Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow
This addresses the challenge of handling randomness in inverse problems for physical or biological sciences, representing an incremental advancement by extending deterministic methods to probability spaces.
The paper tackled the stochastic inverse problem of recovering an unknown random parameter's probability distribution from measurements, finding that the choice of metric in loss function design and optimization significantly impacts stability and optimizer properties.
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.