{"corpus_id":1007626,"paper_sha":"2b07d735ecb18dfac731c51bba162cfba8cf0666","doi":"10.1109/ACSSC.2017.8335427","arxiv_id":"1612.06339","pmid":null,"pmcid":null,"mag_id":2666829939,"dblp_id":"conf/acssc/EftekhariWLCW17","acl_id":null,"title":"Learning the second-moment matrix of a smooth function from point samples","year":2016,"publication_date":"2016-12-19","venue":"Asilomar Conference on Signals, Systems and Computers","journal":{"name":"2017 51st Asilomar Conference on Signals, Systems, and Computers","pages":"671-675","volume":null},"journal_issn":null,"journal_title":null,"publication_types":["JournalArticle","Conference"],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics","Physics","Computer Science"],"reference_count":69,"citation_count":3,"influential_citation_count":0,"is_open_access":false,"arxiv_categories":["cs.IT","math.IT"],"arxiv_license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":null,"s2_open_access_landing_url":null,"s2_open_access_license":null,"s2_open_access_status":null,"pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"Consider an open set D ⊆ R<sup>n</sup>, equipped with a probability measure μ. An important characteristic of a smooth function f : D → R is its second-moment matrix Σ<inf>μ</inf> := ∫ ∇ f(x) (∇f(x))<sup>∗</sup> μ(dx) ∊ R<sup>n × n</sup>, where ∇ f(x) ∊ R<sup>n</sup> is the gradient of f(·) at × ∊ D. For instance, the span of the leading r eigenvectors of Σ<inf>μ</inf> forms an active subspace of f(·), thereby extending the concept of principal component analysis to the problem of ridge approximation. In this work, we propose and analyze a simple algorithm for estimating Σ<inf>μ</inf> from point values of f(·) without imposing any structural assumptions on Σ<inf>μ</inf>.","claims":[{"public_id":"cl_12646f4248c56d0bf42a715d2242bde6","status":"active","text":"A simple algorithm is proposed and analyzed for estimating the second-moment matrix of a smooth function from point values only.","confidence":0.98,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_12646f4248c56d0bf42a715d2242bde6"},{"public_id":"cl_291d3c3edb153a8dc0674800c86863a2","status":"active","text":"The estimation method places no structural assumptions on the second-moment matrix.","confidence":0.93,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous 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