A PTAS for k-means clustering based on weak coresets
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Bi-criteria linear-time approximations for generalized k-mean/median/center
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Sampling-based dimension reduction for subspace approximation
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Efficient subspace approximation algorithms
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Sampling algorithms and coresets for ℓp regression
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Facility Location in Dynamic Geometric Data Streams
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Proceedings of the forty-first annual ACM symposium on Theory of computing
Coresets and sketches for high dimensional subspace approximation problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
A unified framework for approximating and clustering data
Proceedings of the forty-third annual ACM symposium on Theory of computing
Near-optimal private approximation protocols via a black box transformation
Proceedings of the forty-third annual ACM symposium on Theory of computing
A near-linear algorithm for projective clustering integer points
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Data reduction for weighted and outlier-resistant clustering
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Learning Big (Image) Data via Coresets for Dictionaries
Journal of Mathematical Imaging and Vision
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We develop efficient (1 + \varepsilon)-approximation algorithms for generalized facility location problems. Such facilities are not restricted to being points in \mathbb{R}^d, and can represent more complex structures such as linear facilities (lines in \mathbb{R}^d, j-dimensional flats), etc. We introduce coresets for weighted (point) facilities. These prove to be useful for such generalized facility location problems, and provide efficient algorithms for their construction. Applications include: k-mean and k-median generalizations, i.e., find k lines that minimize the sum (or sum of squares) of the distances from each input point to its nearest line. Other applications are generalizations of linear regression problems to multiple regression lines, new SVD/PCA generalizations, and many more. The results significantly improve on previous work, which deals efficiently only with special cases. Open source code for the algorithms in this paper is also available.