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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)

- Takayuki Iguchi, Dustin G. Mixon, Jesse Peterson, Soledad Villar
- ArXiv
- 2015

Recently, [3] introduced an SDP relaxation of the k-means problem in R m. In this work, we consider a random model for the data points in which k balls of unit radius are deterministically distributed throughout R m , and then in each ball, n points are drawn according to a common rotationally invariant probability distribution. For any fixed ball… (More)

- Dustin G. Mixon, Soledad Villar, Rachel Ward
- ArXiv
- 2016

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)

- Takayuki Iguchi, Dustin G. Mixon, Jesse Peterson, Soledad Villar
- ArXiv
- 2015

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)

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)

- Alex Nowak, Soledad Villar, Afonso S. Bandeira, Joan Bruna
- ArXiv
- 2017

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)

- L. A. Cardona, S. de la Fe, B. Lorente, S. Villar, C. Ferrer
- 2013 47th International Carnahan Conference on…
- 2013

Wireless Sensor Networks present the challenge of including robust security mechanisms in resource-constrained and low-power devices. Cryptography is at the heart of security and key management is a fundamental part of it. Adequate analysis and selection of efficient cryptographic algorithms and key management schemes are required. This paper presents a key… (More)

- Dustin G. Mixon, Soledad Villar, Rachel Ward
- 2016 IEEE Information Theory Workshop (ITW)
- 2016

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)

- Efe Onaran, Soledad Villar
- ArXiv
- 2017

The network alignment problem asks for the best correspondence between two given graphs, so that the largest possible number of edges are matched. This problem appears in many scientific problems (like the study of protein-protein interactions) and it is very closely related to the quadratic assignment problem which has graph isomorphism, traveling salesman… (More)

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