Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Recently, Bandeira (C R Math, 2015) 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. ,  devise such an algorithm for the problem of k-means clustering. First,  prove that Peng and i’s semidefinite relaxation of k-means Peng and i (SIAM J Optim 18(1):186–205, 2007) is tight with high probability under a distribution of planted clusters called the stochastic ball model. Our proof follows from a new dual certificate for integral solutions of this semidefinite program. Next,  show how to test the optimality of a proposed k-means solution using this dual certificate in quasilinear time. Finally,  analyze a version of spectral clustering from Peng and i (SIAM J Optim 18(1):186–205, 2007) that is designed to solve k-means in the case of two clusters. In particular,  show that this quasilinear-time method typically recovers planted clusters under the stochastic ball model.
['Takayuki Iguchi', 'Dustin G. Mixon', 'Jesse Peterson', 'Soledad Villar']
I am writing a research paper on a idea Probably certifiably correct k-means clustering
[{"from": "human", "value": "I am writing a research paper on a idea Probably certifiably correct k-means clustering"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Recently, Bandeira (C R Math, 2015) 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. ,  devise such an algorithm for the problem of k-means clustering. First,  prove that Peng and i\u2019s semidefinite relaxation of k-means Peng and i (SIAM J Optim 18(1):186\u2013205, 2007) is tight with high probability under a distribution of planted clusters called the stochastic ball model. Our proof follows from a new dual certificate for integral solutions of this semidefinite program. Next,  show how to test the optimality of a proposed k-means solution using this dual certificate in quasilinear time. Finally,  analyze a version of spectral clustering from Peng and i (SIAM J Optim 18(1):186\u2013205, 2007) that is designed to solve k-means in the case of two clusters. In particular,  show that this quasilinear-time method typically recovers planted clusters under the stochastic ball model."}]