Adapting to unknown sparsity by controlling the false discovery rate

Abstract

We attempt to recover an n-dimensional vector observed in white noise, where n is large and the vector is known to be sparse, but the degree of sparsity is unknown. We consider three different ways of defining sparsity of a vector: using the fraction of nonzero terms; imposing power-law decay bounds on the ordered entries; and controlling the ℓ p norm for p small. We obtain a procedure which is asymptotically minimax for ℓ r loss, simultaneously throughout a range of such sparsity classes. The optimal procedure is a data-adaptive thresholding scheme, driven by control of the false discovery rate (FDR). FDR control is a relatively recent innovation in simultaneous testing, ensuring that at most a certain expected fraction of the rejected null hypotheses will correspond to false rejections. In our treatment, the FDR control parameter q n also plays a determining role in asymptotic minimaxity. If q = lim q n ∈[0, 1/2] and also q n > γ/ log(n), we get sharp asymptotic minimaxity, simultaneously, over a wide range of sparse parameter spaces and loss functions. On the other hand, q = lim q n ∈ (1/2, 1] forces the risk to exceed the minimax risk by a factor growing with q. To our knowledge, this relation between ideas in simultaneous inference and asymptotic decision theory is new. Our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors. These new rules impose complexity penalization of the form 2 · log(potential model size/ actual model sizes). We exhibit a close connection with FDR-controlling procedures under stringent control of the false discovery rate. © Institute of Mathematical Statistics, 2006.