the proposed Softer framework.
• Introduced a "soft" PARAFAC approximation allowing variations in component contributions.
• Adapted Softer for symmetric and semi-symmetric tensor predictors.

Main Results:

• Softer demonstrated increased model flexibility and improved estimation of coefficient tensors.
• The framework led to more accurate identification of important predictor entries and precise predictions.
• Theoretical analysis confirmed weak consistency of the posterior distribution, irrespective of tensor rank.

Conclusions:

• Softer offers a more robust and flexible alternative to classic PARAFAC for tensor regression.
• The method provides improved performance in estimation, prediction, and identification of key predictors.
• Softer's theoretical guarantees and practical advantages make it suitable for analyzing complex data, including brain network characteristics.