Least-Squares Problem Over Probability Measure Space
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909457162698752 |
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| author | Li, Qin Wang, Li Yang, Yunan |
| author_facet | Li, Qin Wang, Li Yang, Yunan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09097 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Least-Squares Problem Over Probability Measure Space Li, Qin Wang, Li Yang, Yunan Optimization and Control Functional Analysis Probability 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. |
| title | Least-Squares Problem Over Probability Measure Space |
| topic | Optimization and Control Functional Analysis Probability |
| url | https://arxiv.org/abs/2501.09097 |