{"http:\/\/dx.doi.org\/10.14288\/1.0388215":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Non UBC","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Neeraj Kayal","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2020-01-09T09:41:47Z","type":"literal","lang":"en"},{"value":"2019-07-12T09:52","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/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.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/73237?expand=metadata","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/extent":[{"value":"56.0 minutes","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/elements\/1.1\/format":[{"value":"video\/mp4","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"","type":"literal","lang":"en"},{"value":"Author affiliation: MSR India","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/spatial":[{"value":"Banff (Alta.)","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0388215","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#peerReviewStatus":[{"value":"Unreviewed","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"Banff International Research Station for Mathematical Innovation and Discovery","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"Attribution-NonCommercial-NoDerivatives 4.0 International","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#rightsURI":[{"value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Faculty","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/isPartOf":[{"value":"BIRS Workshop Lecture Videos (Banff, Alta)","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/subject":[{"value":"Mathematics","type":"literal","lang":"en"},{"value":"Computer Science, Theoretical Computer Science","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/title":[{"value":"Reconstructing arithmetic formulas using lower bound proof techniques","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/type":[{"value":"Moving Image","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#identifierURI":[{"value":"http:\/\/hdl.handle.net\/2429\/73237","type":"literal","lang":"en"}]}}