Constrained Gaussian Wasserstein Optimal Transport with Commutative Covariance Matrices

TL;DR

This work investigates Gaussian Wasserstein optimal transport under three practical constraints—rate, dimension, and channel—when source and reconstruction covariances commute. By exploiting this commutativity, it derives explicit, closed-form distortion expressions and optimal rate allocations for rate-constrained OT with and without common randomness, extends principal component analysis to a generative transport framework under dimension constraints, and analyzes channel-constrained OT including a no-common-randomness setting where a novel hybrid scheme outperforms separation. The results illuminate when separation-based, uncoded, or hybrid strategies are optimal and quantify the distortion tradeoffs as resource constraints vary, showing that the hybrid approach often yields the best performance. The Gaussian setting serves as a tractable, insightful benchmark and a potential worst-case reference for more general OT problems, with implications for perception-aware compression and joint source-channel coding.

Abstract

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.

Figures (4)

• Figure 1: Plots of $\underline{D}_r(R)$ and $\overline{D}_r(R)$ for the case where $(\lambda_1,\lambda_2,\lambda_3)=(2,3,1)$ and $(\hat{\lambda}_1,\hat{\lambda}_2,\hat{\lambda}_3)=(3,1,1)$.
• Figure 2: Plot of $D_d(\Gamma)$ for the case where $(\lambda_1,\lambda_2,\lambda_3)=(2,3,1)$ and $(\hat{\lambda}_1,\hat{\lambda}_2,\hat{\lambda}_3)=(3,1,1)$.
• Figure 3: Plots of $\underline{D}_c(P)$, $D^{(s)}_c(P)$, $D^{(u)}_c(P)$, and $D^{(h)}_c(P)$ for the case where $(\lambda_1,\lambda_2,\lambda_3)=(2,3,1)$ and $(\hat{\lambda}_1,\hat{\lambda}_2,\hat{\lambda}_3)=(3,1,1)$.
• Figure 4: Plots of $\overline{D}_r(R)$ and $\overline{D}'_r(R)$ for the case where $(\lambda_1,\lambda_2,\lambda_3)=(2,3,1)$ and $(\hat{\lambda}_1,\hat{\lambda}_2,\hat{\lambda}_3)=(3,1,1)$.

Theorems & Definitions (11)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Definition 3
• Theorem 3
• Theorem 4
• Definition 4
• Definition 5
• Theorem 5