{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","Extent":"http:\/\/purl.org\/dc\/terms\/extent","FileFormat":"http:\/\/purl.org\/dc\/elements\/1.1\/format","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","GeographicLocation":"http:\/\/purl.org\/dc\/terms\/spatial","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Notes":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","PeerReviewStatus":"https:\/\/open.library.ubc.ca\/terms#peerReviewStatus","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","RightsURI":"https:\/\/open.library.ubc.ca\/terms#rightsURI","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Series":"http:\/\/purl.org\/dc\/terms\/isPartOf","Subject":"http:\/\/purl.org\/dc\/terms\/subject","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Non UBC","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Creator":[{"@value":"Neeraj Kayal","@language":"en"}],"DateAvailable":[{"@value":"2020-01-09T09:41:47Z","@language":"en"}],"DateIssued":[{"@value":"2019-07-12T09:52","@language":"en"}],"Description":[{"@value":"What is the smallest formula computing a given multivariate polynomial f(x)=\n In this talk I will present a paradigm for translating the known lower\nbound proofs for various subclasses of formulas into efficient proper learn=\ning algorithms for the same subclass.\n\nMany lower bounds proofs for various subclasses of arithmetic formulas redu=\nce the problem to showing that any expression for f(x) as a sum of =93simpl=\ne=94 polynomials T_i(x):\n f(x) =3D T_1(x) + T_2(x) + =85 + T_s(x),\nthe number s of simple summands is large. For example, each simple summand =\nT_i could be a product of linear forms or a power of a low degree polynomia=\nl and so on.\nThe lower bound consists of constructing a vector space of linear maps M, e=\nach L in M being a linear map from the set of polynomials F[x] to some vect=\nor space W\n(typically W is F[X] itself) with the following two properties:\n\n(i) For every simple polynomial T, dim(M*T) is small, say =\nthat dim(M*T) <=3D r.\n\n(ii) For the candidate hard polynomial f, dim(M*f) is large,=\n say that dim(M*f) >=3D R.\nThese two properties immediately imply a lower bound: s >=3D R\/r.\n\nThe corresponding reconstruction\/proper learning problem is the following: =\ngiven f(x) we want to find the simple summands T_1(x), T_2(x), =85, T_s(x) =\nwhich add up to f(x).\nWe will see how such a lower bound proof can often be used to solve the rec=\nonstruction problem. Our main tool will be an efficient algorithmic solutio=\nn\nto the problem of decomposing a pair of vector spaces (U, V) under the simu=\nltaneous action of a vector space of linear maps from U to V.\n\nAlong the way we will also obtain very precise bounds on the size of formul=\nas computing certain explicit polynomials. For example, we will obtain for =\nevery s, an explicit\npolynomial f(x) that can be computed by a depth three formula of size s but=\n not by any depth three formula of size (s-1).\n\nBased on joint works with Chandan Saha and Ankit Garg.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/73237?expand=metadata","@language":"en"}],"Extent":[{"@value":"56.0 minutes","@language":"en"}],"FileFormat":[{"@value":"video\/mp4","@language":"en"}],"FullText":[{"@value":"","@language":"en"}],"GeographicLocation":[{"@value":"Banff (Alta.)","@language":"en"}],"IsShownAt":[{"@value":"10.14288\/1.0388215","@language":"en"}],"Language":[{"@value":"eng","@language":"en"}],"Notes":[{"@value":"Author affiliation: MSR India","@language":"en"}],"PeerReviewStatus":[{"@value":"Unreviewed","@language":"en"}],"Provider":[{"@value":"Vancouver : University of British Columbia Library","@language":"en"}],"Publisher":[{"@value":"Banff International Research Station for Mathematical Innovation and Discovery","@language":"en"}],"Rights":[{"@value":"Attribution-NonCommercial-NoDerivatives 4.0 International","@language":"en"}],"RightsURI":[{"@value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","@language":"en"}],"ScholarlyLevel":[{"@value":"Faculty","@language":"en"}],"Series":[{"@value":"BIRS Workshop Lecture Videos (Banff, Alta)","@language":"en"}],"Subject":[{"@value":"Mathematics","@language":"en"},{"@value":"Computer Science, Theoretical Computer Science","@language":"en"}],"Title":[{"@value":"Reconstructing arithmetic formulas using lower bound proof techniques","@language":"en"}],"Type":[{"@value":"Moving Image","@language":"en"}],"URI":[{"@value":"http:\/\/hdl.handle.net\/2429\/73237","@language":"en"}],"SortDate":[{"@value":"2019-07-12 AD","@language":"en"}],"@id":"doi:10.14288\/1.0388215"}