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This paper addresses the case where data come as point sets, or moregenerally as discrete measures. Our motivation is twofold: first we intend toapproximate with a compactly supported measure the mean of the measuregenerating process, that coincides with the intensity measure in the pointprocess framework, or with the expected persistence diagram in the framework ofpersistence-based topological data analysis. To this aim we provide twoalgorithms that we prove almost minimax optimal. Second we build from theestimator of the mean measure a vectorization map, that sends every measureinto a finite-dimensional Euclidean space, and investigate its propertiesthrough a clustering-oriented lens. In a nutshell, we show that in a mixture ofmeasure generating process, our technique yields a representation in$\mathbb{R}^k$, for $k \in \mathbb{N}^*$ that guarantees a good clustering ofthe data points with high probability. Interestingly, our results apply in theframework of persistence-based shape classification via the ATOL proceduredescribed in \cite{Royer19}.

Optimal quantization of the mean measure and applications to statistical learning