Scalable Approximate Algorithms for Optimal Transport Linear Models
This work addresses the scalability and generalization of optimal transport linear models for researchers and practitioners in fields like spectroscopy and signal processing, but it is incremental as it builds on existing Sinkhorn-like methods.
The authors tackled the problem of generalizing optimal transport linear models beyond task-specific applications by proposing a scalable algorithmic framework for non-negative linear regression with entropy-regularized optimal transport loss and convex penalties, resulting in simple multiplicative updates suitable for large-scale implementation.
Recently, linear regression models incorporating an optimal transport (OT) loss have been explored for applications such as supervised unmixing of spectra, music transcription, and mass spectrometry. However, these task-specific approaches often do not generalize readily to a broader class of linear models. In this work, we propose a novel algorithmic framework for solving a general class of non-negative linear regression models with an entropy-regularized OT datafit term, based on Sinkhorn-like scaling iterations. Our framework accommodates convex penalty functions on the weights (e.g. squared-$\ell_2$ and $\ell_1$ norms), and admits additional convex loss terms between the transported marginal and target distribution (e.g. squared error or total variation). We derive simple multiplicative updates for common penalty and datafit terms. This method is suitable for large-scale problems due to its simplicity of implementation and straightforward parallelization.