Least-Squares Problem Over Probability Measure Space

Abstract

In this work, we investigate the variational problem $$\ddot{I}{}_x^{\ast }={\text{argmin}}_{\ddot{I}{}_x}D(G\#\ddot{I}{}_x,\ddot{I}{}_y),$$ where $D$ quantifies the difference between two probability measures, and $G$ is a forward operator that maps a variable $x$ to $y=G(x)$. This problem can be regarded as an analogue of its counterpart in linear spaces (e.g., Euclidean spaces), ${\text{argmin}}_x\Vert G(x)-y{\Vert }^2$. Similar to how the choice of norm $\Vert \cdot \Vert$ influences the optimizer in ${\mathbb{R}}^d$ or other linear spaces, the minimizer in the probabilistic variational problem also depends on the choice of $D$. Our findings reveal that using a $\ddot{I}•$-divergence for $D$ leads to the recovery of a conditional distribution of $\ddot{I}{}_y$, while employing the Wasserstein distance results in the recovery of a marginal distribution.

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