Online nonnegative tensor factorization and CP-dictionary learning for Markovian data

Nonnegative Matrix Factorization (NMF) algorithms are fundamental tools in learning low-dimensional features from vector-valued data, Nonnegative Tensor Factorization (NTF) algorithms serve a similar role for dictionary learning problems for multi-modal data. Also, there is often a critical interest in \textit{online} versions of such factorization algorithms to learn progressively from minibatches, without requiring the full data as in conventional algorithms. However, the current theory of Online NTF algorithms is quite nascent, especially compared to the comprehensive literature on online NMF algorithms. In this work, we introduce a novel online NTF algorithm that learns a CP basis from a given stream of tensor-valued data under general constraints. In particular, using nonnegativity constraints, the learned CP modes also give localized dictionary atoms that respect the tensor structure in multi-model data. On the application side, we demonstrate the utility of our algorithm on a diverse set of examples from image, video, and time series data, illustrating how one may learn qualitatively different CP-dictionaries by not needing to reshape tensor data before the learning process. On the theoretical side, we prove that our algorithm converges to the set of stationary points of the objective function under the hypothesis that the sequence of data tensors have functional Markovian dependence. This assumption covers a wide range of application contexts including data streams generated by independent or MCMC sampling.