Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow

• URL: http://arxiv.org/abs/2410.00229v1
• Date: Mon, 30 Sep 2024 20:56:34 GMT
• Title: Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow
• Authors: Qin Li, Maria Oprea, Li Wang, Yunan Yang,
• Abstract summary: Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. In this paper, we explore three aspects of this problem: direct formulation, variational with regularization, and optimization via gradient flows.
• Score: 7.110337170229741
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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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