  Efficient recovery of a low-dimensional structure from high-dimensional data
has been pursued in various settings including wavelet denoising, generalized
linear models and low-rank matrix estimation. By thresholding some parameters
to zero, estimators such as lasso, elastic net and subset selection allow to
perform not only parameter estimation but also variable selection, leading to
sparsity. Yet one crucial step challenges all these estimators: the choice of
the threshold parameter~$\lambda$. If too large, important features are
missing; if too small, incorrect features are included.
  Within a unified framework, we propose a new selection of $\lambda$ at the
detection edge under the null model. To that aim, we introduce the concept of a
zero-thresholding function and a null-thresholding statistic, that we
explicitly derive for a large class of estimators. The new approach has the
great advantage of transforming the selection of $\lambda$ from an unknown
scale to a probabilistic scale with the simple selection of a probability
level. Numerical results show the effectiveness of our approach in terms of
model selection and prediction.
