. .. The-deterministic-framework,

. , The k-PDTM: a coreset for the DTM

D. .. Proximity-to-the,

. .. Wasserstein-stability-for-the-k-pdtm,

. , Geometric inference with the k-PDTM

. .. The-statistical-framework, 151 4.2.2 Proximity between the k-PDTM and its empirical version

. , The set of local means-A strong connection with the distance-to-measure

. .. Proofs,

E. Aamari, Vitesses de convergence en inférence géométrique, 2017.

D. Charalambos, K. C. Aliprantis, and . Border, Infinite Dimensional Analysis: a Hitchhiker's Guide, 2006.

C. Aaron and A. Cholaquidis, On boundary detection. working paper or preprint, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01291996

D. A. Edwards, The structure of superspace, Studies in Topology, vol.12, pp.121-133, 1975.

A. Araujo and E. Giné, The Central Limit Theorem for Real and Banach Valued Random Variables, 1980.

S. Arlot, Rééchantillonnage et Sélection de Modèles, 2007.

M. Akahira and K. Takeuchi, Asymptotic Efficiency of Statistical Estimators: Concepts and Higher Order Asymptotic Efficiency: Concepts and Higher Order Asymptotic Efficiency, Lecture Notes in Statistics, 2011.

D. Arthur and S. Vassilvitskii, k-means++: the advantages of careful seeding, Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.1027-1035, 2007.

S. Boucheron, O. Bousquet, and G. Lugosi, Theory of classification: A survey of some recent advances, ESAIM: Probability and Statistics, vol.9, p.323, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00017923

T. Bednarski and B. R. Clarke, Trimmed likelihood estimation of location and scale of the normal distribution, Australian Journal of Statistics, vol.35, issue.2, pp.141-153, 1993.

M. Buchet, F. Chazal, S. Y. Oudot, and D. R. Sheehy, Efficient and robust persistent homology for measures, Proceedings of the Twenty-sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pp.168-180, 2015.
DOI : 10.1137/1.9781611973730.13
URL : https://hal.archives-ouvertes.fr/hal-01074566

H. Bensmail, G. Celeux, E. A. Raftery, and C. P. Robert, Inference in model-based cluster analysis, Statistics and Computing, vol.7, pp.1-10, 1997.

J. Boissonnat, F. Chazal, and M. Yvinec, Geometric and Topological Inference, 2017.
DOI : 10.1017/9781108297806
URL : https://hal.archives-ouvertes.fr/hal-01615863

G. Biau, L. Devroye, and G. Lugosi, On the performance of clustering in hilbert spaces, IEEE Trans. Information Theory, vol.54, issue.2, pp.781-790, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00290855

C. Brécheteau, A. Fischer, and C. Levrard, Robust bregman clustering, 2018.

A. Banerjee, X. Guo, and H. Wang, On the optimality of conditional expectation as a bregman predictor, IEEE Transactions on Information Theory, vol.51, issue.7, pp.2664-2669, 2005.

P. Billingsley, Convergence of Probability Measures, 1999.

M. Boutin and G. Kemper, On reconstructing n-point configurations from the distribution of distances or areas, Advances in Applied Mathematics, vol.32, pp.709-735, 2004.

S. Bobkov and M. Ledoux, One-dimensional empirical measures, order statistics and kantorovich transport distances. the Memoirs of the Amer, 2014.

B. Buet and G. P. Leonardi, Recovering measures from approximate values on balls, Annales Academiae Scientiarum Fennicae Mathematica, p.41, 2016.
DOI : 10.5186/aasfm.2016.4160
URL : https://doi.org/10.5186/aasfm.2016.4160

C. Brécheteau and C. Levrard, The k-pdtm: A coreset for robust geometric inference, 2017.

S. Boucheron, G. Lugosi, and P. Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00794821

S. Gary and . Bloom, A counterexample to a theorem of s.piccard, Journal of Combinatorial Theory, Series A, vol.22, pp.378-379, 1977.

A. Banerjee, S. Merugu, S. Inderjit, J. Dhillon, and . Ghosh, Clustering with bregman divergences, J. Mach. Learn. Res, vol.6, pp.1705-1749, 2005.
DOI : 10.1137/1.9781611972740.22
URL : http://www.lans.ece.utexas.edu/~abanerjee/papers/04/sdm-breg.ps

J. Boissonnat, F. Nielsen, and R. Nock, Bregman voronoi diagrams. Discrete & Computational Geometry, vol.44, pp.281-307, 2010.
DOI : 10.1007/s00454-010-9256-1
URL : https://hal.archives-ouvertes.fr/hal-00488441

É. Borel, Leçons sur la théorie des fonctions. Collection de monographies sur la théorie des fonctions. Gauthier-Villars et fils, 1898.

É. Borel, Traité du calcul des probabilités et de ses applications, volume I of Traité du calcul des probabilités et de ses applications, Gautier-Villars et cie, 1930.

L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Computational Mathematics and Mathematical Physics, vol.7, issue.3, pp.200-217, 1967.

C. Brécheteau, The dtm-signature for a geometric comparison of metricmeasure spaces from samples. Unpublished, 2016.

C. Brécheteau, A statistical test of isomorphism between metric-measure spaces using the distance-to-a-measure signatures, 2018.

M. Buchet, Topological inference from measures, 2014.
URL : https://hal.archives-ouvertes.fr/tel-01108521

J. A. Cuesta-albertos, A. Gordaliza, and C. Matrán, Trimmed k-means: an attempt to robustify quantizers, Ann. Statist, vol.25, issue.2, pp.553-576, 1997.
DOI : 10.1214/aos/1031833664
URL : https://doi.org/10.1214/aos/1031833664

J. A. Cuesta-albertos, C. Matrán, A. Mayo-iscar-;-frédéric, D. Chazal, L. J. Cohen-steiner et al., Robust estimation in the normal mixture model based on robust clustering, Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol.70, issue.4, pp.1393-1403, 2008.

F. Chazal, D. Cohen-steiner, and Q. Mérigot, Geometric inference for probability measures, Foundations of Computational Mathematics archive, vol.11, pp.733-751, 2011.
DOI : 10.1007/s10208-011-9098-0
URL : https://hal.archives-ouvertes.fr/hal-00772444

F. Chazal, . Vin-de, M. Silva, S. Y. Glisse, and . Oudot, The Structure and Stability of Persistence Modules, Briefs in Mathematics, 2016.
DOI : 10.1007/978-3-319-42545-0
URL : https://hal.archives-ouvertes.fr/hal-01107617

F. Chazal, V. D. Silva, S. Oudot-;-frédéric-chazal, B. T. Fasy, F. Lecci et al., On the bootstrap for persistence diagrams and landscapes. Modeling and Analysis of Information Systems, Geometriae Dedicata, vol.173, issue.1, pp.96-105, 2013.
DOI : 10.18255/1818-1015-2013-6-111-120
URL : https://hal.archives-ouvertes.fr/hal-00879982

G. Celeux and G. Govaert, Gaussian parsimonious clustering models, Pattern Recognition, vol.28, issue.5, pp.781-793, 1995.
DOI : 10.1016/0031-3203(94)00125-6
URL : https://hal.archives-ouvertes.fr/inria-00074643

F. Chazal, M. Glisse, C. Labruère, and B. Michel, Convergence rates for persistence diagram estimation in topological data analysis, J. Mach. Learn. Res, vol.16, issue.1, pp.3603-3635, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01073072

F. Chazal, L. J. Guibas, S. Y. Oudot, and P. Skraba, Persistence-based clustering in riemannian manifolds, J. ACM, vol.60, issue.6, 2013.
DOI : 10.1145/1998196.1998212
URL : https://hal.archives-ouvertes.fr/inria-00389390

F. Chazal and A. Lieutier, The ?-medial axis, Graphical Models, vol.67, pp.304-331, 2005.

N. A. Campbell and R. J. Mahon, A multivariate study of variation in two species of rock crab of genus emphLeptograpsus, Australian Journal of Zoology, vol.22, pp.417-425, 1974.

F. Chazal, P. Massart, and B. Michel, Rates of convergence for robust geometric inference, Electron. J. Statist, vol.10, issue.2, pp.2243-2286, 2016.
DOI : 10.1214/16-ejs1161
URL : https://hal.archives-ouvertes.fr/hal-01157551

D. L. Cohn, Measure Theory. Birkhäuser, 1980.

D. Cohen-steiner, H. Edelsbrunner, and J. Harer, Stability of persistence diagrams, Discrete & Computational Geometry, vol.37, issue.1, pp.103-120, 2007.

A. De-acosta and E. Giné, Convergence Of Moments And Related Functionals In The Central Limit Theorem In Banach Spaces, Z. Wahrsch.ver.Geb, 1979.

E. Eustasio-del-barrio, C. Giné, and . Matrán, Central limit theorems for the Wasserstein distance between the empirical and the true distributions, The Annals of Probability, vol.27, pp.1009-1071, 1999.

E. Del-barrio, H. Lescornel, and J. Loubes, A statistical analysis of a deformation model with wasserstein barycenters : estimation procedure and goodness of fit test. unpublished, 2015.

F. Dotto, A. Farcomeni, L. A. García-escudero, and A. Mayo-iscar, A reweighting approach to robust clustering, Statistics and Computing, vol.28, issue.2, pp.477-493, 2018.

R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Theory of Probability & Its Applications, vol.15, pp.458-486, 1970.

A. Dasgupta and A. E. Raftery, Detecting features in spatial point processes with clutter via model-based clustering, Journal of the American Statistical Association, vol.93, issue.441, pp.294-302, 1998.

R. Dudley, Balls in rk do not cut all subsets of k + 2 points, Advances in Mathematics, vol.31, issue.3, pp.306-308, 1979.

A. Eddie and L. Clément, Stability and minimax optimality of tangential delaunay complexes for manifold reconstruction. Discrete and Computational Geometry, 2017.

. Herbert and . Edelsbrunner, Weighted Alpha Shapes. Number num 1760 in Report, 1992.

R. Fortet and E. Mourier, Convergence de la répartition empirique vers la répartition théorique, Ann. Scient. École Norm. Sup, vol.70, pp.266-285, 1953.

B. Efron, Bootstrap methods: Another look at the jackknife, Annals of Statistics, vol.7, pp.1-26, 1979.

L. Edelsbrunner and Z. , Topological persistence and simplification, Discrete & Computational Geometry, vol.28, issue.4, pp.511-533, 2002.

H. Federer, Curvature measures, Trans. Amer. Math. Soc, vol.93, 1959.

H. Federer, Geometric Measure Theory, 1996.

N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure. Probability Theory and Related Fields, vol.162, p.707, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00915365

H. Fritz, L. A. García-escudero, and A. Mayo-iscar, tclust: An r package for a trimming approach to cluster analysis, 2012.

R. A. Fisher, On the mathematical foundations of theoretical statistics, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol.222, pp.309-368, 1922.

A. Fischer, Quantization and clustering with bregman divergences, Journal of Multivariate Analysis, vol.101, issue.9, pp.2207-2221, 2010.

B. T. Fasy, J. Kim, F. Lecci, and C. Maria, , 2014.

M. Fromont, B. Laurent, M. Lerasle, and P. Reynaudbouret, Kernels based tests with non-asymptotic bootstrap approaches for twosample problems, Journal of Machine Learning Research: Workshop and Conference proceedings, vol.23, pp.23-24, 2012.

K. Florek, J. ?ukaszewicz, J. Perkal, H. Steinhaus, and S. Zubrzycki, Sur la liaison et la division des points d'un ensemble fini, Colloquium Mathematicae, vol.2, issue.3-4, pp.282-285, 1951.

B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan et al., Confidence sets for persistence diagrams, Ann. Statist, vol.42, issue.6, pp.2301-2339, 2014.

E. W. Forgy, Cluster analysis of multivariate data: efficiency versus interpretability of classifications, Biometrics, vol.21, pp.768-769, 1965.

C. Fraley and A. E. Raftery, How many clusters? which clustering method? answers via model-based cluster analysis, The Computer Journal, vol.41, pp.578-588, 1998.

C. Fraley and A. E. Raftery, Model-based clustering, discriminant analysis, and density estimation, Journal of the American Statistical Association, vol.97, issue.458, pp.611-631, 2002.
DOI : 10.1198/016214502760047131

J. G. Sheals, The application of computer techniques to acarine taxonomy: a preliminary examination with species of the hypoaspis-androlaelaps complex (acarina), Proceedings of the Linnean Society of London, vol.176, pp.11-21, 1965.

L. García-escudero, A. Gordaliza, C. Matrán, and A. Mayo, A review of robust clustering methods, Advances in Data Analysis and Classification, vol.4, pp.89-109, 2010.

A. Luis, A. García-escudero, C. Gordaliza, A. Matrán, and . Mayoiscar, A general trimming approach to robust cluster analysis, Ann. Statist, vol.36, issue.3, pp.1324-1345, 2008.

S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, 2000.

M. Gromov, J. Lafontaine, and P. Pansu, Structures métriques pour les variétés riemanniennes. Textes mathématiques. CEDIC/Fernand Nathan, 1981.

M. Gromov, J. Lafontaine, and P. Pansu, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics-Birkhäuser. Birkhäuser, 1999.

A. Gretton, K. M. Borgwardt, M. J. Rasch, B. Schölkopf, and A. Smola, A kernel two-sample test, Journal of Machine Learning Research, vol.13, pp.723-773, 2012.

L. J. Guibas, Q. Mérigot, and D. Morozov, Witnessed kdistance, Proceedings of the Twenty-seventh Annual Symposium on Computational Geometry, SoCG '11, pp.57-64, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00872490

A. Gordaliza, Best approximations to random variables based on trimming procedures, Journal of Approximation Theory, vol.64, issue.2, pp.162-180, 1991.
DOI : 10.1016/0021-9045(91)90072-i
URL : https://doi.org/10.1016/0021-9045(91)90072-i

T. María, G. Gallegos, and . Ritter, A robust method for cluster analysis, Ann. Statist, vol.33, issue.1, pp.347-380, 2005.

M. Gromov, Groups of polynomial growth and expanding maps, vol.53, pp.53-78, 1981.

C. Hennig, Breakdown points for maximum likelihood estimators of location-scale mixtures, Ann. Statist, vol.32, issue.4, pp.1313-1340, 2004.

A. S. Hadi and A. Luceño, Maximum trimmed likelihood estimators: a unified approach, examples, and algorithms, Computational Statistics & Data Analysis, vol.25, issue.3, pp.251-272, 1997.

J. Peter and . Huber, Robust estimation of a location parameter, Ann. Math. Statist, vol.35, issue.1, pp.73-101, 1964.

. Pavel?í?ekpavel?pavel?í?ek, General trimmed estimation: robust approach to nonlinear and limited dependent variable models, Econometric Theory, vol.24, issue.6, pp.1500-1529, 2008.

P. Jansser, J. Jureckova, and N. Veraverbeke, Rate of convergence of one-and two-step m-estimators with applications to maximum likelihood and pitman estimators, Annals of Statistics, vol.13, issue.3, pp.1222-1229, 1985.

S. Johnson, Hierarchical clustering schemes, Psychometrika, vol.32, issue.3, pp.241-254, 1967.

B. Kloeckner, Approximation by finitely supported measures. ESAIM: Control, Optimisation and Calculus of Variations, vol.18, pp.343-359, 2012.
DOI : 10.1051/cocv/2010100
URL : https://hal.archives-ouvertes.fr/hal-00461329

A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, 1933.

H. Komiya, Elementary proof for sion's minimax theorem, Kodai Math. J, vol.11, issue.1, pp.5-7, 1988.
DOI : 10.2996/kmj/1138038812
URL : https://doi.org/10.2996/kmj/1138038812

L. V. Kantorovich and G. Sh, Rubinshte ? in. On a space of completely additive functions, Vestnik Leningrad. Univ, vol.13, pp.52-59, 1958.

P. Lévy, Théorie de l'addition des variables aléatoires, Monographies des probabilite's. Paris : Gautier-Villars, 1937.

C. Levrard, Quantification vectorielle en grande dimension : vitesses de convergence et sélection de variables, 2014.

C. Levrard, Nonasymptotic bounds for vector quantization in hilbert spaces, Ann. Statist, vol.43, issue.2, pp.592-619, 2015.
DOI : 10.1214/14-aos1293
URL : http://arxiv.org/pdf/1405.6672

A. Lieutier, Any open bounded subset of rn has the same homotopy type as its medial axis, Computer-Aided Design, vol.36, pp.1029-1046, 2004.

T. Linder, Learning-theoretic methods in vector quantization, Principles of nonparametric learning, vol.434, pp.163-210, 2001.
DOI : 10.1007/978-3-7091-2568-7_4
URL : http://magenta.mast.queensu.ca/~linder/psfiles/cism.ps

S. P. Lloyd, Least squares quantization in PCM, IEEE Transactions on Information Theory, vol.28, pp.129-137, 1982.
DOI : 10.1109/tit.1982.1056489
URL : http://www.cs.toronto.edu/~roweis/csc2515-2006/readings/lloyd57.pdf

G. Lugosi, Pattern classification and learning theory, Principles of nonparametric learning, vol.434, pp.1-56, 2001.
DOI : 10.1007/978-3-7091-2568-7_1
URL : http://www.econ.upf.es/~lugosi/lecturenotes.ps

A. Müller, Integral probability metrics and their generating classes of functions, Advances in Applied Probability, vol.29, issue.2, pp.429-443, 1997.

J. Macqueen, Some methods for classification and analysis of multivariate observations, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol.1, pp.281-297, 1967.

M. Markatou, Mixture models, robustness, and the weighted likelihood methodology, Biometrics, vol.56, issue.2, pp.483-486, 2000.
DOI : 10.1111/j.0006-341x.2000.00483.x

P. Massart, The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality, The Annals of Probability, vol.18, issue.3, pp.1269-1283, 1990.
DOI : 10.1214/aop/1176990746
URL : http://doi.org/10.1214/aop/1176990746

P. Massart, Concentration Inequalities and Model Selection, 2007.

F. Mémoli, Gromov-Wasserstein distances and the metric approach to object matching, Foundations of Computational Mathematics, vol.11, issue.4, pp.417-487, 2011.

Q. Mérigot, Lower bounds for k-distance approximation, SoCG '1329th ACM Symposium on Computational Geometry, vol.15, pp.435-440, 2013.

G. J. Mclachlan and D. Peel, Finite mixture models, Wiley Series in Probability and Statistics, 2000.

. H. Ch and . Müller, Trimmed likelihood estimators in generalized linear models, Sixth International Conference on Computer Data Analysis and Modeling, vol.2, pp.142-150, 2001.

S. Mendelson and R. Vershynin, Entropy and the combinatorial dimension. Inventiones mathematicae, vol.152, pp.37-55, 2003.

F. Nielsen, J. Boissonnat, and R. Nock, Bregman voronoi diagrams: Properties, algorithms and applications, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00137865

N. M. Neykov, P. Filzmoser, R. Dimova, and P. N. Neytchev, Robust fitting of mixtures using the trimmed likelihood estimator, Computational Statistics & Data Analysis, vol.52, pp.299-308, 2007.

N. Neykov and P. Neytchev, A robust alternative of the maximum likelihood estimators, COMPSTAT 1990-Short Communications, vol.01, pp.99-100, 1990.

P. Niyogi, S. Smale, and S. Weinberger, Finding the homology of submanifolds with high confidence from random samples. Discrete and Computational Geometry, vol.39, pp.419-441, 2008.

R. Osada, T. Funkhouser, B. Chazelle, and D. Dobkin, Shape distributions, ACM Transactions on Graphics, vol.21, pp.807-832, 2002.

D. Peel and G. J. Mclachlan, Robust mixture modelling using the t distribution, Statistics and Computing, vol.10, issue.4, pp.339-348, 2000.

D. Pollard, A central limit theorem for k-means clustering, The Annals of Probability, vol.10, issue.4, pp.919-945, 1982.

D. Pollard, Quantization and the method of k-means, IEEE Transactions on Information Theory, vol.28, issue.2, 1982.

D. N. Politis and J. P. Romano, Large sample confidence regions based on subsamples under minimal assumptions, The Annals of Statistics, vol.22, issue.4, pp.2031-2050, 1994.

Y. V. Prokhorov, Convergence of Random Processes and Limit Theorems in Probability Theory. Theory of Probability and its Applications, vol.1, pp.157-214, 1956.

J. M. Phillips, B. Wang, and Y. Zheng, Geometric inference on kernel density estimates, 2015.

G. R. Shorack and J. A. Wellner, Empirical processes with applications to statistics, Society for Industrial & Applied Mathematics, 2009.

J. Peter, A. M. Rousseeuw, and . Leroy, Robust Regression & Outlier Detection, 1987.

R. Tyrrell-rockafellar, Princetown Landmarks in Mathematics, 1970.

J. Peter and . Rousseeuw, Least median of squares regression, Journal of the American Statistical Association, vol.79, issue.388, pp.871-880, 1984.

P. J. Rousseeuw, Multivariate estimation with high breakdown point. Mathematical statistics and applications, vol.8, 1985.

A. Richard, H. F. Redner, and . Walker, Mixture densities, maximum likelihood and the em algorithm, SIAM Review, vol.26, issue.2, pp.195-239, 1984.

A. Strehl and J. Ghosh, Cluster ensembles-a knowledge reuse framework for combining multiple partitions, Journal of Machine Learning Research, vol.3, pp.583-617, 2002.

H. Steinhaus, Sur la division des corp materiels en parties, Bull. Acad. Polon. Sci, vol.1, pp.801-804, 1956.

M. Stephens, Bayesian analysis of mixture models with an unknown number of components-an alternative to reversible jump methods, Ann. Statist, vol.28, issue.1, pp.40-74, 2000.

Q. Tang and R. J. Karunamuni, Robust variable selection for finite mixture regression models, Annals of the Institute of Statistical Mathematics, vol.70, issue.3, pp.489-521, 2018.

V. Vapnik, Estimation of Dependences Based on Empirical Data, Springer Series in Statistics (Springer Series in Statistics), 1982.

V. S. Varadarajan, On a problem in measure-spaces, Ann. Math. Statist, vol.29, issue.4, p.1958

V. S. Varadarajan, On the convergence of sample probability distributions, The Indian Journal of Statistics, vol.19, pp.23-26, 1933.

S. Van-de and . Geer, Rates of convergence for the maximum likelihood estimator in mixture models, Journal of Nonparametric Statistics, vol.6, issue.4, pp.293-310, 1996.

W. Aad, J. A. Van-der-vaart, and . Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics, Springer Series in Statistics, 1996.

C. Villani, Topics in Optimal Transportation, 2003.

C. Villani, Optimal transport : old and new. Grundlehren der mathematischen Wissenschaften, 2009.

D. L. Vandev, N. M. Neykov, and *. , About regression estimators with high breakdown point, Statistics, vol.32, issue.2, pp.111-129, 1998.

C. Yu, W. Yao, and K. Chen, A new method for robust mixture regression, Canadian Journal of Statistics, vol.45, issue.1, pp.77-94, 2016.