# Testing noisy linear functions for sparsity

We consider the following basic inference problem: there is an unknown high-dimensional vector w ∈R^n, and an algorithm is given access to labeled pairs (x,y) where x ∈R^n is a measurement and y = w · x + noise. What is the complexity of deciding whether the target vector w is (approximately) k-sparse? The recovery analogue of this problem — given the promise that w is sparse, find or approximate the vector w— is the famous sparse recovery problem, with a rich body of work in signal processing, statistics, and computer science. We study the decision version of this problem (i.e. deciding whether the unknown w is k-sparse) from the vantage point of property testing. Our focus is on answering the following high-level question: when is it possible to efficiently test whether the unknown target vector w is sparse versus far-from-sparse using a number of samples which is completely independent of the dimension n? We consider the natural setting in which x is drawn from a i.i.d. product distribution D over R^n and the noise process is independent of the input x. As our main result, we give a general algorithm which solves the above-described testing problem using a number of samples which is completely independent of the ambient dimension n, as long as D is not a Gaussian. In fact, our algorithm is fully noise tolerant, in the sense that for an arbitrary w, it approximately computes the distance of w to the closest k-sparse vector. To complement this algorithmic result, we show that weakening any of our condition makes it information-theoretically impossible for any algorithm to solve the testing problem with fewer than essentially log n samples.

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