SLOPE is Adaptive to Unknown Sparsity and Asymptotically Minimax

Weijie Su; Emmanuel Candes

Paper

SLOPE is Adaptive to Unknown Sparsity and Asymptotically Minimax

Abstract

We consider high-dimensional sparse regression problems in which we observe

, where

is an

design matrix and

is an

-dimensional vector of independent Gaussian errors, each with variance

. Our focus is on the recently introduced SLOPE estimator ((Bogdan et al., 2014)), which regularizes the least-squares estimates with the rank-dependent penalty

, where

is the

th largest magnitude of the fitted coefficients. Under Gaussian designs, where the entries of

are i.i.d.~

, we show that SLOPE, with weights

just about equal to

(

is the

th quantile of a standard normal and

is a fixed number in

) achieves a squared error of estimation obeying

as the dimension

increases to

, and where

is an arbitrary small constant. This holds under a weak assumption on the

-sparsity level, namely,

and

, and is sharp in the sense that this is the best possible error any estimator can achieve. A remarkable feature is that SLOPE does not require any knowledge of the degree of sparsity, and yet automatically adapts to yield optimal total squared errors over a wide range of

-sparsity classes. We are not aware of any other estimator with this property.