Online nonnegative CP-dictionary learning for Markovian data
This work addresses a gap in convergence guarantees for online tensor factorization, which is important for applications in image, video, and time-series data analysis, though it is incremental in extending matrix factorization results to tensors.
The paper tackles the problem of online tensor factorization for streaming multi-modal data by introducing a novel algorithm that learns a CP basis under general constraints, including nonnegativity for interpretability, and proves almost sure convergence to stationary points for Markovian data, covering i.i.d. cases and showing faster convergence in experiments.
Online Tensor Factorization (OTF) is a fundamental tool in learning low-dimensional interpretable features from streaming multi-modal data. While various algorithmic and theoretical aspects of OTF have been investigated recently, a general convergence guarantee to stationary points of the objective function without any incoherence or sparsity assumptions is still lacking even for the i.i.d. case. In this work, we introduce a novel algorithm that learns a CANDECOMP/PARAFAC (CP) basis from a given stream of tensor-valued data under general constraints, including nonnegativity constraints that induce interpretability of the learned CP basis. We prove that our algorithm converges almost surely to the set of stationary points of the objective function under the hypothesis that the sequence of data tensors is generated by an underlying Markov chain. Our setting covers the classical i.i.d. case as well as a wide range of application contexts including data streams generated by independent or MCMC sampling. Our result closes a gap between OTF and Online Matrix Factorization in global convergence analysis \commHL{for CP-decompositions}. Experimentally, we show that our algorithm converges much faster than standard algorithms for nonnegative tensor factorization tasks on both synthetic and real-world data. Also, 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 from the same tensor data by exploiting the tensor structure in multiple ways.