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<sub>p</sub> norm for p small. We obtain a procedure which is asymptotically minimax for ℓ<sup>r</sup> loss, simultaneously throughout a range of such sparsity classes.\n¶ 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.\n¶ In our treatment, the FDR control parameter q<sub>n</sub> also plays a determining role in asymptotic minimaxity. If q=lim q<sub>n</sub>∈[0,1/2] and also q<sub>n</sub>>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<sub>n</sub>∈(1/2,1] forces the risk to exceed the minimax risk by a factor growing with q.\n¶ To our knowledge, this relation between ideas in simultaneous inference and asymptotic decision theory is new.\n¶ 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.

Keywords

Affiliated Institutions

• Tel Aviv University IL
• Stanford University US

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