Density Estimation via Discrepancy

TL;DR

This paper introduces a non-parametric density estimation method on hyper-rectangles using adaptive discrepancy control, enabling effective pattern recognition tasks like mode seeking and density exploration.

Contribution

It proposes a novel piecewise constant density estimator based on binary splits and discrepancy criteria, with theoretical guarantees and practical applications.

Findings

Estimator preserves most of the estimation power.

Method is applicable to mode seeking and density landscape exploration.

Demonstrated effectiveness through simulations and examples.

Abstract

Given i.i.d samples from some unknown continuous density on hyper-rectangle $[0, 1]^d$, we attempt to learn a piecewise constant function that approximates this underlying density non-parametrically. Our density estimate is defined on a binary split of $[0, 1]^d$ and built up sequentially according to discrepancy criteria; the key ingredient is to control the discrepancy adaptively in each sub-rectangle to achieve overall bound. We prove that the estimate, even though simple as it appears, preserves most of the estimation power. By exploiting its structure, it can be directly applied to some important pattern recognition tasks such as mode seeking and density landscape exploration. We demonstrate its applicability through simulations and examples.