  We address the problem of signal denoising via transform-domain shrinkage
based on a novel $\textit{risk}$ criterion called the minimum probability of
error (MPE), which measures the probability that the estimated parameter lies
outside an $\epsilon$-neighborhood of the actual value. However, the MPE,
similar to the mean-squared error (MSE), depends on the ground-truth parameter,
and has to be estimated from the noisy observations. We consider linear
shrinkage-based denoising functions, wherein the optimum shrinkage parameter is
obtained by minimizing an estimate of the MPE. When the probability of error is
integrated over $\epsilon$, it leads to the expected $\ell_1$ distortion. The
proposed MPE and $\ell_1$ distortion formulations are applicable to various
noise distributions by invoking a Gaussian mixture model approximation. Within
the realm of MPE, we also develop an extension of the transform-domain
shrinkage by grouping transform coefficients, resulting in $\textit{subband
shrinkage}$. The denoising performance obtained within the proposed framework
is shown to be better than that obtained using the minimum MSE-based approaches
formulated within $\textbf{$\textit {Stein's unbiased risk estimation}$}$
(SURE) framework, especially in the low measurement signal-to-noise ratio (SNR)
regime. Performance comparison with three state-of-the-art denoising
algorithms, carried out on electrocardiogram signals and two test signals taken
from the $\textit{Wavelab}$ toolbox, exhibits that the MPE framework results in
consistent SNR gains for input SNRs below $5$ dB.
