[{"key":"dc.contributor.author","value":"Neeraj Kayal","language":null},{"key":"dc.coverage.spatial","value":"Banff (Alta.)","language":null},{"key":"dc.date.accessioned","value":"2020-01-09T06:00:46Z","language":null},{"key":"dc.date.available","value":"2020-01-09T09:41:47Z","language":null},{"key":"dc.date.issued","value":"2019-07-12T09:52","language":null},{"key":"dc.identifier.other","value":"BIRS-VIDEO-201907120952-Kayal","language":null},{"key":"dc.identifier.other","value":"BIRS-VIDEO-19w5088-33630","language":null},{"key":"dc.identifier.uri","value":"http:\/\/hdl.handle.net\/2429\/73237","language":null},{"key":"dc.description.abstract","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":null},{"key":"dc.format.extent","value":"56.0 minutes","language":null},{"key":"dc.format.mimetype","value":"video\/mp4","language":null},{"key":"dc.language.iso","value":"eng","language":null},{"key":"dc.publisher","value":"Banff International Research Station for Mathematical Innovation and Discovery","language":null},{"key":"dc.relation","value":"19w5088: Algebraic Techniques in Computational Complexity","language":null},{"key":"dc.relation.ispartofseries","value":"BIRS Workshop Lecture Videos (Banff, Alta)","language":null},{"key":"dc.rights","value":"Attribution-NonCommercial-NoDerivatives 4.0 International","language":null},{"key":"dc.rights.uri","value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","language":null},{"key":"dc.subject","value":"Mathematics","language":null},{"key":"dc.subject","value":"Computer Science, Theoretical Computer Science","language":null},{"key":"dc.title","value":"Reconstructing arithmetic formulas using lower bound proof techniques","language":null},{"key":"dc.type","value":"Moving Image","language":null},{"key":"dc.description.affiliation","value":"Non UBC","language":null},{"key":"dc.description.reviewstatus","value":"Unreviewed","language":null},{"key":"dc.description.notes","value":"Author affiliation: MSR India","language":null},{"key":"dc.description.scholarlevel","value":"Faculty","language":null},{"key":"dc.date.updated","value":"2020-01-09T06:00:46Z","language":null}]