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We study the problem of testing if a function depends on a small number of linear directions of its input data. We call a function f a linear k-junta if it is completely determined by some k-dimensional subspace of the input space. In this paper, we study the problem of testing whether a given n variable function f : R^n →{0,1}, is a linear k-junta or ϵ-far from all linear k-juntas, where the closeness is measured with respect to the Gaussian measure on R^n. This problems is a common generalization of (i) The combinatorial problem of junta testing on the hypercube which tests whether a Boolean function is dependent on at most k of its variables and (ii) Geometric testing problems such as testing if a function is an intersection of k halfspaces. We prove the existence of a poly(k · s/ϵ)-query non-adaptive tester for linear k-juntas with surface area at most s. The polynomial dependence on s is necessary as we provide a poly(s) lower bound on the query complexity of any non-adaptive test for linear juntas. Moreover, we show that if the function is a linear k-junta with surface area at most s, then there is a (s · k)^O(k)-query non-adaptive algorithm to learn the function up to a rotation of the basis. We also use this procedure to obtain a non-adaptive tester (with the same query complexity) for subclasses of linear k-juntas closed under rotation.
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