Rates of convergence of the constrained least squares estimator in high-dimensional monotone single-index models

In the high-dimensional monotone single index model, one observes i.i.d. copies of (X,Y) where the real response variable Y is linked to the d-dimensional covariate X through the relationship E[Y|X]=Ψ0(α0TX) almost surely, where the dimension d is allowed to grow with the sample size. The index parameter α0∈ℝd and the monotone ridge function Ψ0 are both unknown. The main focus of the article is to compute convergence rates of the least squares estimator (LSE) of the bundled regression function g0=Ψ0(α0T⋅) under the constraints that Ψ0 is monotone and α0 belongs to a certain set 𝒮. Special attention is paid to the case where 𝒮 describes sparsity constraints on α0, that is when 𝒮 is the set of vectors in ℝd whose number of non zero components does not exceed some threshold. Our results cover also the case of a misspecified model, when α0∉𝒮. Since the LSE of the index is computationally expensive for large d, we have implemented a forward selection algorithm, which runs much faster, for the case of sparcity constraints. Simulations indicate that the forward selection estimator (for both the index parameter and monotone ridge function) is a quite satisfactory alternative in the situation of sparsity.