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Recognizing that rendering and inverse rendering broadly involve vector-valued integrands, we identify the limitations of classical variance reduction methods in this context. To address this, we introduce ratio control variates, an estimator that leverages a ratio-based approach instead of the conventional difference-based control variates. Our analysis and experiments demonstrate that ratio control variables can significantly reduce the mean squared error of vector-valued integration compared to existing methods and are broadly applicable to various rendering and inverse rendering tasks.<\/jats:p>","DOI":"10.1145\/3731175","type":"journal-article","created":{"date-parts":[[2025,7,27]],"date-time":"2025-07-27T04:02:22Z","timestamp":1753588942000},"page":"1-16","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Vector-Valued Monte Carlo Integration Using Ratio Control Variates"],"prefix":"10.1145","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-2595-2493","authenticated-orcid":false,"given":"Haolin","family":"Lu","sequence":"first","affiliation":[{"name":"UC San Diego, Saarbr\u00fccken, Germany"},{"name":"MPI for Informatics, Saarbr\u00fccken, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6330-6849","authenticated-orcid":false,"given":"Delio","family":"Vicini","sequence":"additional","affiliation":[{"name":"Google Inc., Zurich, Switzerland"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-3833-3875","authenticated-orcid":false,"given":"Wesley","family":"Chang","sequence":"additional","affiliation":[{"name":"UC San Diego, La Jolla, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5443-470X","authenticated-orcid":false,"given":"Tzu-Mao","family":"Li","sequence":"additional","affiliation":[{"name":"UC San Diego, La Jolla, USA"}]}],"member":"320","published-online":{"date-parts":[[2025,7,27]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/218380.218500"},{"key":"e_1_2_2_2_1","doi-asserted-by":"crossref","unstructured":"Dejan Azinovi\u0107 Tzu-Mao Li Anton Kaplanyan and Matthias Nie\u00dfner. 2019. 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