2025/01/15 by Qin Li, Li Wang, Li, Qin +3 · 1 citation
Computer Science · Decision Sciences · Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Fuzzy Systems and Optimization #Multi-Criteria Decision Making #Optimization and Control (math.OC) #Probability (math.PR) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2501.09097
openalex publication_date 2025/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we investigate the variational problem ρx^∗ = argminρx D(G#ρx, ρy) , 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), argminx ‖G(x) - y‖2. Similar to how the choice of norm ‖⋅‖ influences the optimizer in \mathbb Rd 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 ρy, while employing the Wasserstein distance results in the recovery of a marginal distribution.