{"paper":{"title":"When Can We Solve the Weighted Low Rank Approximation Problem in Truly Subquadratic Time?","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI","cs.LG"],"primary_cat":"cs.CC","authors_text":"Chenyang Li, Yingyu Liang, Zhao Song, Zhenmei Shi","submitted_at":"2025-02-24T07:18:24Z","abstract_excerpt":"The weighted low-rank approximation problem is a fundamental numerical linear algebra problem and has many applications in machine learning. Given a $n \\times n$ weight matrix $W$ and a $n \\times n$ matrix $A$, the goal is to find two low-rank matrices $U, V \\in \\mathbb{R}^{n \\times k}$ such that the cost of $\\| W \\circ (U V^\\top - A) \\|_F^2$ is minimized. Previous work has to pay $\\Omega(n^2)$ time when matrices $A$ and $W$ are dense, e.g., having $\\Omega(n^2)$ non-zero entries. In this work, we show that there is a certain regime, even if $A$ and $W$ are dense, we can still hope to solve the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.16912","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/2502.16912/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"}