研究随机反问题的稳定性和优化方法，提出基于测度传输的新框架。
Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow
- 在概率空间中建模未知参数，用测度传输理论分析反问题
- 不同度量选择显著影响解的稳定性与优化性能
- 适用于物理、生物中参数不确定的反演场景
物理或生物科学中的反问题常需恢复一个随机未知参数。目标是求解该参数的概率分布，使其生成的数据与观测值一致。这类问题自然属于随机反问题。本文探讨三个方面：直接反演、带正则化的变分形式，以及通过梯度流进行优化，类比确定性反问题。关键区别在于所处空间：此处为概率空间，而非欧几里得或Sobolev空间，因此需借助测度传输理论工具。研究发现，损失函数设计和优化过程中所选度量对解的稳定性与优化器性质有显著影响。
原文摘要 · Abstract (English)
Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with measurements. Consequently, these problems are naturally framed as stochastic inverse problems. In this paper, we explore three aspects of this problem: direct inversion, variational formulation with regularization, and optimization via gradient flows, drawing parallels with deterministic inverse problems. A key difference from the deterministic case is the space in which we operate. Here, we work within probability space rather than Euclidean or Sobolev spaces, making tools from measure transport theory necessary for the study. Our findings reveal that the choice of metric -- both in the design of the loss function and in the optimization process -- significantly impacts the stability and properties of the optimizer.
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