Minimax Rates for Estimating the Dimension of a Manifold

Jisu Kim (1) , Alessandro Rinaldo (2) , Larry Wasserman (2, 3)

1 DATASHAPE - Understanding the Shape of Data
2 Statistics Department, Carnegie Mellon University
3 Machine Learning Department [Carnegie Mellon Univ.]

Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be estimated. We characterize the statistical difficulty of this problem by deriving upper and lower bounds on the minimax rate for estimating the dimension. First, we consider the problem of testing the hypothesis that the support of the data-generating probability distribution is a well-behaved manifold of intrinsic dimension $d_1$ versus the alternative that it is of dimension $d_2$, with $d_{1}

Domaines: Statistiques [math.ST]; Géométrie algorithmique [cs.CG]

Jisu Kim, Alessandro Rinaldo, Larry Wasserman. Minimax Rates for Estimating the Dimension of a Manifold. Journal of Computational Geometry, 2019, 10 (1), ⟨10.20382/jocg.v10i1a3⟩. ⟨hal-02425684⟩