Multiple hypothesis testing is a significant problem in nearly all neuroimaging studies. In order to correct for this phenomena, we require a reliable estimate of the Family-Wise Error Rate (FWER). The well known Bonferroni correction method, while simple to implement, is quite conservative, and can substantially under-power a study because it ignores dependencies between test statistics. Permutation testing, on the other hand, is an exact, non-parametric method of estimating the FWER for a given $\alpha$-threshold, but for acceptably low thresholds the computational burden can be prohibitive. In this paper, we show that permutation testing in fact amounts to populating the columns of a very large matrix ${\bf P}$. By analyzing the spectrum of this matrix, under certain conditions, we see that ${\bf P}$ has a low-rank plus a low-variance residual decomposition which makes it suitable for highly sub--sampled --- on the order of $0.5\%$ --- matrix completion methods. Based on this observation, we propose a novel permutation testing methodology which offers a large speedup, without sacrificing the fidelity of the estimated FWER. Our evaluations on four different neuroimaging datasets show that a computational speedup factor of roughly $50\times$ can be achieved while recovering the FWER distribution up to very high accuracy. Further, we show that the estimated $\alpha$-threshold is also recovered faithfully, and is stable.
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