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Life sciences · Preprint
arXiv · August 13, 2026
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This is a theoretical and mathematical paper developing the statistical and algorithmic foundations of feature-parameterized inverse optimal transport through Sinkhorn linearization and spectral analysis. Four theorems and one observation establish identifiability, sparsistency, well-posedness, convergence guarantees, and misspecification behaviour under specified mathematical conditions. The work is purely theoretical with numerical illustration of one property; no clinical, biological, or empirical application evidence is provided.
Preprint.
Theta is globally injective on the quotient of the gauge kernel with dimension bound F <= (K-1)^2 (T1: identifiability) L1-penalized estimator recovers true support under irrepresentability and score concentration with exponential failure probability (T2: sparsistency) Feature-moment map is strongly monotone with Lipschitz inverse constant L <= epsilon ||Phi^T S_a||_op / (pi_min lambda_min(Sigma)) (T3: well-posedness)
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This is a theoretical paper establishing mathematical foundations and spectral properties for inverse optimal transport without empirical validation, clinical outcomes, or real-world application data.
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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).
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