发表机构
School of Medicine, Nankai University; College of Artificial Intelligence, Nankai University（南开大学医学院; 南开大学人工智能学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对特征参数化逆最优传输，提出Sinkhorn线性化与谱代理，建立谱三明治核心界，推导可识别性等四项定理及误设观察，统一其统计与算法理论。
AI 中文摘要
我们研究特征参数化代价函数C_θ(i,j) = -θ^T φ(i,j)下的逆最优传输（IOT）的统计与算法理论。核心技术贡献是Sinkhorn线性化——熵最优传输计划对代价的隐函数敏感性——及其谱代理，一个谱精确且几何透明的公式。切空间上的受限海森矩阵满足谱三明治（π_min/ε）I ≤ H_T^{-1} ≤（π_max/ε）I，由此得到驱动整个理论的核心界σ_min ≥（π_min/(a_max ε)）√λ_min(Σ)。基于此核心，我们建立四个定理和一个观察：T1（可识别性）：θ在规范核的商空间上是全局单射，维度界F ≤ (K-1)^2；T2（稀疏一致性）：在不可表示性与得分集中条件下，l1惩罚估计量以指数级失败概率恢复真实支撑；T3（适定性）：特征矩映射M(θ) = Φ^T x_θ是强单调的，其逆的利普希茨常数L ≤ ε ||Φ^T S_a||_op /（π_min λ_min(Σ)）；T4（收敛性）：局部强凸性μ ≥ π_min² λ_min(Σ) / ε²保证梯度下降单调收敛；O5（误设）：估计量收敛到真实的最优传输模型投影，对投影映射的赫尔德连续性进行数值评估，得到依赖于设定的经验指数α_eff ∈ (0,1)。
英文摘要
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).
Comments32 pages, 16 figures
