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Chi, Stefan Steinerberger","submitted_at":"2018-06-28T17:36:23Z","abstract_excerpt":"Convex clustering refers, for given $\\left\\{x_1, \\dots, x_n\\right\\} \\subset \\mathbb{R}^p$, to the minimization of \\begin{eqnarray*} u(\\gamma) & = & \\underset{u_1, \\dots, u_n }{\\arg\\min}\\;\\sum_{i=1}^{n}{\\lVert x_i - u_i \\rVert^2} + \\gamma \\sum_{i,j=1}^{n}{w_{ij} \\lVert u_i - u_j\\rVert},\\\\ \\end{eqnarray*} where $w_{ij} \\geq 0$ is an affinity that quantifies the similarity between $x_i$ and $x_j$. We prove that if the affinities $w_{ij}$ reflect a tree structure in the $\\left\\{x_1, \\dots, x_n\\right\\}$, then the convex clustering solution path reconstructs the tree exactly. 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