De-noising by thresholding operator adapted wavelets

TL;DR

This paper extends wavelet thresholding techniques to operator-adapted wavelets (gamblets) for denoising functions with prior information on their derivatives, achieving near-minimax optimality and efficient computation.

Contribution

It introduces a new denoising method using gamblets that adapts to the regularity of an operator applied to the unknown function, not just the function itself.

Findings

Near-minimax optimal approximation with high probability.

Energy norm of the estimate bounded by that of the true function.

Method has near-linear computational complexity.

Abstract

Donoho and Johnstone proposed a method from reconstructing an unknown smooth function $u$ from noisy data $u+\zeta$ by translating the empirical wavelet coefficients of $u+\zeta$ towards zero. We consider the situation where the prior information on the unknown function $u$ may not be the regularity of $u$ but that of $ \L u$ where $\L$ is a linear operator (such as a PDE or a graph Laplacian). We show that the approximation of $u$ obtained by thresholding the gamblet (operator adapted wavelet) coefficients of $u+\zeta$ is near minimax optimal (up to a multiplicative constant), and with high probability, its energy norm (defined by the operator) is bounded by that of $u$ up to a constant depending on the amplitude of the noise. Since gamblets can be computed in $\mathcal{O}(N \operatorname{polylog} N)$ complexity and are localized both in space and eigenspace, the proposed method is of near-linear complexity and generalizable to non-homogeneous noise.