Constrained Gaussian Wasserstein Optimal Transport with Commutative Covariance Matrices
This work addresses constraints in optimal transport for applications like lossy compression and generative PCA, but it is incremental as it focuses on a specific commutative case.
The paper tackles the problem of Gaussian Wasserstein optimal transport under constraints (rate, dimension, channel) when source and reconstruction covariance matrices commute, deriving explicit minimum achievable mean squared error results.
Optimal transport has found widespread applications in signal processing and machine learning. Among its many equivalent formulations, optimal transport seeks to reconstruct a random variable/vector with a prescribed distribution at the destination while minimizing the expected distortion relative to a given random variable/vector at the source. However, in practice, certain constraints may render the optimal transport plan infeasible. In this work, we consider three types of constraints: rate constraints, dimension constraints, and channel constraints, motivated by perception-aware lossy compression, generative principal component analysis, and deep joint source-channel coding, respectively. Special attenion is given to the setting termed Gaussian Wasserstein optimal transport, where both the source and reconstruction variables are multivariate Gaussian, and the end-to-end distortion is measured by the mean squared error. We derive explicit results for the minimum achievable mean squared error under the three aforementioned constraints when the covariance matrices of the source and reconstruction variables commute.