Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow
Authors: Qin Li, Maria Oprea, Li Wang, Yunan Yang
Organizations: Department of Mathematics, University of Wisconsin-Madison, Madison, WI · Center for Applied Mathematics, Cornell University, Ithaca, NY · School of Mathematics, University of Minnesota Twin Cities, Minneapolis, MN · Department of Mathematics, Cornell University, Ithaca, NY
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 a 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.
Figures & tables
Figure 1. A diagram showing the relations between deterministic inverse problem ( 1 ) and the stochastic inverse problem ( 4 ) formulated based on the push-forward map.
Figure 2. Orthogonal decomposition of the domain of A .
Figure 3. In the over-determined case, the W2 gradient flow of KL divergence and the squared W2 metric between A#ρu and ρyδ have two different steady states ρy∞ . The KL divergence recovers the conditional distribution of ρyδ on Col(A) while the squared W2 metric recovers the marginal distribution of ρyδ on Col(A) .