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Theses

Convergence Rates for Geometric Inference

Eddie Aamari 1, 2, 3
2 SELECT - Model selection in statistical learning
LMO - Laboratoire de Mathématiques d'Orsay, Inria Saclay - Ile de France
3 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : Some datasets exhibit non-trivial geometric or topological features that can be interesting to infer. This thesis deals with non-asymptotic rates for various geometric quantities associated with submanifolds $M \subset \mathbb{R}^D$. In all the settings, we are given an i.i.d. $n$-sample with common distribution $P$ having support $M$. We study the optimal rates of estimation of the submanifold $M$ for the loss given by the Hausdorff metric, of the reach $\tau_{M}$, of the tangent space $T_{X} M$ and the second fundamental form $II_{X}^M$, for $X \in M$ both deterministic and random. The rates are given in terms of the sample size $n$, the instrinsic dimension of $M$ and of its regularity. In the process, we obtain stability results for existing reconstruction techniques, a denoising procedure and results on the geometry of the reach $\tau_{M}$. An extension of Assouad's lemma is presented, allowing to derive minimax lower bounds in singular frameworks.
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Eddie Aamari. Convergence Rates for Geometric Inference. Statistics [math.ST]. Université Paris-Saclay, 2017. English. ⟨NNT : 2017SACLS203⟩. ⟨tel-01607782⟩

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