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In this paper, we propose an ensemble learning algorithm named \textit{bagged$k$-distance for mode-based clustering} (\textit{BDMBC}) by putting forward anew measurement called the \textit{probability of localized level sets}(\textit{PLLS}), which enables us to find all clusters for varying densitieswith a global threshold. On the theoretical side, we show that with a properlychosen number of nearest neighbors $k_D$ in the bagged $k$-distance, thesub-sample size $s$, the bagging rounds $B$, and the number of nearestneighbors $k_L$ for the localized level sets, BDMBC can achieve optimalconvergence rates for mode estimation. It turns out that with a relativelysmall $B$, the sub-sample size $s$ can be much smaller than the number oftraining data $n$ at each bagging round, and the number of nearest neighbors$k_D$ can be reduced simultaneously. Moreover, we establish optimal convergenceresults for the level set estimation of the PLLS in terms of Hausdorffdistance, which reveals that BDMBC can find localized level sets for varyingdensities and thus enjoys local adaptivity. On the practical side, we conductnumerical experiments to empirically verify the effectiveness of BDMBC for modeestimation and level set estimation, which demonstrates the promising accuracyand efficiency of our proposed algorithm.

Bagged $k$-Distance for Mode-Based Clustering Using the Probability of Localized Level Sets