Least-Squares Problem Over Probability Measure Space

TL;DR

We study a variational framework over probability measures to invert a forward map $\mathcal{G}$ by matching $\mathcal{G}\#ρ_x$ to a data measure $ρ_y$ via a discrepancy $\mathcal{D}$. When $\mathcal{D}$ is a $\phi$-divergence, the recovered push-forward equals the conditional distribution of $ρ_y$ on the forward range $\mathcal{R}=\mathcal{G}(\Theta)$; when $\mathcal{D}$ is the $p$-Wasserstein distance, it equals the projection of $ρ_y$ onto $\mathcal{R}$, via a projection operator $\mathcal{P_G}$. The results rely on measure disintegration and projection arguments and delineate a clear distinction from Bayesian inversion, highlighting that the reconstructed object remains a probability measure rather than collapsing to a Dirac delta in the deterministic limit. This provides a principled, metric-dependent way to perform inverse inference under nontrivial data support, with potential implications for uncertainty quantification in nonlinear inverse problems.

Abstract

In this work, we investigate the variational problem $$ρ_x^\ast = \text{argmin}_{ρ_x} D(G\#ρ_x, ρ_y)\,, $$ 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), $\text{argmin}_x \|G(x) - y\|^2$. Similar to how the choice of norm $\|\cdot\|$ 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 $φ$-divergence for $D$ leads to the recovery of a conditional distribution of $ρ_y$, while employing the Wasserstein distance results in the recovery of a marginal distribution.

Theorems & Definitions (6)

• Definition 3.1
• Theorem 3.2
• proof : Proof of Theorem \ref{['thm:phi_divergence']}
• Definition 4.1
• Theorem 4.2
• proof