Minimax theory for a class of nonlinear statistical inverse problems

Abstract

We study a class of statistical inverse problems with non-linear pointwise operators motivated by concrete statistical applications. A two-step procedure is proposed, where the first step smoothes the data and inverts the non-linearity. This reduces the initial non-linear problem to a linear inverse problem with deterministic noise, which is then solved in a second step. The noise reduction step is based on wavelet thresholding and is shown to be minimax optimal (up to logarithmic factors) in a pointwise functiondependent sense. Our analysis is based on a modified notion of Hölder smoothness scales that are natural in this setting.

A regularity class for the roots of nonnegative functions

“…The decay of the wavelet coefficients | f α , ψ j,k | characterizes the Besov norms f α ∞∞ equals the slightly larger Hölder-Zygmund space for integer β, resulting in a slight suboptimality when αβ ∈ N. While Theorem 1 provides a more concise statement, the extra local information provided by the above wavelet bounds can be crucial to obtain sharp results, for example in certain nonlinear statistical inverse problems [22]. Recall that f ∈ C γ if and…”

“…[16]). In the context of this problem, one must necessarily restrict to positive square roots and it is not enough to find flatness conditions ensuring that √ f has some regularity: some control of the Hölder norm is also required [22].…”

The Le Cam distance between density estimation, Poisson processes and Gaussian white noise

“…If f is allowed to depend on n and 0 < β ≤ 2, the pointwise rate of estimation at any x ∈ (0, 1) over the parameter space C β (R) is given by (1.3), up to log n-factors (see Theorems 3.1 and 3.3 of [26] and Theorems 1 and 2 of [29]). This rate of convergence does not extend beyond β = 2 using the usual definition of Hölder smoothness (Theorem 3 of [29]). To take advantage of higher order smoothness, we must therefore modify our function class.…”

“…Such a notion is widely used in high-dimensional statistics and turns out to be natural in our setting as well. Density estimation is a qualitatively different problem for densities taking values near zero, both in terms of estimation rates [26,29] and asymptotic equivalence, as we show below. Indeed, an n-dependent threshold turns out to be the correct notion to characterize "small densities", much as in the case of high-dimensional statistics.…”

“…As a density approaches zero however, the Hellinger distance behaves more like the L 1 -distance, leading to the same rates occurring in irregular models, such as in nonparametric regression with one-sided errors [15]. For further discussion see [26,29].…”