Learning from Low Rank Tensor Data: A Random Tensor Theory Perspective

Mohamed El Amine Seddik, Malik Tiomoko, Alexis Decurninge, Maxim Panov, Maxime Guillaud

Keywords: Random tensor theory, random matrix theory, spiked models, tensors, supervised learning, unsupervised learning, clustering, ridge

Abstract: Under a simplified data model, this paper provides a theoretical analysis of learning from data that have an underlying low-rank tensor structure in both supervised and unsupervised settings. For the supervised setting, we provide an analysis of a Ridge classifier (with high regularization parameter) with and without knowledge of the low-rank structure of the data. Our results quantify analytically the gain in misclassification errors achieved by exploiting the low-rank structure for denoising purposes, as opposed to treating data as mere vectors. We further provide a similar analysis in the context of clustering, thereby quantifying the exact performance gap between tensor methods and standard approaches which treat data as simple vectors.