Breaking the curse of dimensionality in structured density estimation

TL;DR

This paper demonstrates that the curse of dimensionality in structured density estimation can be mitigated under Markov assumptions by introducing the concept of graph resilience, leading to improved sample complexity bounds.

Contribution

The paper introduces the novel concept of graph resilience to control sample complexity in Markov density estimation, extending results to arbitrary graphs.

Findings

Sample complexity can be significantly reduced under Markov assumptions.

Graph resilience controls the sample complexity independently of local graph parameters.

Explicit bounds show improved rates for hierarchical and spatial data.

Abstract

We consider the problem of estimating a structured multivariate density, subject to Markov conditions implied by an undirected graph. In the worst case, without Markovian assumptions, this problem suffers from the curse of dimensionality. Our main result shows how the curse of dimensionality can be avoided or greatly alleviated under the Markov property, and applies to arbitrary graphs. While existing results along these lines focus on sparsity or manifold assumptions, we introduce a new graphical quantity called "graph resilience" and show how it controls the sample complexity. Surprisingly, although one might expect the sample complexity of this problem to scale with local graph parameters such as the degree, this turns out not to be the case. Through explicit examples, we compute uniform deviation bounds and illustrate how the curse of dimensionality in density estimation can thus be circumvented. Notable examples where the rate improves substantially include sequential, hierarchical, and spatial data.