Sinkhorn Linearization and the Spectral Proxy: Unifying the Statistical and Algorithmic Theory of Feature-Parameterized Inverse Optimal Transport via a Single Spectral Sandwich

2026-08-13

AI Summary

This work addresses the lack of a unified theoretical foundation for parameter identifiability, estimation consistency, and algorithmic convergence in feature-parameterized inverse optimal transport (IOT). It proposes an analytical framework based on Sinkhorn linearization and spectral surrogates, which—through the introduction of spectral sandwich inequalities, restricted Hessian analysis, and ℓ₁ regularization—establishes, for the first time, four core theoretical guarantees for IOT: global parameter identifiability, exact support recovery, a Lipschitz continuous inverse mapping, and monotonic convergence of gradient descent. The framework combines geometric transparency with spectral precision and further quantifies the Hölder stability of the projection map under model misspecification. Numerical experiments corroborate the theoretical findings.

Abstract

We develop the statistical and algorithmic theory of inverse optimal transport (IOT) under the feature-parameterized cost C_theta(i,j) = -theta^T phi(i,j). The core technical contribution is the Sinkhorn linearization -- the implicit-function sensitivity of the entropic OT plan to the cost -- together with its spectral proxy, a formula that is spectrally exact yet geometrically transparent. The restricted Hessian on the tangent space satisfies the spectral sandwich (pi_min/epsilon) I <= H_T^{-1} <= (pi_max/epsilon) I, yielding the single core bound sigma_min >= (pi_min/(a_max epsilon)) sqrt(lambda_min(Sigma)) that drives the entire theory. On this core we establish four theorems and one observation. T1 (identifiability): theta is globally injective on the quotient of the gauge kernel, with dimension bound F <= (K-1)^2. T2 (sparsistency): the l1-penalized estimator recovers the true support under irrepresentability and score concentration, with exponential failure probability. T3 (well-posedness): the feature-moment map M(theta) = Phi^T x_theta is strongly monotone, and the inverse is Lipschitz with constant L <= epsilon ||Phi^T S_a||_op / (pi_min lambda_min(Sigma)). T4 (convergence): local strong convexity with mu >= pi_min^2 lambda_min(Sigma) / epsilon^2 guarantees monotone gradient descent convergence. O5 (misspecification): the estimator converges to the OT-model projection of the truth; the Holder continuity of the projection map is assessed numerically, yielding setting-dependent empirical exponents alpha_eff in (0,1).

Problem

• inverse optimal transport
• feature-parameterized cost
• parameter identifiability
• sparsistency
• model misspecification

Innovation

• Sinkhorn linearization
• spectral proxy
• inverse optimal transport
• feature-parameterized cost
• spectral sandwich