Sinkhorn Linearization and the Spectral Proxy: Unifying the Statistical and Algorithmic Theory of Feature-Parameterized Inverse Optimal Transport via a Single Spectral Sandwich
Introduces Sinkhorn linearization and spectral proxy, unifying statistical and algorithmic theory of feature-parameterized inverse OT with a core spectral bound.
Key Findings
Methodology
This paper models inverse optimal transport (IOT) with feature-parameterized cost C_θ(i,j) = -θ^T φ(i,j), framing it as a statistical inference problem. The core technical innovation is the Sinkhorn linearization, which derives the exact sensitivity of entropic OT plans to cost perturbations via implicit differentiation of KKT conditions. The authors introduce a spectral proxy formula, δxSSP = -(1/ε) P^T D_π P δc, which preserves spectral bounds (π_min, π_max) while providing geometric transparency. They establish a restricted Hessian H_T with a spectral sandwich (π_min/ε) I ≤ H_T^{-1} ≤ (π_max/ε) I, leading to a fundamental spectral bound σ_min(J_θ) ≥ (π_min / (a_max ε)) √λ_min(Σ). Building on this, four theorems are proved: (T1) identifiability of θ in the quotient space, (T2) support recovery via ℓ1-penalized estimation with exponential rate, (T3) strong monotonicity and Lipschitz stability of the feature-moment map M(θ), and (T4) convergence guarantees under local strong convexity. An additional observation (O5) characterizes estimator behavior under model misspecification, with numerical assessments of Holder continuity. The framework unifies geometric, spectral, and statistical analyses, providing a comprehensive understanding of feature-parameterized inverse OT.
Key Results
- The proposed method achieves support recovery with exponential probability decay as sample size n increases, with empirical support accuracy exceeding 95% in simulated and real datasets. The spectral bounds ensure numerical stability in high-dimensional feature spaces. Support errors decrease exponentially with n, validating the theoretical rate. The analysis confirms that the spectral proxy accurately captures the sensitivity of the OT plan, enabling robust support estimation even under model misspecification.
- Varying the regularization parameter λ reveals a critical balance: too large λ induces bias, too small causes support leakage. Numerical experiments show that the support support error diminishes exponentially with sample size, consistent with theoretical predictions. The spectral bounds underpin the stability of the inverse problem, ensuring that the estimator remains well-behaved in high-dimensional settings.
- In cases of model misspecification, the estimator converges to the projection of the true data-generating process onto the OT model set. Numerical results demonstrate that the residuals align with the predicted Holder continuity, confirming the robustness of the approach. These findings highlight the method’s capacity to handle real-world data deviations from ideal OT assumptions.
Significance
This work bridges the gap between geometric, spectral, and statistical perspectives in inverse OT, providing a unified framework that is both theoretically rigorous and practically applicable. The spectral bounds offer explicit guarantees for support recovery and stability, crucial for high-dimensional applications like transfer learning, economic matching, and biological lineage tracing. By establishing the fundamental limits and robustness properties, the paper paves the way for scalable, reliable inverse OT algorithms capable of handling complex, real-world data with high-dimensional features. Its theoretical insights also deepen understanding of the geometric structure underlying OT plans and their sensitivities, contributing significantly to the mathematical foundations of optimal transport and its inverse problems.
Technical Contribution
The paper’s key technical contribution is the derivation of the Sinkhorn linearization, which provides an exact expression for the OT plan’s sensitivity to cost perturbations via the implicit differentiation of KKT conditions. The introduction of the spectral proxy formula simplifies the analysis by maintaining spectral bounds while offering geometric clarity. The authors prove a spectral sandwich for the restricted Hessian, establishing a core bound that underpins all subsequent results. This spectral framework enables explicit support recovery rates, stability analysis, and convergence guarantees. The approach extends classical spectral analysis of OT to the feature-parameterized inverse setting, offering a unified, scalable, and geometrically interpretable methodology that advances both theoretical understanding and practical algorithms.
Novelty
This work is the first to unify the statistical and algorithmic analysis of feature-parameterized inverse OT through a spectral sandwich framework derived from Sinkhorn linearization. Unlike prior studies limited to low-dimensional or explicit cost models, this paper handles high-dimensional feature spaces with explicit bounds on support recovery and stability. The spectral proxy formula provides a novel geometric perspective on plan sensitivity, and the core spectral bound σ_min(J_θ) offers a fundamental limit on parameter identifiability. These innovations collectively form a comprehensive, scalable framework that bridges geometric, spectral, and statistical aspects of inverse OT, filling critical gaps in the literature.
Limitations
- The spectral bounds and theoretical guarantees rely on the assumption of fixed regularization ε and positive plan entries; in scenarios where π_min approaches zero, stability may deteriorate, limiting applicability in sparse or degenerate cases.
- Support recovery exponential rates depend on spectral properties of feature matrices, which may be difficult to verify or achieve in practice, especially with highly correlated features or complex data distributions.
- The analysis of model misspecification assumes Holder continuity of the projection map, which may not hold in highly nonlinear or non-stationary environments, potentially affecting robustness in real-world applications.
Future Work
未来的研究将集中在扩展光谱界限到非线性特征空间和动态模型，探索更复杂的模型失配情形，以及在大规模高维数据中提升算法效率。此外，结合深度学习技术以增强模型表达能力，研究非参数化的逆运输问题，以及在实际应用中实现实时、鲁棒的支持恢复，将是重要的研究方向。
AI Executive Summary
Inverse Optimal Transport (IOT) has emerged as a powerful framework for understanding complex transfer and matching phenomena across diverse fields such as economics, biology, and machine learning. Traditional approaches to IOT often rely on explicit, low-dimensional cost models, which struggle to scale to high-dimensional feature spaces and lack rigorous guarantees on support recovery and stability. Recognizing these limitations, the present work introduces a novel theoretical framework grounded in Sinkhorn linearization and spectral analysis, unifying the statistical and algorithmic aspects of feature-parameterized inverse OT.
At the core of this approach is the derivation of an exact sensitivity expression for entropic OT plans with respect to cost perturbations, achieved through the implicit differentiation of KKT conditions. This sensitivity is encapsulated in a spectral proxy formula, which preserves the spectral bounds dictated by the minimal and maximal plan entries, π_min and π_max. This spectral proxy not only simplifies computational procedures but also provides deep geometric insight into how the OT plan responds locally to changes in the cost parameters.
Building on this foundation, the authors establish a core spectral bound for the restricted Hessian of the entropic OT problem, forming a 'spectral sandwich' that tightly constrains the inverse problem’s stability. This bound underpins four key theorems: (T1) the global identifiability of the feature parameter θ in the quotient space, (T2) the exponential support recovery rate under ℓ1 regularization, (T3) the strong monotonicity and Lipschitz stability of the feature-moment map, and (T4) the convergence guarantees of gradient-based algorithms. An additional observation (O5) characterizes the behavior of the estimator under model misspecification, with numerical experiments confirming the Holder continuity of the projection map.
Empirically, the method demonstrates high support recovery accuracy (>95%) in simulated and real datasets, with exponential convergence rates as sample size increases. The spectral bounds ensure robustness in high-dimensional settings, and the support recovery remains stable under model deviations. These results significantly advance the theoretical understanding of inverse OT, providing explicit bounds and stability guarantees that were previously lacking.
This unified spectral framework not only deepens the mathematical understanding of inverse OT but also offers practical tools for scalable, reliable inference in complex, high-dimensional applications. Future directions include extending the analysis to nonlinear feature spaces, dynamic models, and integrating deep learning techniques to handle real-world data complexities. Overall, this work marks a substantial step forward in the theory and practice of inverse optimal transport, promising broad impacts across scientific and engineering disciplines.
Deep Dive
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).
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