{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:40:45Z","timestamp":1776840045817,"version":"3.51.2"},"reference-count":104,"publisher":"American Mathematical Society (AMS)","issue":"354","license":[{"start":{"date-parts":[[2025,9,4]],"date-time":"2025-09-04T00:00:00Z","timestamp":1756944000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["FA9550- 20-1-0358"],"award-info":[{"award-number":["FA9550- 20-1-0358"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DMS-208535"],"award-info":[{"award-number":["DMS-208535"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["FA9550- 20-1-035"],"award-info":[{"award-number":["FA9550- 20-1-035"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0021226"],"award-info":[{"award-number":["DE-SC0021226"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0023188"],"award-info":[{"award-number":["DE-SC0023188"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["FA9550-20-1-0397"],"award-info":[{"award-number":["FA9550-20-1-0397"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["N00014-20-1-2595"],"award-info":[{"award-number":["N00014-20-1-2595"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["2022254"],"award-info":[{"award-number":["2022254"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["376319"],"award-info":[{"award-number":["376319"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["FA9550- 20-1-0358"],"award-info":[{"award-number":["FA9550- 20-1-0358"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["DMS-208535"],"award-info":[{"award-number":["DMS-208535"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["FA9550- 20-1-035"],"award-info":[{"award-number":["FA9550- 20-1-035"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["DE-SC0021226"],"award-info":[{"award-number":["DE-SC0021226"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["DE-SC0023188"],"award-info":[{"award-number":["DE-SC0023188"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["FA9550-20-1-0397"],"award-info":[{"award-number":["FA9550-20-1-0397"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["N00014-20-1-2595"],"award-info":[{"award-number":["N00014-20-1-2595"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["2022254"],"award-info":[{"award-number":["2022254"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006831","name":"U.S. Air Force","doi-asserted-by":"publisher","award":["376319"],"award-info":[{"award-number":["376319"]}],"id":[{"id":"10.13039\/100006831","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["FA9550- 20-1-0358"],"award-info":[{"award-number":["FA9550- 20-1-0358"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-208535"],"award-info":[{"award-number":["DMS-208535"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["FA9550- 20-1-035"],"award-info":[{"award-number":["FA9550- 20-1-035"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DE-SC0021226"],"award-info":[{"award-number":["DE-SC0021226"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DE-SC0023188"],"award-info":[{"award-number":["DE-SC0023188"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["FA9550-20-1-0397"],"award-info":[{"award-number":["FA9550-20-1-0397"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["N00014-20-1-2595"],"award-info":[{"award-number":["N00014-20-1-2595"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2022254"],"award-info":[{"award-number":["2022254"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["376319"],"award-info":[{"award-number":["376319"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550- 20-1-0358"],"award-info":[{"award-number":["FA9550- 20-1-0358"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DMS-208535"],"award-info":[{"award-number":["DMS-208535"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550- 20-1-035"],"award-info":[{"award-number":["FA9550- 20-1-035"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DE-SC0021226"],"award-info":[{"award-number":["DE-SC0021226"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DE-SC0023188"],"award-info":[{"award-number":["DE-SC0023188"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-20-1-0397"],"award-info":[{"award-number":["FA9550-20-1-0397"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["N00014-20-1-2595"],"award-info":[{"award-number":["N00014-20-1-2595"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["2022254"],"award-info":[{"award-number":["2022254"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["376319"],"award-info":[{"award-number":["376319"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["FA9550- 20-1-0358"],"award-info":[{"award-number":["FA9550- 20-1-0358"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DMS-208535"],"award-info":[{"award-number":["DMS-208535"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["FA9550- 20-1-035"],"award-info":[{"award-number":["FA9550- 20-1-035"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0021226"],"award-info":[{"award-number":["DE-SC0021226"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0023188"],"award-info":[{"award-number":["DE-SC0023188"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["FA9550-20-1-0397"],"award-info":[{"award-number":["FA9550-20-1-0397"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["N00014-20-1-2595"],"award-info":[{"award-number":["N00014-20-1-2595"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["2022254"],"award-info":[{"award-number":["2022254"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["376319"],"award-info":[{"award-number":["376319"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550- 20-1-0358"],"award-info":[{"award-number":["FA9550- 20-1-0358"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DMS-208535"],"award-info":[{"award-number":["DMS-208535"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550- 20-1-035"],"award-info":[{"award-number":["FA9550- 20-1-035"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DE-SC0021226"],"award-info":[{"award-number":["DE-SC0021226"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DE-SC0023188"],"award-info":[{"award-number":["DE-SC0023188"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-20-1-0397"],"award-info":[{"award-number":["FA9550-20-1-0397"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["N00014-20-1-2595"],"award-info":[{"award-number":["N00014-20-1-2595"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["2022254"],"award-info":[{"award-number":["2022254"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["376319"],"award-info":[{"award-number":["376319"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["FA9550- 20-1-0358"],"award-info":[{"award-number":["FA9550- 20-1-0358"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["DMS-208535"],"award-info":[{"award-number":["DMS-208535"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["FA9550- 20-1-035"],"award-info":[{"award-number":["FA9550- 20-1-035"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["DE-SC0021226"],"award-info":[{"award-number":["DE-SC0021226"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["DE-SC0023188"],"award-info":[{"award-number":["DE-SC0023188"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["FA9550-20-1-0397"],"award-info":[{"award-number":["FA9550-20-1-0397"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["N00014-20-1-2595"],"award-info":[{"award-number":["N00014-20-1-2595"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["2022254"],"award-info":[{"award-number":["2022254"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"publisher","award":["376319"],"award-info":[{"award-number":["376319"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["FA9550- 20-1-0358"],"award-info":[{"award-number":["FA9550- 20-1-0358"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["DMS-208535"],"award-info":[{"award-number":["DMS-208535"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["FA9550- 20-1-035"],"award-info":[{"award-number":["FA9550- 20-1-035"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["DE-SC0021226"],"award-info":[{"award-number":["DE-SC0021226"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["DE-SC0023188"],"award-info":[{"award-number":["DE-SC0023188"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["FA9550-20-1-0397"],"award-info":[{"award-number":["FA9550-20-1-0397"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["N00014-20-1-2595"],"award-info":[{"award-number":["N00014-20-1-2595"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["2022254"],"award-info":[{"award-number":["2022254"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006221","name":"United States - Israel Binational Science Foundation","doi-asserted-by":"publisher","award":["376319"],"award-info":[{"award-number":["376319"]}],"id":[{"id":"10.13039\/100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["FA9550- 20-1-0358"],"award-info":[{"award-number":["FA9550- 20-1-0358"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["DMS-208535"],"award-info":[{"award-number":["DMS-208535"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["FA9550- 20-1-035"],"award-info":[{"award-number":["FA9550- 20-1-035"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["DE-SC0021226"],"award-info":[{"award-number":["DE-SC0021226"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["DE-SC0023187"],"award-info":[{"award-number":["DE-SC0023187"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["DE-SC0023188"],"award-info":[{"award-number":["DE-SC0023188"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["FA9550-20-1-0397"],"award-info":[{"award-number":["FA9550-20-1-0397"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["N00014-20-1-2595"],"award-info":[{"award-number":["N00014-20-1-2595"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["2022254"],"award-info":[{"award-number":["2022254"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["376319"],"award-info":[{"award-number":["376319"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling\u2014a central task in statistical inference and generative modeling. We provide a priori error estimates in the continuum limit, i.e., when the measures (or their densities) are given, but when the transport map is discretized or approximated using a finite-dimensional function space. Our analysis relies on the regularity theory of transport maps and on classical approximation theory for high-dimensional functions. A third element of our analysis, which is of independent interest, is the development of new stability estimates that relate the distance between two maps to the distance\u00a0(or divergence) between the pushforward measures they define. We present a series of applications of our framework, where quantitative convergence rates are obtained for practical problems using Wasserstein metrics, maximum mean discrepancy, and Kullback\u2013Leibler divergence. Specialized rates for approximations of the popular triangular Kn\u00f6the\u2013Rosenblatt maps are obtained, followed by numerical experiments that demonstrate and extend our theory.<\/p>","DOI":"10.1090\/mcom\/4013","type":"journal-article","created":{"date-parts":[[2024,9,4]],"date-time":"2024-09-04T14:16:22Z","timestamp":1725459382000},"page":"1863-1909","source":"Crossref","is-referenced-by-count":7,"title":["An approximation theory framework for measure-transport sampling algorithms"],"prefix":"10.1090","volume":"94","author":[{"given":"Ricardo","family":"Baptista","sequence":"first","affiliation":[]},{"given":"Bamdad","family":"Hosseini","sequence":"additional","affiliation":[]},{"given":"Nikola","family":"Kovachki","sequence":"additional","affiliation":[]},{"given":"Youssef","family":"Marzouk","sequence":"additional","affiliation":[]},{"given":"Amir","family":"Sagiv","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,9,4]]},"reference":[{"key":"1","series-title":"Lectures in Mathematics ETH Z\\\"{u}rich","isbn-type":"print","volume-title":"Gradient flows in metric spaces and in the space of probability measures","author":"Ambrosio, Luigi","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/9783764324285"},{"key":"2","doi-asserted-by":"crossref","unstructured":"C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan, An introduction to MCMC for machine learning, Mach. Learn. 50 (2003), 5\u201343.","DOI":"10.1023\/A:1020281327116"},{"key":"3","unstructured":"M. Arjovsky, S. Chintala, and L. Bottou, Wasserstein generative adversarial networks, in International conference on machine learning, PMLR, 2017, pp. 214\u2013223."},{"key":"4","doi-asserted-by":"crossref","unstructured":"R. Baptista, Y. Marzouk, O. Zahm, On the representation and learning of monotone triangular transport maps, Found. Comput. Math. (2023), 1\u201346, DOI:10.1007\/s10208-023-09630-x.","DOI":"10.1007\/s10208-023-09630-x"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1007\/s002110050002","article-title":"A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem","volume":"84","author":"Benamou, Jean-David","year":"2000","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/j.jcp.2013.12.015","article-title":"Numerical solution of the optimal transportation problem using the Monge-Amp\u00e8re equation","volume":"260","author":"Benamou, Jean-David","year":"2014","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"7","doi-asserted-by":"crossref","first-page":"121","DOI":"10.5802\/smai-jcm.74","article-title":"Model reduction and neural networks for parametric PDEs","volume":"7","author":"Bhattacharya, Kaushik","year":"2021","journal-title":"SMAI J. Comput. Math."},{"key":"8","unstructured":"M. Bi\u0144kowski, D. J. Sutherland, M. Arbel, and A. Gretton, Demystifying MMD GANs, in International Conference on Learning Representations, 2018."},{"key":"9","first-page":"Paper No. [39], 70","article-title":"(\ud835\udc53,\u0393)-divergences: interpolating between \ud835\udc53-divergences and integral probability metrics","volume":"23","author":"Birrell, Jeremiah","year":"2022","journal-title":"J. Mach. Learn. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/1532-4435","issn-type":"print"},{"key":"10","series-title":"Information Science and Statistics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-45528-0","volume-title":"Pattern recognition and machine learning","author":"Bishop, Christopher M.","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387310732"},{"issue":"518","key":"11","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1080\/01621459.2017.1285773","article-title":"Variational inference: a review for statisticians","volume":"112","author":"Blei, David M.","year":"2017","journal-title":"J. Amer. Statist. Assoc.","ISSN":"https:\/\/id.crossref.org\/issn\/0162-1459","issn-type":"print"},{"key":"12","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-34514-5","volume-title":"Measure theory. Vol. I, II","author":"Bogachev, V. I.","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540345138"},{"key":"13","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-34514-5","volume-title":"Measure theory. Vol. I, II","author":"Bogachev, V. I.","year":"2007","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540345138"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1137\/S0040585X97981457","article-title":"Nonlinear transformations of convex measures","volume":"50","author":"Bogachev, V. I.","year":"2005","journal-title":"Teor. Veroyatn. Primen.","ISSN":"https:\/\/id.crossref.org\/issn\/0040-361X","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1070\/SM2005v196n03ABEH000882","article-title":"Triangular transformations of measures","volume":"196","author":"Bogachev, V. I.","year":"2005","journal-title":"Mat. Sb.","ISSN":"https:\/\/id.crossref.org\/issn\/0368-8666","issn-type":"print"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1137\/120874850","article-title":"From Knothe\u2019s rearrangement to Brenier\u2019s optimal transport map","volume":"45","author":"Bonnotte, Nicolas","year":"2013","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"19","key":"17","first-page":"805","article-title":"D\u00e9composition polaire et r\u00e9arrangement monotone des champs de vecteurs","volume":"305","author":"Brenier, Yann","year":"1987","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0249-6291","issn-type":"print"},{"issue":"5","key":"18","doi-asserted-by":"publisher","first-page":"A3523--A3548","DOI":"10.1137\/18M1181675","article-title":"Convergence of probability densities using approximate models for forward and inverse problems in uncertainty quantification","volume":"40","author":"Butler, T.","year":"2018","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"4","key":"19","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1615\/Int.J.UncertaintyQuantification.2022038086","article-title":"\ud835\udc3f^{\ud835\udc5d} convergence of approximate maps and probability densities for forward and inverse problems in uncertainty quantification","volume":"12","author":"Butler, T.","year":"2022","journal-title":"Int. J. Uncertain. Quantif.","ISSN":"https:\/\/id.crossref.org\/issn\/2152-5080","issn-type":"print"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"99","DOI":"10.2307\/2152752","article-title":"The regularity of mappings with a convex potential","volume":"5","author":"Caffarelli, Luis A.","year":"1992","journal-title":"J. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0894-0347","issn-type":"print"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"547","DOI":"10.1007\/s002200000257","article-title":"Monotonicity properties of optimal transportation and the FKG and related inequalities","volume":"214","author":"Caffarelli, Luis A.","year":"2000","journal-title":"Comm. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3616","issn-type":"print"},{"issue":"157","key":"22","doi-asserted-by":"publisher","first-page":"67","DOI":"10.2307\/2007465","article-title":"Approximation results for orthogonal polynomials in Sobolev spaces","volume":"38","author":"Canuto, C.","year":"1982","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"23","doi-asserted-by":"publisher","first-page":"1165","DOI":"10.1214\/15-AOS1401","article-title":"Vector quantile regression: an optimal transport approach","volume":"44","author":"Carlier, Guillaume","year":"2016","journal-title":"Ann. Statist.","ISSN":"https:\/\/id.crossref.org\/issn\/0090-5364","issn-type":"print"},{"issue":"6","key":"24","doi-asserted-by":"publisher","first-page":"2554","DOI":"10.1137\/080740647","article-title":"From Knothe\u2019s transport to Brenier\u2019s map and a continuation method for optimal transport","volume":"41","author":"Carlier, G.","year":"2009","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"2","key":"25","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1590\/S0101-82052003000200002","article-title":"Hermite spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions","volume":"22","author":"Xu, Cheng-Long","year":"2003","journal-title":"Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1807-0302","issn-type":"print"},{"issue":"4","key":"26","first-page":"1491","article-title":"Lipschitz changes of variables between perturbations of log-concave measures","volume":"17","author":"Colombo, Maria","year":"2017","journal-title":"Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)","ISSN":"https:\/\/id.crossref.org\/issn\/0391-173X","issn-type":"print"},{"issue":"3","key":"27","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1214\/13-STS421","article-title":"MCMC methods for functions: modifying old algorithms to make them faster","volume":"28","author":"Cotter, S. L.","year":"2013","journal-title":"Statist. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0883-4237","issn-type":"print"},{"key":"28","doi-asserted-by":"crossref","unstructured":"A. Creswell, T. White, V. Dumoulin, K. Arulkumaran, B. Sengupta, and A. A. Bharath, Generative adversarial networks: An overview, IEEE Signal Process. Mag. 35 (2018), 53\u201365.","DOI":"10.1109\/MSP.2017.2765202"},{"issue":"6","key":"29","doi-asserted-by":"publisher","first-page":"1863","DOI":"10.1007\/s10208-021-09537-5","article-title":"Deep composition of tensor-trains using squared inverse Rosenblatt transports","volume":"22","author":"Cui, Tiangang","year":"2022","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"key":"30","unstructured":"M. Cuturi, Sinkhorn distances: Lightspeed computation of optimal transport, Adv. Neural Inf. Process. Syst. 26 (2013)."},{"key":"31","unstructured":"N. Deb, P. Ghosal, and B. Sen, Rates of estimation of optimal transport maps using plug-in estimators via barycentric projections, Adv. Neural Inf. Process. Syst. 34 (2021), 29736\u201329753."},{"issue":"1","key":"32","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1137\/18M1205959","article-title":"Density estimation in uncertainty propagation problems using a surrogate model","volume":"8","author":"Ditkowski, Adi","year":"2020","journal-title":"SIAM\/ASA J. Uncertain. Quantif."},{"key":"33","unstructured":"V. Divol, J. Niles-Weed, and A.-A. Pooladian, Optimal transport map estimation in general function spaces, Preprint,  arXiv:2212.03722, (2022)."},{"issue":"23","key":"34","doi-asserted-by":"publisher","first-page":"7815","DOI":"10.1016\/j.jcp.2012.07.022","article-title":"Bayesian inference with optimal maps","volume":"231","author":"El Moselhy, Tarek A.","year":"2012","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"2","key":"35","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1051\/m2an\/2011045","article-title":"On the convergence of generalized polynomial chaos expansions","volume":"46","author":"Ernst, Oliver G.","year":"2012","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"key":"36","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/019","volume-title":"Partial differential equations","volume":"19","author":"Evans, Lawrence C.","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821849743","edition":"2"},{"key":"37","series-title":"Textbooks in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1201\/b18333","volume-title":"Measure theory and fine properties of functions","author":"Evans, Lawrence C.","year":"2015","ISBN":"https:\/\/id.crossref.org\/isbn\/9781482242386"},{"key":"38","series-title":"Zurich Lectures in Advanced Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.4171\/170","volume-title":"The Monge-Amp\\`ere equation and its applications","author":"Figalli, Alessio","year":"2017","ISBN":"https:\/\/id.crossref.org\/isbn\/9783037191705"},{"issue":"3","key":"39","doi-asserted-by":"publisher","first-page":"A1432--A1459","DOI":"10.1137\/110822372","article-title":"A numerical method for the elliptic Monge-Amp\u00e8re equation with transport boundary conditions","volume":"34","author":"Froese, Brittany D.","year":"2012","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"2","key":"40","doi-asserted-by":"publisher","first-page":"C1--C11","DOI":"10.1111\/ectj.12083","article-title":"A survey of some recent applications of optimal transport methods to econometrics","volume":"20","author":"Galichon, Alfred","year":"2017","journal-title":"Econom. J.","ISSN":"https:\/\/id.crossref.org\/issn\/1368-4221","issn-type":"print"},{"key":"41","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1515\/9781400883592","volume-title":"Optimal transport methods in economics","author":"Galichon, Alfred","year":"2016","ISBN":"https:\/\/id.crossref.org\/isbn\/9780691172767"},{"issue":"2","key":"42","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/BF02392620","article-title":"The geometry of optimal transportation","volume":"177","author":"Gangbo, Wilfrid","year":"1996","journal-title":"Acta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-5962","issn-type":"print"},{"key":"43","unstructured":"A. Genevay, G. Peyr\u00e9, and M. Cuturi, Learning Generative Models with Sinkhorn Divergences, in International Conference on Artificial Intelligence and Statistics, PMLR, 2018, pp. 1608\u20131617."},{"key":"44","series-title":"Johns Hopkins Studies in the Mathematical Sciences","isbn-type":"print","volume-title":"Matrix computations","author":"Golub, Gene H.","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9781421407944","edition":"4"},{"key":"45","unstructured":"I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, Generative adversarial nets, Adv. Neural Inf. Process. Syst. 27 (2014)."},{"key":"46","unstructured":"J. Gui, Z. Sun, Y. Wen, D. Tao, and J. Ye, A review on generative adversarial networks: Algorithms, theory, and applications, IEEE Trans. Knowl. Data Eng. (2021)."},{"key":"47","doi-asserted-by":"crossref","unstructured":"C. E. Guti\u00e9rrez and H. Brezis, The Monge-Ampere Equation, vol. 44, Springer, 2001.","DOI":"10.1007\/978-1-4612-0195-3"},{"issue":"2","key":"48","doi-asserted-by":"publisher","first-page":"1166","DOI":"10.1214\/20-aos1997","article-title":"Minimax estimation of smooth optimal transport maps","volume":"49","author":"H\u00fctter, Jan-Christian","year":"2021","journal-title":"Ann. Statist.","ISSN":"https:\/\/id.crossref.org\/issn\/0090-5364","issn-type":"print"},{"issue":"2","key":"49","doi-asserted-by":"publisher","first-page":"762","DOI":"10.1137\/070704770","article-title":"Perturbation bounds for determinants and characteristic polynomials","volume":"30","author":"Ipsen, Ilse C. F.","year":"2008","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"key":"50","unstructured":"N. J. Irons, M. Scetbon, S. Pal, and Z. Harchaoui, Triangular Flows for Generative Modeling: Statistical Consistency, Smoothness Classes, and Fast Rates, in International Conference on Artificial Intelligence and Statistics, PMLR, 2022, pp. 10161\u201310195."},{"key":"51","unstructured":"P. Jaini, K. A. Selby, and Y. Yu, Sum-of-Squares Polynomial Flow, in International Conference on Machine Learning, PMLR, 2019, pp. 3009\u20133018."},{"issue":"1","key":"52","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/S0036141096303359","article-title":"The variational formulation of the Fokker-Planck equation","volume":"29","author":"Jordan, Richard","year":"1998","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"53","first-page":"199","article-title":"On the translocation of masses","volume":"37","author":"Kantorovitch, L.","year":"1942","journal-title":"C. R. (Doklady) Acad. Sci. URSS (N.S.)"},{"key":"54","first-page":"39","article-title":"Contributions to the theory of convex bodies","volume":"4","author":"Knothe, Herbert","year":"1957","journal-title":"Michigan Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0026-2285","issn-type":"print"},{"key":"55","doi-asserted-by":"crossref","unstructured":"I. Kobyzev, S. J. Prince, and M. A. Brubaker, Normalizing flows: An introduction and review of current methods, IEEE Tran. Pattern Anal. Mach. Intell. 43 (2020), 3964\u20133979.","DOI":"10.1109\/TPAMI.2020.2992934"},{"issue":"2","key":"56","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1137\/S0040585X97985947","article-title":"On Sobolev regularity of mass transport and transportation inequalities","volume":"57","author":"Kolesnikov, A. V.","year":"2013","journal-title":"Theory Probab. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0040-585X","issn-type":"print"},{"issue":"7","key":"57","doi-asserted-by":"publisher","first-page":"4490","DOI":"10.1016\/j.jfa.2014.01.010","article-title":"On continuity equations in infinite dimensions with non-Gaussian reference measure","volume":"266","author":"Kolesnikov, Alexander V.","year":"2014","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"},{"key":"58","doi-asserted-by":"crossref","unstructured":"R. Baptista, B. Hosseini, N. Kovachki and Y. Marzouk, Conditional sampling with monotone GANs: from generative models to likelihood-free inference, SIAM\/ASA Journal on Uncertainty Quantification (2024), vol 12, 868\u2013900, DOI:10.1137\/23M1581546.","DOI":"10.1137\/23M1581546"},{"key":"59","first-page":"Paper No. [89], 97","article-title":"Neural operator: learning maps between function spaces with applications to PDEs","volume":"24","author":"Kovachki, Nikola","year":"2023","journal-title":"J. Mach. Learn. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/1532-4435","issn-type":"print"},{"issue":"1","key":"60","doi-asserted-by":"publisher","first-page":"tnac001, 141","DOI":"10.1093\/imatrm\/tnac001","article-title":"Error estimates for DeepONets: a deep learning framework in infinite dimensions","volume":"6","author":"Lanthaler, Samuel","year":"2022","journal-title":"Trans. Math. Appl."},{"key":"61","unstructured":"C.-L. Li, W.-C. Chang, Y. Cheng, Y. Yang, and B. P\u00f3czos, MMD GAN: towards deeper understanding of moment matching network, Adv. Neural Inf. Process. Syst. 30 (2017)."},{"issue":"3","key":"62","doi-asserted-by":"publisher","first-page":"1941","DOI":"10.1093\/imanum\/draa045","article-title":"Quantitative stability and error estimates for optimal transport plans","volume":"41","author":"Li, Wenbo","year":"2021","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"4","key":"63","doi-asserted-by":"publisher","first-page":"3073","DOI":"10.1137\/16M1071560","article-title":"Optimal transport via a Monge-Amp\u00e8re optimization problem","volume":"49","author":"Lindsey, Michael","year":"2017","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"64","unstructured":"Y. Lu and J. Lu, A universal approximation theorem of deep neural networks for expressing probability distributions, Adv. Neural Inf. Process. Syst. 33 (2020), 3094\u20133105."},{"key":"65","isbn-type":"print","first-page":"785","article-title":"Sampling via measure transport: an introduction","author":"Marzouk, Youssef","year":"2017","ISBN":"https:\/\/id.crossref.org\/isbn\/9783319123844"},{"issue":"2","key":"66","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1016\/S0016-0032(96)00063-4","article-title":"The Jensen-Shannon divergence","volume":"334","author":"Men\u00e9ndez, M. L.","year":"1997","journal-title":"J. Franklin Inst. B","ISSN":"https:\/\/id.crossref.org\/issn\/0016-0032","issn-type":"print"},{"key":"67","unstructured":"G. Monge, M\u00e9moire sur la th\u00e9orie des d\u00e9blais et des remblais, De l\u2019Imprimerie Royale, 1781."},{"key":"68","doi-asserted-by":"crossref","unstructured":"K. Muandet, K. Fukumizu, B. Sriperumbudur, and B. Sch\u00f6lkopf, Kernel mean embedding of distributions: A review and beyond, Found. Trends Mach. Learn. 10 (2017), 1\u2013141.","DOI":"10.1561\/2200000060"},{"key":"69","unstructured":"B. Muzellec and M. Cuturi, Subspace detours: Building transport plans that are optimal on subspace projections, Adv. Neural Inf. Process. Syst. 32 (2019)."},{"key":"70","series-title":"Springer Series in Operations Research and Financial Engineering","isbn-type":"print","volume-title":"Numerical optimization","author":"Nocedal, Jorge","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387303031","edition":"2"},{"issue":"1","key":"71","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1007\/s00211-018-0988-9","article-title":"Pointwise rates of convergence for the Oliker-Prussner method for the Monge-Amp\u00e8re equation","volume":"141","author":"Nochetto, Ricardo H.","year":"2019","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"72","doi-asserted-by":"publisher","first-page":"1003","DOI":"10.1214\/23-aap1983","article-title":"On the difference between entropic cost and the optimal transport cost","volume":"34","author":"Pal, Soumik","year":"2024","journal-title":"Ann. Appl. Probab.","ISSN":"https:\/\/id.crossref.org\/issn\/1050-5164","issn-type":"print"},{"key":"73","series-title":"SpringerBriefs in Probability and Mathematical Statistics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-38438-8","volume-title":"An invitation to statistics in Wasserstein space","author":"Panaretos, Victor M.","year":"[2020] \\copyright2020","ISBN":"https:\/\/id.crossref.org\/isbn\/9783030384371"},{"key":"74","first-page":"Paper No. 57, 64","article-title":"Normalizing flows for probabilistic modeling and inference","volume":"22","author":"Papamakarios, George","year":"2021","journal-title":"J. Mach. Learn. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/1532-4435","issn-type":"print"},{"key":"75","unstructured":"G. Papamakarios, T. Pavlakou, and I. Murray, Masked autoregressive flow for density estimation, Adv. Neural Inf. Process. Syst. 30 (2017)."},{"issue":"2","key":"76","doi-asserted-by":"publisher","first-page":"645","DOI":"10.1137\/17M1134640","article-title":"Transport map accelerated Markov chain Monte Carlo","volume":"6","author":"Parno, Matthew D.","year":"2018","journal-title":"SIAM\/ASA J. Uncertain. Quantif."},{"key":"77","doi-asserted-by":"crossref","unstructured":"G. Peyr\u00e9 and M. Cuturi, Computational optimal transport: With applications to data science, Found. Trends Mach. Learn. 11 (2019), 355\u2013607.","DOI":"10.1561\/2200000073"},{"key":"78","isbn-type":"print","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1017\/S0962492900002919","article-title":"Approximation theory of the MLP model in neural networks","author":"Pinkus, Allan","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0521770882"},{"key":"79","unstructured":"A.-A. Pooladian, V. Divol, and J. Niles-Weed, Minimax estimation of discontinuous optimal transport maps: The semi-discrete case, Preprint,  arXiv:2301.11302, (2023)."},{"key":"80","unstructured":"A.-A. Pooladian and J. Niles-Weed, Entropic estimation of optimal transport maps, Preprint,  arXiv:2109.12004, (2021)."},{"key":"81","unstructured":"D. Rezende and S. Mohamed, Variational Inference with Normalizing Flows, in International conference on machine learning, PMLR, 2015, pp. 1530\u20131538."},{"key":"82","series-title":"Springer Texts in Statistics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3071-5","volume-title":"Monte Carlo statistical methods","author":"Robert, Christian P.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/038798707X"},{"key":"83","doi-asserted-by":"publisher","first-page":"470","DOI":"10.1214\/aoms\/1177729394","article-title":"Remarks on a multivariate transformation","volume":"23","author":"Rosenblatt, Murray","year":"1952","journal-title":"Ann. Math. Statistics","ISSN":"https:\/\/id.crossref.org\/issn\/0003-4851","issn-type":"print"},{"issue":"3","key":"84","doi-asserted-by":"publisher","first-page":"707","DOI":"10.4310\/CMS.2020.v18.n3.a6","article-title":"The Wasserstein distances between pushed-forward measures with applications to uncertainty quantification","volume":"18","author":"Sagiv, Amir","year":"2020","journal-title":"Commun. Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/1539-6746","issn-type":"print"},{"issue":"4","key":"85","doi-asserted-by":"publisher","first-page":"1165","DOI":"10.1007\/s00211-022-01281-4","article-title":"Spectral convergence of probability densities for forward problems in uncertainty quantification","volume":"150","author":"Sagiv, Amir","year":"2022","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"86","series-title":"Progress in Nonlinear Differential Equations and their Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-20828-2","volume-title":"Optimal transport for applied mathematicians","volume":"87","author":"Santambrogio, Filippo","year":"2015","ISBN":"https:\/\/id.crossref.org\/isbn\/9783319208275"},{"key":"87","unstructured":"V. Seguy, B. B. Damodaran, R. Flamary, N. Courty, A. Rolet, and M. Blondel, Large-scale Optimal Transport and Mapping Estimation, in International Conference on Learning Representations, 2018, pp. 1\u201315."},{"issue":"5","key":"88","doi-asserted-by":"publisher","first-page":"1768","DOI":"10.4208\/cicp.oa-2020-0149","article-title":"Deep network approximation characterized by number of neurons","volume":"28","author":"Shen, Zuowei","year":"2020","journal-title":"Commun. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/1815-2406","issn-type":"print"},{"issue":"4","key":"89","doi-asserted-by":"publisher","first-page":"921","DOI":"10.1137\/20M1312204","article-title":"Coupling techniques for nonlinear ensemble filtering","volume":"64","author":"Spantini, Alessio","year":"2022","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"key":"90","first-page":"Paper No. 71","article-title":"Inference via low-dimensional couplings","volume":"19","author":"Spantini, Alessio","year":"2018","journal-title":"J. Mach. Learn. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/1532-4435","issn-type":"print"},{"key":"91","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1017\/S0962492910000061","article-title":"Inverse problems: a Bayesian perspective","volume":"19","author":"Stuart, A. M.","year":"2010","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"key":"92","series-title":"American Mathematical Society Colloquium Publications, Vol. 23","doi-asserted-by":"crossref","DOI":"10.1090\/coll\/023","volume-title":"Orthogonal Polynomials","author":"Szeg\u00f6, Gabor","year":"1939"},{"issue":"2","key":"93","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1002\/cpa.21423","article-title":"A family of nonparametric density estimation algorithms","volume":"66","author":"Tabak, E. G.","year":"2013","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"1","key":"94","doi-asserted-by":"publisher","first-page":"217","DOI":"10.4310\/cms.2010.v8.n1.a11","article-title":"Density estimation by dual ascent of the log-likelihood","volume":"8","author":"Tabak, Esteban G.","year":"2010","journal-title":"Commun. Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/1539-6746","issn-type":"print"},{"key":"95","series-title":"Cambridge Series in Statistical and Probabilistic Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/9781108231596","volume-title":"High-dimensional probability","volume":"47","author":"Vershynin, Roman","year":"2018","ISBN":"https:\/\/id.crossref.org\/isbn\/9781108415194"},{"key":"96","series-title":"Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71050-9","volume-title":"Optimal transport","volume":"338","author":"Villani, C\u00e9dric","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540710493"},{"key":"97","doi-asserted-by":"crossref","unstructured":"M. J. Wainwright and M. I. Jordan, Graphical models, exponential families, and variational inference, Found. Trends Mach. Learn. 1 (2008), 1\u2013305.","DOI":"10.1561\/2200000001"},{"key":"98","unstructured":"S. Wang and Y. Marzouk, On minimax density estimation via measure transport, Preprint,  arXiv:2207.10231, (2022)."},{"key":"99","unstructured":"A. Wehenkel and G. Louppe, Unconstrained monotonic neural networks, Adv. Neural Inf. Process. Syst. 32 (2019)."},{"key":"100","unstructured":"J. Westermann and J. Zech, Measure transport via polynomial density surrogates, Preprint,  arXiv:2311.04172, (2023)."},{"key":"101","isbn-type":"print","volume-title":"Numerical methods for stochastic computations","author":"Xiu, Dongbin","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780691142128"},{"issue":"3","key":"102","doi-asserted-by":"publisher","first-page":"919","DOI":"10.1007\/s00365-022-09569-2","article-title":"Sparse approximation of triangular transports, Part I: The finite-dimensional case","volume":"55","author":"Zech, Jakob","year":"2022","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"issue":"3","key":"103","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1007\/s00365-022-09570-9","article-title":"Sparse approximation of triangular transports, Part II: The infinite-dimensional case","volume":"55","author":"Zech, Jakob","year":"2022","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"key":"104","doi-asserted-by":"crossref","unstructured":"C. Zhang, J. B\u00fctepage, H. Kjellstr\u00f6m, and S. Mandt, Advances in variational inference, IEEE Tran. Pattern Anal. Mach. Intell. 41 (2018), 2008\u20132026.","DOI":"10.1109\/TPAMI.2018.2889774"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-354\/S0025-5718-2024-04013-9\/mcom4013_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-354\/S0025-5718-2024-04013-9\/S0025-5718-2024-04013-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:43:52Z","timestamp":1776836632000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2025-94-354\/S0025-5718-2024-04013-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,4]]},"references-count":104,"journal-issue":{"issue":"354","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["S0025-5718-2024-04013-9"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4013","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2024,9,4]]}}}