{"paper":{"title":"Least-Squares Problem Over Probability Measure Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.OC","authors_text":"Li Wang, Qin Li, Yunan Yang","submitted_at":"2025-01-15T19:30:48Z","abstract_excerpt":"In this work, we investigate the variational problem $$\\rho_x^\\ast = \\text{argmin}_{\\rho_x} D(G\\#\\rho_x, \\rho_y)\\,, $$ 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 $D$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.09097","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.09097/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}