Samopriya Basu
Department of Statistics & Actuarial Science, 尤物视频
(Joint work with Faezeh Yazdi and Don Estep)
Title: An optimal-transport solution to marginal-constrained stochastic inverse problems
Date: Friday, March 7th, 2025
Time: 1:30PM (PDT)
Location: AQ 5037
Abstract: Of central importance to scientific inference and experimental design is the inverse problem of determining uncertainty in input characteristics from observed variation in data produced as output of a physical system running on the inputs. In practice, given observations {di = Q(位i , vi): i = 1, . . . , M} from the output of a smooth map Q: V 脳 V 鈫� R m, we want to construct a probability measure P藛 over the argument space V 脳V that is consistent with the observations. In the present work, the input space is factored as a product because we also assume that the variation over part of it is known, possibly also in the form of a sample v1, v2, . . . , vN 鈭� V. This serves as a marginal constraint on the probability measure P藛 to be reconstructed. There are two variations to this problem. If M = N and the (v, d)-data is paired, by augmenting the map Q with the projection onto V, we show that the present marginal-constrained inverse problem can be subsumed under the standard framework of (unconstrained) stochastic inverse problems developed previously in the literature. If, however, the vj-s are known from a separate experiment, we will not know the joint distribution of this aforementioned augmented map despite knowing its (empirical) marginals over the output space and the V-margin of the input. For the latter case, we propose to couple the observed marginals by (entropic) optimal transport with an asymmetric cost designed to fit the observed vj-s to di-s at fixed 位-s. This yields a joint distribution for the augmented map which can then be used to solve the inverse problem. In this talk, summarizing ongoing work, I lay out the theory behind our proposed methodology and demonstrate its efficacy on a simple synthetic example.