anyakrakusuma: A Python Library for Entropic Schrödinger Bridges on Idealized Geometries

2026-07-20

AI Summary

This work addresses the numerical instability of discrete static Schrödinger bridge problems—equivalently, entropy-regularized optimal transport—under high cost-to-regularization ratios by introducing a log-domain Sinkhorn–Knopp iteration algorithm. The proposed method achieves high-precision solutions even in extreme parameter regimes where the Gibbs kernel suffers severe underflow. A diagnostic framework based on information entropy, covariance analysis, and geometric metrics is developed to precisely characterize the structural properties of optimal couplings and intermediate distributions. The algorithm maintains marginal residuals as low as 10⁻⁹ and perfect marginal fidelity at cost-to-regularization ratios up to 400, demonstrating convergence across approximately eight orders of magnitude. It accurately recovers geometric transformations such as 90-degree rotations with an error of merely 0.07°, and exhibits empirical convergence rates significantly surpassing theoretical worst-case bounds.

Abstract

We present anyakrakusuma, an open-source Python library that solves the discrete static Schrödinger bridge problem, the entropically regularized counterpart of optimal transport, through a log-domain Sinkhorn--Knopp iteration and reconstructs the entropic interpolation between two empirical point clouds. The solver is paired with a diagnostic pipeline that characterizes the optimal coupling and the intermediate distributions through information-theoretic and geometric measures. We exercise the library on four idealized planar cases spanning a circle-to-circle dilation, a spiral-to-mixture fragmentation, a rigid reorientation of two moons, and a Lissajous-to-trefoil deformation. The log-domain formulation is necessary rather than merely convenient at the parameters studied, where the cost-to-regularization ratio reaches four hundred and the Gibbs kernel underflows double precision across most of its range; the iteration nonetheless attains a marginal residual of $10^{-9}$ and unit marginal fidelity in every case. Residual histories decay geometrically over approximately eight decades at per-iteration contraction factors between $0.966$ and $0.976$, which are local rates near the fixed point that lie many orders of magnitude below the worst-case Hilbert-metric bound. The covariance analysis recovers an imposed ninety-degree reorientation to within $0.07^\circ$, roughly forty times smaller than its uncertainty, across a masked interval of near-isotropy on which the principal axis is unobservable. The diagnostics are reported with explicit attention to the regimes in which each is well defined, including the differential entropy, which is meaningful only on the open interpolation interval. The presented cases are constructed rather than measured; quantitative application to empirical point clouds requires further study.

Problem

• Schrödinger bridge
• optimal transport
• entropic regularization
• point clouds
• interpolation

Innovation

• entropic Schrödinger bridge
• log-domain Sinkhorn iteration
• optimal transport
• information-theoretic diagnostics
• numerical stability