Sinkhorn linearization unifies inverse optimal transport theory with spectral sandwich bound

Inverse optimal transport (IOT) asks: given an observed transport plan, can we recover the underlying cost function? This paper, by Han Dong and four co-authors, tackles the feature-parameterized setting C_θ(i,j) = -θ^T φ(i,j), where costs are linear in features. The authors' core innovation is the Sinkhorn linearization—an implicit-function sensitivity analysis of the entropic OT plan to cost perturbations—paired with a spectral proxy that is spectrally exact but geometrically transparent. This combination yields a 'spectral sandwich' bound, (π_min/ε)I ≤ H_T^{-1} ≤ (π_max/ε)I, on the restricted Hessian, from which a single core inequality drives the entire theory.

On this foundation, the paper proves four theorems and one observation. T1 establishes global identifiability of θ up to a gauge kernel, with a dimension bound F ≤ (K-1)^2. T2 shows that an l1-penalized estimator recovers the true support (sparsistency) under irrepresentability and score concentration, with exponentially decaying failure probability. T3 proves the feature-moment map is strongly monotone and its inverse is Lipschitz with constant L ≤ ε||Φ^T S_a||_op / (π_min λ_min(Σ)). T4 guarantees local strong convexity (μ ≥ π_min^2 λ_min(Σ)/ε^2), ensuring monotone convergence of gradient descent. Finally, O5 analyzes model misspecification, showing convergence to the OT-model projection of the truth and numerically assessing Hölder continuity of the projection map with exponents α_eff in (0,1). This unified framework—covering statistical identifiability, optimization guarantees, and robustness—gives IOT practitioners a single theoretical backbone for designing estimators and solvers.