Fast rates for noisy clustering

TL;DR

This paper establishes fast convergence rates for noisy clustering by analyzing the impact of measurement errors and deriving bounds that depend on the regularity of the data density and the noise level.

Contribution

It introduces a general upper bound for empirical minimization with noisy data using deconvolution kernels and applies it to derive convergence rates in noisy clustering.

Findings

Achieves a convergence rate of (1/n^{rac{\u03b3}{ 7+ 7})}) depending on data regularity and noise.

Extends existing bounds to the noisy setting, showing how noise affects convergence rates.

Provides theoretical insights into the impact of measurement errors on clustering performance.

Abstract

The effect of errors in variables in empirical minimization is investigated. Given a loss $l$ and a set of decision rules $\mathcal{G}$, we prove a general upper bound for an empirical minimization based on a deconvolution kernel and a noisy sample $Z_i=X_i+\epsilon_i,i=1,...,n$. We apply this general upper bound to give the rate of convergence for the expected excess risk in noisy clustering. A recent bound from \citet{levrard} proves that this rate is $\mathcal{O}(1/n)$ in the direct case, under Pollard's regularity assumptions. Here the effect of noisy measurements gives a rate of the form $\mathcal{O}(1/n^{\frac{\gamma}{\gamma+2\beta}})$, where $\gamma$ is the H\"older regularity of the density of $X$ whereas $\beta$ is the degree of illposedness.