Quadratic Multilinear Discriminant Analysis for Tensorial Data Classification

Over the past decades, there has been an increase of attention to adapting machine learning methods to fully exploit the higher order structure of tensorial data. One problem of great interest is tensor classification, and in particular the extension of linear discriminant analysis to the multilinear setting. We propose a novel method for multilinear discriminant analysis that is radically different from the ones considered so far, and it is the first extension to tensors of quadratic discriminant analysis. Our proposed approach uses invariant theory to extend the nearest Mahalanobis distance classifier to the higher-order setting, and to formulate a well-behaved optimization problem. We extensively test our method on a variety of synthetic data, outperforming previously proposed MDA techniques. We also show how to leverage multi-lead ECG data by constructing tensors via taut string, and use our method to classify healthy signals versus unhealthy ones; our method outperforms state-of-the-art MDA methods, especially after adding significant levels of noise to the signals. Our approach reached an AUC of $0.95\left(0.03\right)$ on clean signals—where the second best method reached $0.91\left(0.03\right)$—and an AUC of $0.89\left(0.03\right)$ after adding noise to the signals (with a signal-to-noise-ratio of $-30$)—where the second best method reached $0.85\left(0.05\right)$. Our approach is fundamentally different than previous work in this direction, and proves to be faster, more stable, and more accurate on the tests we performed.