Spectral Shinkage of Gaussian Entropic Optimal Transport

Abstract: We present a functional calculus treatment of Entropic Optimal Transport (EOT) between Gaussian measures on separable Hilbert spaces, providing a unified framework that handles infinite-dimensional degeneracy. By leveraging the notion of proper alignment and the Schur complement, we reveal that the Gaussian EOT solution operates as a precise \textit{spectral shrinkage}: the optimal coupling is uniquely determined by contracting the spectrum of the correlation operator via a universal scalar function. This geometric insight facilitates an algorithmic shift from iterative fixed-point schemes (e.g., Sinkhorn) to direct algebraic computation, enabling efficient multi-scale analysis, where a single spectral decomposition allows for the exact evaluation of entropic costs across arbitrary regularization parameters $\varepsilon > 0$ at negligible additional cost. Furthermore, we investigate the asymptotic behavior as $\varepsilon \downarrow 0$ in settings where the unregularized Optimal Transport problem admits non-unique solutions. We establish a selection principle that the regularized limit converges to the most diffusive optimal coupling --characterized as the centroid of the convex set of optimal Kantorovich plans. This demonstrates that in degenerate regimes, the entropic limit systematically rejects deterministic Monge solutions (extremal points) in favor of the optimal solution with minimal Hilbert-Schmidt correlation, effectively filtering out spurious correlations in the null space. Finally, we derive stability bounds and convergence rates, recovering established parametric rates ($\varepsilon \log(1/\varepsilon)$) in finite dimensions while identifying distinct non-parametric rates dependent on spectral decay in infinite-dimensional settings.