Uncertainty quantification through adaptive and honest confidence sets
Empirical uncertainty quantification of estimation procedures can be simple in parametric,
low dimensional situations. However, it becomes challenging and often problematic in
high and infinite dimensional models. Indeed, adaptivity to the unknown model complexity
becomes key in this case, and uncertainty quantification becomes akin to model estimation, see [3, 4].
Such model-adaptive uncertainty quantification can be formalised through the concept of adaptive
and honest confidence sets . Recent own results [2, 3] related to this concept will be presented.
 T. Cai and M. Low. An adaptation theory for nonparametric confidence intervals. The Annals of Statistics , volume 32:5, pp. 1805-1840, 2004.
 A. Carpentier, J. Eisert, D. Gross, and R. Nickl. Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems. arXiv:1504.03234.
 A. Carpentier, O. Klopp, M. Löffer, and R. Nickl. Adaptive confidence sets for matrix completion. arXiv:1608.04861.
 M. Hoffmann and R. Nickl. On adaptive inference and confidence bands. The Annals of Statistics , volume 39:5, pp. 2383-2409, 2011.
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Registration to Low-Dimensional Function Spaces
Registration aims to decompose amplitude and phase variation of samples of curves. Phase variation is captured by warping functions which monotonically transform the domains. Resulting registered curves should then only exhibit amplitude variation. Most existing registration method rely on aligning typical shape features like peaks or valleys to be found in each sample function. It is shown that this is not necessarily an optimal strategy for subsequent statistical data exploration and inference. In this context a major goal is to identify low dimensional linear subspaces of functions that are able to provide accurate approximations of the observed functional data. In this paper we present a registration method where warping functions are defined in such a way that the resulting registered curves span a low dimensional linear function space. Problems of identifiability are discussed in detail, and connections to established registration procedures are analyzed. The method is applied to real and simulated data.