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Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schr\"{o}dinger Bridge Estimation
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Analyzes the statistical and algorithmic convergence of intermediate Sinkhorn iterations for estimating Schrödinger bridges (Sinkhorn bridge) in the two-sample setting, and connects it to several existing SB solvers.
Problem
Statistical guarantees for Schrödinger bridge estimation from finite samples of both source and target distributions, and for intermediate Sinkhorn iterates of the Sinkhorn bridge method.
Approach
Extends the one-sample Sinkhorn bridge analysis to the two-sample setting, bounding squared TV error by O(1/m+1/n) and showing exponential decay in Sinkhorn iterations (rate tanh(R^2/eps)^{4k}); shows SF2M, DSBM-IMF, BM2 and lightSB(-M) reduce to the Sinkhorn bridge estimator.
Key results
- Two-sample Sinkhorn bridge estimator error is O(1/m+1/n) in expected squared TV, improving the n-rate from n^{-1/2} to n^{-1} of the one-sample bound.
- Intermediate Sinkhorn iterates converge exponentially: E[TV^2] bounded by (tau/(1-tau)) R^6/eps^2 * tanh(R^2/eps)^{4k}.
- Numerical experiments show MSE scales roughly as (1/m+1/n) and integrated MSE decays exponentially in k.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| increase source and target sample sizes m and n | Sinkhorn bridge (optimal potentials) · 3D Gaussian source and target distributions | drift MSE | Helped |
| increase the number of Sinkhorn iterations k | Sinkhorn bridge (k iterations) · 3D Gaussian source and target distributions, [0, τ] | integrated drift MSE | Helped |
| extend the analysis to two-sample estimation | Sinkhorn bridge estimation · Two-sample Schrödinger bridge estimation | expected squared total variation error rate in n: n^{-1/2} -> n^{-1} | Helped |
Limitations
- Bound depends linearly on ambient dimension d, unlike intrinsic-dimension-only one-sample bound
- Requires compact smooth manifold support and bounded supports; restricts time interval to [0,tau] with tau<1 because estimates degenerate near t=1
- Constants deteriorate in epsilon and R compared with one-sample bound
Experimental setup
Datasets: synthetic 3-dimensional Gaussians eight-Gaussians moons
Metrics: mean squared error of drift integrated MSE vs Sinkhorn iterations
Baselines: SF2M DSBM-IMF BM2 lightSB lightSB-M
Builds on: Sinkhorn bridge (Pooladian and Niles-Weed, 2024) Sinkhorn algorithm Girsanov theorem Markovian projection
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