29 research outputs found
Signal detection for inverse problems in a multidimensional framework
International audienceThis paper is devoted to multi-dimensional inverse problems. In this setting, we address a goodness-of-fit testing problem. We investigate the separation rates associated to different kinds of smoothness assumptions and different degrees of ill-posedness
General regularization schemes for signal detection in inverse problems
The authors discuss how general regularization schemes, in particular linear regularization schemes and projection schemes, can be used to design tests for signal detection in statistical inverse problems. It is shown that such tests can attain the minimax separation rates when the regularization parameter is chosen appropriately. It is also shown how to modify these tests in order to obtain (up to a factor) a test which adapts to the unknown smoothness in the alternative. Moreover, the authors discuss how the so-called \emph{direct} and \emph{indirect} tests are related via interpolation properties
Feature selection by Higher Criticism thresholding: optimal phase diagram
We consider two-class linear classification in a high-dimensional, low-sample
size setting. Only a small fraction of the features are useful, the useful
features are unknown to us, and each useful feature contributes weakly to the
classification decision -- this setting was called the rare/weak model (RW
Model). We select features by thresholding feature -scores. The threshold is
set by {\it higher criticism} (HC). Let \pee_i denote the -value
associated to the -th -score and \pee_{(i)} denote the -th order
statistic of the collection of -values. The HC threshold (HCT) is the order
statistic of the -score corresponding to index maximizing (i/n -
\pee_{(i)})/\sqrt{\pee_{(i)}(1-\pee_{(i)})}. The ideal threshold optimizes the
classification error. In \cite{PNAS} we showed that HCT was numerically close
to the ideal threshold. We formalize an asymptotic framework for studying the
RW model, considering a sequence of problems with increasingly many features
and relatively fewer observations. We show that along this sequence, the
limiting performance of ideal HCT is essentially just as good as the limiting
performance of ideal thresholding. Our results describe two-dimensional {\it
phase space}, a two-dimensional diagram with coordinates quantifying "rare" and
"weak" in the RW model. Phase space can be partitioned into two regions -- one
where ideal threshold classification is successful, and one where the features
are so weak and so rare that it must fail. Surprisingly, the regions where
ideal HCT succeeds and fails make the exact same partition of the phase
diagram. Other threshold methods, such as FDR threshold selection, are
successful in a substantially smaller region of the phase space than either HCT
or Ideal thresholding.Comment: 4 figures, 24 page
Data-driven efficient score tests for deconvolution problems
We consider testing statistical hypotheses about densities of signals in
deconvolution models. A new approach to this problem is proposed. We
constructed score tests for the deconvolution with the known noise density and
efficient score tests for the case of unknown density. The tests are
incorporated with model selection rules to choose reasonable model dimensions
automatically by the data. Consistency of the tests is proved