{"id":165841,"date":"2014-01-01T00:00:00","date_gmt":"2014-01-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/learning-sparse-polynomial-functions\/"},"modified":"2018-10-16T20:02:43","modified_gmt":"2018-10-17T03:02:43","slug":"learning-sparse-polynomial-functions","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/learning-sparse-polynomial-functions\/","title":{"rendered":"Learning sparse polynomial functions"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We study the question of learning a sparse multi-variate polynomial over the real domain. In particular, for some unknown polynomial <i>f(vx )<\/i> of degree-<i>d<\/i> and <i>k<\/i> monomials, we show how to reconstruct <i>f<\/i>, within error <i>\u03b5<\/i>, given only a set of examples <i>bar x<sub>i<\/sub><\/i> drawn uniformly from the <i>n<\/i>-dimensional cube (or an <i>n<\/i>-dimensional Gaussian distribution), together with evaluations <i>f(bar x<sub>i<\/sub>)<\/i> on them. The result holds even in the \u201cnoisy setting\u201d, where we have only values <i>f(bar x<sub>i<\/sub>)+g<\/i> where <i>g<\/i> is noise (say, modeled as a Gaussian random variable). The runtime of our algorithm is polynomial in <i>n,k,1\/\u03b5<\/i> and <i>C<sub>d<\/sub><\/i> where <i>C<sub>d<\/sub><\/i> depends only on <i>d<\/i>. Note that, in contrast, in the \u201cboolean version\u201d of this problem, where <i>bar x<\/i> is drawn from the hypercube, the problem is at least as hard as the \u201cnoisy parity problem,\u201d where we do not know how to break the <i>n<sup>\u03a9(d)<\/sup><\/i> time barrier, even for <i>k=1<\/i>, and some believe it may be impossible to do so.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We study the question of learning a sparse multi-variate polynomial over the real domain. In particular, for some unknown polynomial f(vx ) of degree-d and k monomials, we show how to reconstruct f, within error \u03b5, given only a set of examples bar xi drawn uniformly from the n-dimensional cube (or an n-dimensional Gaussian distribution), 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