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De-noising by thresholding operator adapted wavelets
Donoho and Johnstone proposed a method from reconstructing an unknown smooth
function from noisy data by translating the empirical wavelet
coefficients of towards zero. We consider the situation where the
prior information on the unknown function may not be the regularity of
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 obtained by thresholding the
gamblet (operator adapted wavelet) coefficients of 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 up to a
constant depending on the amplitude of the noise. Since gamblets can be
computed in complexity and are
localized both in space and eigenspace, the proposed method is of near-linear
complexity and generalizable to non-homogeneous noise
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