振荡逆问题的对称Sinkhorn-Gibbs推断
Symmetrized Sinkhorn-Gibbs Inference for Oscillatory Inverse Problems
浏览论文内容
中文总结 AI 辅助
针对振荡逆问题信号带符号无法直接应用经典最优传输的问题，提出对称Sinkhorn-Gibbs推断框架，提升了后验推断准确性与噪声鲁棒性。
中文摘要 AI 辅助
振荡逆问题常因信号失配和跳周期现象呈现高度非凸的失配景观，给不确定性量化带来重大挑战。Gibbs后验为融入问题特定的失配度量提供了灵活框架，但其性能高度依赖所选失配度量诱导的经验风险景观。最优传输在比较振荡信号时会考虑潜在特征域的空间和时间结构，但经典最优传输无法直接应用，因为观测信号和预测信号通常带有符号，不满足经典传输公式的正性要求。我们为振荡逆问题引入对称Sinkhorn-Gibbs推断框架，该方法结合了带符号信号的归一化过程与对称Sinkhorn损失，利用归一化信号及其归一化负信号的互补传输信息。所得损失被融入Gibbs后验框架，形成适配振荡数据的Gibbs推断方法。我们确立了该损失的光滑性性质，证明了所得Gibbs后验的适定性，推导了鲁棒性保证，并开发了后验计算的自适应采样策略。数值实验表明，与基于欧氏距离和逐迹Wasserstein损失的Gibbs后验相比，该方法的经验风险景观振荡更少、虚假局部最小值更少、后验推断更准确、对观测噪声的鲁棒性更强，且总体恢复效果更好。
英文摘要
Oscillatory inverse problems often exhibit highly nonconvex discrepancy landscapes due to signal misalignment and cycle-skipping phenomena, posing significant challenges for uncertainty quantification. While Gibbs posteriors provide a flexible framework for incorporating problem-specific discrepancy measures, their performance depends strongly on the empirical risk landscape induced by the chosen discrepancy measure. Optimal transport accounts for the spatial and temporal structure of the underlying feature domain when comparing oscillatory signals. However, the direct application of classical optimal transport is precluded because observed and predicted signals are typically signed and therefore do not satisfy the positivity requirements of classical transport formulations. We introduce a symmetrized Sinkhorn-Gibbs inference framework for oscillatory inverse problems. The proposed approach combines a normalization procedure for signed signals with a symmetrized Sinkhorn loss that exploits complementary transport information from the normalized signals and their normalized negations. The resulting loss is incorporated into the Gibbs posterior framework, yielding a Gibbs inference methodology tailored to oscillatory data. We establish smoothness properties of the proposed loss, prove well-definedness of the resulting Gibbs posterior, derive robustness guarantees, and develop an adaptive sampling strategy for posterior computation. Numerical experiments demonstrate less oscillatory empirical risk landscapes with fewer spurious local minima, more accurate posterior inference, greater robustness to observational noise, and improved population-level recovery than Gibbs posteriors based on Euclidean and trace-wise Wasserstein losses.
