Given a data set of size n in d'-dimensional Euclidean space, the k-means problem asks for a set of k points (called centers) such that the sum of the l_2^2-distances between the data points and the set of centers is minimized. Previous work on this problem in the local differential privacy setting shows how to achieve multiplicative approximation factors arbitrarily close to optimal, but suffers high additive error. The additive error has also been seen to be an issue in implementations of differentially private k-means clustering algorithms in both the central and local settings. In this work, we introduce a new locally private k-means clustering algorithm that achieves near-optimal additive error whilst retaining constant multiplicative approximation factors and round complexity. Concretely, given any c>sqrt(2), our algorithm achieves O(k^(1 + O(1/(2c^2-1))) * sqrt(d' n) * log d' * poly log n) additive error with an O(c^2) multiplicative approximation factor.
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Fast Optimal Circular Clustering and Applications on Round Genomes
Round genomes are found in bacteria, plant chloroplasts, and mitochondria. Genetic or epigenetic marks can present biologically interesting clusters along a circular genome. The circular data clustering problem groups N points on a circle into K clusters to minimize the within-cluster sum of squared distances. Repeatedly applying the K-means algorithm takes quadratic time, impractical for large circular datasets. To overcome this issue, we developed a fast, reproducible, and optimal circular clustering (FOCC) algorithm of worst-case O(KN log^2 N) time. The core is a fast optimal framed clustering algorithm, which we designed by integrating two divide-and-conquer and one bracket dynamic programming strategies. The algorithm is optimal based on a property of monotonic increasing cluster borders over frames on linearized data. On clustering 50,000 circular data points, FOCC outruns brute-force or heuristic circular clustering by three orders of magnitude. We produced clusters of CpG sites and genes along three round genomes, exhibiting higher quality than heuristic clustering. More broadly, the presented subquadratic-time algorithms offer the fastest known solution to not only framed and circular clustering, but also angular, periodical, and looped clustering. We implemented these algorithms in the R package OptCirClust (https://CRAN.R-project.org/package=OptCirClust)
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- Award ID(s):
- 1661331
- NSF-PAR ID:
- 10236465
- Date Published:
- Journal Name:
- IEEE/ACM Transactions on Computational Biology and Bioinformatics
- ISSN:
- 1545-5963
- Page Range / eLocation ID:
- 1 to 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TCBB.2021.3077573
