Stochastic Mirror Descent for Low-Rank Tensor Decomposition Under Non-Euclidean Losses
This work addresses scalability and convergence challenges in tensor decomposition for statistical machine learning and signal processing, offering an incremental improvement over existing methods.
The paper tackles the problem of low-rank tensor decomposition under non-Euclidean losses, such as for count and binary data, by proposing a stochastic mirror descent framework with tensor fiber sampling, achieving promising performance and substantial computational savings compared to state-of-the-art methods.
This work considers low-rank canonical polyadic decomposition (CPD) under a class of non-Euclidean loss functions that frequently arise in statistical machine learning and signal processing. These loss functions are often used for certain types of tensor data, e.g., count and binary tensors, where the least squares loss is considered unnatural.Compared to the least squares loss, the non-Euclidean losses are generally more challenging to handle. Non-Euclidean CPD has attracted considerable interests and a number of prior works exist. However, pressing computational and theoretical challenges, such as scalability and convergence issues, still remain. This work offers a unified stochastic algorithmic framework for large-scale CPD decomposition under a variety of non-Euclidean loss functions. Our key contribution lies in a tensor fiber sampling strategy-based flexible stochastic mirror descent framework. Leveraging the sampling scheme and the multilinear algebraic structure of low-rank tensors, the proposed lightweight algorithm ensures global convergence to a stationary point under reasonable conditions. Numerical results show that our framework attains promising non-Euclidean CPD performance. The proposed framework also exhibits substantial computational savings compared to state-of-the-art methods.