Minimax Rate of Testing in Sparse Linear Regression

Abstract

We consider the problem of testing the hypothesis that the parameter of linear regression model is 0 against an s-sparse alternative separated from 0 in the ℓ 2 -distance. We show that, in Gaussian linear regression model with p < n, where p is the dimension of the parameter and n is the sample size, the non-asymptotic minimax rate of testing has the form (s/n) log(1 + √ p/s). We also show that this is the minimax rate of estimation of the ℓ 2 -norm of the regression parameter.MSC 2010 subject classifications: 62J05, 62G10.

Estimation of the $l_2$-norm and testing in sparse linear regression with unknown variance

“…The setting in [2] is different from ours since it does not consider the alternative defined by separation in the ℓ 2 -norm. It does not allow one to compare [2] directly with [18,25,6] and with the present work. Note also that [2] explores an asymptotic setting as p, N, s tend to ∞, while in this paper we are interested in non-asymptotic results.…”

“…In the Gaussian mean model, which corresponds to an orthogonal nonrandom design X, the problem of signal detection has been extensively studied in the last fifteen years (see, e.g., [16,17,3,13,8,9] and the references therein). More recently, this problem has also been investigated in the random design linear regression model, which is most related to the present paper [18,2,25,6]. Among these, [2] and [6] deal with the case of known σ, while [18,25] consider both known and unknown σ.…”

“…More recently, this problem has also been investigated in the random design linear regression model, which is most related to the present paper [18,2,25,6]. Among these, [2] and [6] deal with the case of known σ, while [18,25] consider both known and unknown σ. The setting in [2] is different from ours since it does not consider the alternative defined by separation in the ℓ 2 -norm.…”

Near-Optimal Procedures for Model Discrimination with Non-Disclosure Properties

“…Still, one important direction is signal testing in linear regression [19,20,21]. Variants of this problem were considered when the signal is sparse and the noise either known [22] or unknown [23]. Other works test, in the same flavor, sparsity of the signal [24] or some component of the signal [25].…”

Minimax optimal estimators for general additive functional estimation

“…The distinct elements problem, i.e., finding how many different colors are present among at most d colored balls in an urn, can also be expressed in this form Polyanskiy and Wu (2019); Wu and Yang (2018). Moreover, the quadratic functional defined by F (t) = t 2 is key in the problem of signal detection Carpentier et al (2018), and when the vector θ is assumed to be sparse, i.e., when most of its coefficients are assumed to be exactly 0, it also plays a crucial role for noise variance estimation . Finally, robust estimation of the mean is shown in Collier and Dalalyan (2019) to be related with a linear functional of the outliers.…”