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The Information-Theoretic Requirements of Subspace Clustering with Missing Data
"... Abstract Subspace clustering with missing data (SCMD) is a useful tool for analyzing incomplete datasets. Let d be the ambient dimension, and r the dimension of the subspaces. Existing theory shows that N k = O(rd) columns per subspace are necessary for SCMD, and r+1 }) are sufficient. We close thi ..."
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Abstract Subspace clustering with missing data (SCMD) is a useful tool for analyzing incomplete datasets. Let d be the ambient dimension, and r the dimension of the subspaces. Existing theory shows that N k = O(rd) columns per subspace are necessary for SCMD, and r+1 }) are sufficient. We close this gap, showing that N k = O(rd) is also sufficient. To do this we derive deterministic sampling conditions for SCMD, which give precise information-theoretic requirements and determine sampling regimes. These results explain the performance of SCMD algorithms from the literature. Finally, we give a practical algorithm to certify the output of any SCMD method deterministically.
k-variates++: more pluses in the k-means++
, 2016
"... Abstract k-means++ seeding has become a de facto standard for hard clustering algorithms. In this paper, our first contribution is a two-way generalisation of this seeding, k-variates++, that includes the sampling of general densities rather than just a discrete set of Dirac densities anchored at t ..."
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Abstract k-means++ seeding has become a de facto standard for hard clustering algorithms. In this paper, our first contribution is a two-way generalisation of this seeding, k-variates++, that includes the sampling of general densities rather than just a discrete set of Dirac densities anchored at the point locations, and a generalisation of the well known Arthur-Vassilvitskii (AV) approximation guarantee, in the form of a bias+variance approximation bound of the global optimum. This approximation exhibits a reduced dependency on the "noise" component with respect to the optimal potential -actually approaching the statistical lower bound. We show that kvariates++ reduces to efficient (biased seeding) clustering algorithms tailored to specific frameworks; these include distributed, streaming and on-line clustering, with direct approximation results for these algorithms. Finally, we present a novel application of k-variates++ to differential privacy. For either the specific frameworks considered here, or for the differential privacy setting, there is little to no prior results on the direct application of k-means++ and its approximation bounds -state of the art contenders appear to be significantly more complex and / or display less favorable (approximation) properties. We stress that our algorithms can still be run in cases where there is no closed form solution for the population minimizer. We demonstrate the applicability of our analysis via experimental evaluation on several domains and settings, displaying competitive performances vs state of the art.
