Soledad Villar

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We study exact recovery conditions for convex relaxations of point cloud clustering problems, focusing on two of the most common optimization problems for unsupervised clustering: <i>k</i>-means and <i>k</i>-median clustering. Motivations for focusing on convex relaxations are: (a) they come with a certificate of optimality, and (b) they are generic tools(More)
We introduce a model-free relax-and-round algorithm for k-means clustering based on a semidefinite relaxation due to Peng and Wei [PW07]. The algorithm interprets the SDP output as a denoised version of the original data and then rounds this output to a hard clustering. We provide a generic method for proving performance guarantees for this algorithm, and(More)
Recently, Bandeira [5] introduced a new type of algorithm (the so-called probably certifiably correct algorithm) that combines fast solvers with the optimality certificates provided by convex relaxations. In this paper, we devise such an algorithm for the problem of k-means clustering. First, we prove that Peng and Wei’s semidefinite relaxation of k-means(More)
Recently, [3] introduced an SDP relaxation of the k-means problem in R. In this work, we consider a random model for the data points in which k balls of unit radius are deterministically distributed throughout R, and then in each ball, n points are drawn according to a common rotationally invariant probability distribution. For any fixed ball configuration(More)
The Gromov-Hausdorff distance provides a metric on the set of isometry classes of compact metric spaces. Unfortunately, computing this metric directly is believed to be computationally intractable. Motivated by applications in shape matching and point-cloud comparison, we study a semidefinite programming relaxation of the Gromov-Hausdorff metric. This(More)
We introduce a model-free, parameter-free relax-and-round algorithm for k-means clustering, based on a semidefinite programming relaxation (SDP) due to Peng and Wei [1]. The algorithm interprets the SDP output as a denoised version of the original data and then rounds this output to a hard clustering. We analyze the performance of this algorithm in the(More)
Many inverse problems are formulated as optimization problems over certain appropriate input distributions. Recently, there has been a growing interest in understanding the computational hardness of these optimization problems, not only in the worst case, but in an average-complexity sense under this same input distribution. In this note, we are interested(More)
We introduce a manifold optimization relaxation for k-means clustering that generalizes spectral clustering. We show how to implement it as gradient descent in a compact manifold. We also present numerical simulations of the algorithm using Manopt [5]. An extended version of this article, with further theory and numerical simulations will be available as(More)
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