A unified framework for non-negative matrix and tensor factorisations with a smoothed Wasserstein loss
This work provides a novel approach for dimensionality reduction in metric-structured data like imaging, though it appears incremental as it extends existing factorization methods with a smoothed Wasserstein loss.
The authors tackled the problem of computing non-negative matrix and tensor factorizations using a Wasserstein distance loss to incorporate data geometry, resulting in a general framework with an efficient computational method demonstrated through numerical examples.
Non-negative matrix and tensor factorisations are a classical tool for finding low-dimensional representations of high-dimensional datasets. In applications such as imaging, datasets can be regarded as distributions supported on a space with metric structure. In such a setting, a loss function based on the Wasserstein distance of optimal transportation theory is a natural choice since it incorporates the underlying geometry of the data. We introduce a general mathematical framework for computing non-negative factorisations of both matrices and tensors with respect to an optimal transport loss. We derive an efficient computational method for its solution using a convex dual formulation, and demonstrate the applicability of this approach with several numerical illustrations with both matrix and tensor-valued data.