Fast rates for noisy clustering

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 $Z_i=X_i+ε_i,i=1,...,n$. 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 $\mathcal{O}(1/n^{\fracγ{γ+2β}})$, where $γ$ is the Hölder regularity of the density of $X$ whereas $β$ is the degree of illposedness.

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