研究高斯分布下带约束的最优传输，给出最小均方误差的显式解。
Constrained Gaussian Wasserstein Optimal Transport with Commutative Covariance Matrices
- 在协方差矩阵可交换条件下，推导三类约束下的最优传输解
- 在速率、维度和信道约束下，给出最小均方误差的闭式表达
- 适用于感知感知压缩、生成主成分分析等场景
最优传输在信号处理与机器学习中应用广泛。其核心目标是在源端随机变量已知的前提下，以最小期望失真重建具有指定分布的目标随机变量。然而实际中，某些约束会使最优传输方案不可行。本文研究三类约束：速率约束（源于感知感知有损压缩）、维度约束（源于生成主成分分析）和信道约束（源于深度联合源信道编码）。重点关注高斯沃尔什斯坦最优传输情形，即源与重建变量均为多元高斯分布，且端到端失真以均方误差衡量。当源与重建变量的协方差矩阵可交换时，本文推导出三种约束下可达最小均方误差的显式结果。
原文摘要 · Abstract (English)
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.
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