Some problems in geometric data analysis

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The analysis of datasets is increasingly geometrical. In some applications, such as in the analysis of the cosmic web in astrophysics, this is arguably the most natural approach. In others, modeling datasets via geometric structures allows to bypass the use of functions, which are in general difficult to deal with in high-dimensions because of the so-called "curse of dimensionality". The project aims at making contributions in this general area of geometrical data analysis, and in particular in fields like clustering, dimensionality reduction, and surface estimation, via the development of new methodology and new theory.Geometrical approaches to data analysis are well-established. Clustering, dimensionality reduction, and manifold/surface estimation, are well-developed, with ongoing work in the form of robust PCA, subspace clustering, manifold or surface clustering, manifold learning, geometric statistics, computational geometry, etc. The vast majority of this research is methodological or applied to a particular problem in a specific field, and theory is by and large lagging behind. This is, for example, the case in important areas such as subspace clustering, manifold embedding and sensor localization. This project has the ambition to contribute theoretical insights in those areas. While methodology tends to be well ahead of theory, good and timely theoretical analyses can shed some light on applied problems, and can sometimes inform the design of more effective methodology. And such methodology is missing in some areas of geometric statistics. Thus the project also includes the development of practical methodology that provably matches the minimax performance bounds available for the problem at hand, particularly in the areas of manifold estimation and the estimation of geometric characteristics of a distribution.