arXiv subjects

Dena Marie Asta

Publications and source records attributed to Dena Marie Asta.

4 recordsLinked to original sources

arXiv2024

Lower Bounds for Kernel Density Estimation on Symmetric Spaces

We prove that kernel density estimation on symmetric spaces of non-compact type, whose L2-risk was bounded above in previous work (Asta,2021), in fact achieves a minimax rate of convergence. With this result, the story for kernel density estimation on all symmetric spaces is completed. The idea in adapting the proof for Euclidean space is to suitably abstract vector space operations on Euclidean space to both actions of symmetric groups and reparametrizations of Helgason-Fourier transforms and to use the fact that the exponential map for symmetric spaces of non-compact type defines a diffeomorphism.

math.ST↗

arXiv2023

Non-Parametric Manifold Learning

We introduce an estimator for distances in a compact Riemannian manifold based on graph Laplacian estimates of the Laplace-Beltrami operator. We upper bound the error in the estimate of manifold distances, or more precisely an estimate of a spectrally truncated variant of manifold distance of interest in non-commutative geometry (cf. [Connes and Suijelekom, 2020]), in terms of spectral errors in the graph Laplacian estimates and, implicitly, several geometric properties of the manifold. A consequence is a proof of consistency for (untruncated) manifold distances. The estimator resembles, and in fact its convergence properties are derived from, a special case of the Kontorovic dual reformulation of Wasserstein distance known as Connes' Distance Formula.

math.ST↗

arXiv2022

Kernel Density Estimation on Symmetric Spaces of Non-Compact Type

We construct a kernel density estimator on symmetric spaces of non-compact type and establish an upper bound for its convergence rate, analogous to the minimax rate for classical kernel density estimators on Euclidean space. Symmetric spaces of non-compact type include hyperboloids of constant negative curvature and spaces of symmetric positive definite matrices. This paper obtains a simplified formula in the special case when the symmetric space is the space of normal distributions, a 2-dimensional hyperboloid.

math.ST↗

arXiv2022

Bootstrapping Exchangeable Random Graphs

We introduce two new bootstraps for exchangeable random graphs. One, the "empirical graphon bootstrap", is based purely on resampling, while the other, the "histogram bootstrap", is a model-based "sieve" bootstrap. We show that both of them accurately approximate the sampling distributions of motif densities, i.e., of the normalized counts of the number of times fixed subgraphs appear in the network. These densities characterize the distribution of (infinite) exchangeable networks. Our bootstraps therefore give a valid quantification of uncertainty in inferences about fundamental network statistics, and so of parameters identifiable from them.

stat.ME↗