Recovering Trees with Convex Clustering
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Convex clustering refers, for given {x_1, ..., x_n}⊂^p, to the minimization of u(γ) & = & u_1, ..., u_n ∑_i=1^n x_i - u_i ^2 + γ∑_i,j=1^nw_ij u_i - u_j, where w_ij≥ 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 {x_1, ..., x_n}, then the convex clustering solution path reconstructs the tree exactly. The main technical ingredient implies the following combinatorial byproduct: for every set {x_1, ..., x_n }⊂^p of n ≥ 2 distinct points, there exist at least n/6 points with the property that for any of these points x there is a unit vector v ∈^p such that, when viewed from x, `most' points lie in the direction v 1/n-1∑_i=1 x_i ≠ x^n〈x_i - x/ x_i - x , v 〉 & ≥ & 1/4.
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