The mean shift (MS) algorithm seeks a mode of the kernel density estimate (KDE). This study presents a convergence guarantee of the mode estimate sequence generated by the MS algorithm and an evaluation of the convergence rate, under fairly mild conditions, with the help of the argument concerning the {\L}ojasiewicz inequality. Our findings, which extend existing ones covering analytic kernels and the Epanechnikov kernel, are significant in that they cover the biweight kernel that is optimal among non-negative kernels in terms of the asymptotic statistical efficiency for the KDE-based mode estimation.

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