Quadratic functional estimation in inverse problems

TL;DR

This paper develops a Pinsker type estimator for quadratic functionals in inverse problems modeled by Gaussian sequences, analyzing its risk and efficiency across different smoothness classes of signals.

Contribution

It introduces a new projection estimator for quadratic functionals in inverse problems and provides risk bounds and efficiency analysis for various smoothness conditions.

Findings

Achieves parametric rates for sufficiently smooth signals.

Provides upper bounds for the second order risk term.

Conjectures asymptotic sharpness of risk bounds.

Abstract

We consider in this paper a Gaussian sequence model of observations $Y_i$, $i\geq 1$ having mean (or signal) $\theta_i$ and variance $\sigma_i$ which is growing polynomially like $i^\gamma$, $\gamma >0$. This model describes a large panel of inverse problems. We estimate the quadratic functional of the unknown signal $\sum_{i\geq 1}\theta_i^2$ when the signal belongs to ellipsoids of both finite smoothness functions (polynomial weights $i^\alpha$, $\alpha>0$) and infinite smoothness (exponential weights $e^{\beta i^r}$, $\beta >0$, $0<r \leq 2$). We propose a Pinsker type projection estimator in each case and study its quadratic risk. When the signal is sufficiently smoother than the difficulty of the inverse problem ($\alpha>\gamma+1/4$ or in the case of exponential weights), we obtain the parametric rate and the efficiency constant associated to it. Moreover, we give upper bounds of the second order term in the risk and conjecture that they are asymptotically sharp minimax. When the signal is finitely smooth with $\alpha \leq \gamma +1/4$, we compute non parametric upper bounds of the risk of and we presume also that the constant is asymptotically sharp.