Academic

In this talk, I will present a unified theoretical framework for clustering methods based on hill-climbing strategies, originally introduced by Fukunaga and Hostetler in the 1970s. These methods assign data points to density modes by following gradient ascent paths, offering an intuitive and powerful approach to nonparametric clustering. I will discuss several algorithmic variants—including Euler Shift, Mean Shift, Max Shift, and Max Slope Shift—and show how they can be rigorously analyzed for consistency, both in idealized settings and when applied to estimated densities from data. The results establish conditions under which these algorithms reliably recover the underlying cluster structure as tuning parameters shrink and sample sizes grow. This work contributes to the mathematical foundation of modal clustering and highlights its robustness and versatility in statistical learning.