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We address the problem of recovering a sparse $n$-vector within a givensubspace. This problem is a subtask of some approaches to dictionary learningand sparse principal component analysis. Hence, if we can prove scaling lawsfor recovery of sparse vectors, it will be easier to derive and prove recoveryresults in these applications. In this paper, we present a scaling law forrecovering the sparse vector from a subspace that is spanned by the sparsevector and $k$ random vectors. We prove that the sparse vector will be theoutput to one of $n$ linear programs with high probability if its support size$s$ satisfies $s \lesssim n/\sqrt{k \log n}$. The scaling law still holds whenthe desired vector is approximately sparse. To get a single estimate for thesparse vector from the $n$ linear programs, we must select which output is thesparsest. This selection process can be based on any proxy for sparsity, andthe specific proxy has the potential to improve or worsen the scaling law. Ifsparsity is interpreted in an $\ell_1/\ell_\infty$ sense, then the scaling lawcan not be better than $s \lesssim n/\sqrt{k}$. Computer simulations show thatselecting the sparsest output in the $\ell_1/\ell_2$ or thresholded-$\ell_0$senses can lead to a larger parameter range for successful recovery than thatgiven by the $\ell_1/\ell_\infty$ sense.

Scaling Law for Recovering the Sparsest Element in a Subspace