Empirical geometry of multivariate data: a deconvolution approach

Let ⱼ : j = 1,…,n\ be independent observations in Rᵐ, m ≥ 1 with common distribution Q. Suppose that Yⱼ = Xⱼ + ξⱼ, j = 1,…,n, where ⱼ, ξⱼ, j = 1,…,n\ are independent, X_ j, j = 1,…,n have common distribution P and ξ_ j, j = 1, …,n have common distribution μ, so that Q = P * μ. The problem is to recover hidden geometric structure of the support of P based on the independent observations Yⱼ. Assuming that the distribution of the errors μ is known, deconvolution statistical estimates of the fractal dimension and the hierarchical cluster tree of the support that converge with exponential rates are suggested. Moreover, the exponential rates of convergence hold for adaptive versions of the estimates even in the case of normal noise ξ_ j with unknown covariance. In the case of the dimension estimation, though, the exponential rate holds only when the set of all possible values of the dimension is finite (e.g., when the dimension is known to be integer). If this set is infinite, the optimal convergence rate of the estimator becomes very slow (typically, logarithmic), even when there is no noise in the observations.