Approximate Sparsity Class and Minimax Estimation

TL;DR

This paper introduces the approximate sparsity class for functions based on Fourier coefficient decay, establishes its metric entropy and minimax rate, and proposes an adaptive estimator that nearly attains this optimal rate.

Contribution

It defines a new approximate sparsity class, derives its metric entropy and minimax rate, and develops an adaptive estimator achieving near-optimal convergence.

Findings

The approximate sparsity class has a calculable metric entropy.

The minimax rate of convergence for this class is established.

An adaptive estimator nearly attains the minimax rate.

Abstract

Motivated by the orthogonal series density estimation in $L^2([0,1],\mu)$, in this project we consider a new class of functions that we call the approximate sparsity class. This new class is characterized by the rate of decay of the individual Fourier coefficients for a given orthonormal basis. We establish the $L^2([0,1],\mu)$ metric entropy of such class, with which we show the minimax rate of convergence. For the density subset in this class, we propose an adaptive density estimator based on a hard-thresholding procedure that achieves this minimax rate up to a $\log$ term.