Expectation-maximization alternating least squares for tensor network logistic regression

In recent years, a learning method for classifiers using tensor networks (TNs) has attracted attention. When constructing a classification function for high-dimensional data using a basis function model, a huge number of basis functions and coefficients are generally required, but the TN model makes it possible to avoid the curse of dimensionality by representing the huge coefficients using TNs. However, there is a problem with TN learning, namely the gradient vanishing, and learning using the gradient method cannot be performed efficiently. In this study, we propose a novel optimization algorithm for learning TN classifiers by using alternating least square (ALS) algorithm. Unlike conventional gradient-based methods, which suffer from vanishing gradients and inefficient training, our proposed approach can effectively minimize squared loss and logistic loss. To make ALS applicable to logistic regression, we introduce an auxiliary function derived from Pólya-Gamma augmentation, allowing logistic loss to be minimized as a weighted squared loss. We apply the proposed method to the MNIST classification task and discuss the effectiveness of the proposed method.