Computing a set of points meeting every cell defined by a family of polynomials on a variety
WAFR Proceedings of the workshop on Algorithmic foundations of robotics
Automatic subspace clustering of high dimensional data for data mining applications
SIGMOD '98 Proceedings of the 1998 ACM SIGMOD international conference on Management of data
Fast algorithms for projected clustering
SIGMOD '99 Proceedings of the 1999 ACM SIGMOD international conference on Management of data
Entropy-based subspace clustering for mining numerical data
KDD '99 Proceedings of the fifth ACM SIGKDD international conference on Knowledge discovery and data mining
Finding generalized projected clusters in high dimensional spaces
SIGMOD '00 Proceedings of the 2000 ACM SIGMOD international conference on Management of data
Approximate clustering via core-sets
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
A Monte Carlo algorithm for fast projective clustering
Proceedings of the 2002 ACM SIGMOD international conference on Management of data
Approximation Algorithms for k-Line Center
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
On coresets for k-means and k-median clustering
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Approximating extent measures of points
Journal of the ACM (JACM)
High-Dimensional Shape Fitting in Linear Time
Discrete & Computational Geometry
PODS '04 Proceedings of the twenty-third ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Subgradient and sampling algorithms for l1 regression
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Coresets forWeighted Facilities and Their Applications
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
How to get close to the median shape
Computational Geometry: Theory and Applications - Special issue on the 21st European workshop on computational geometry (EWCG 2005)
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
Robust Shape Fitting via Peeling and Grating Coresets
Discrete & Computational Geometry
Sampling Algorithms and Coresets for $\ell_p$ Regression
SIAM Journal on Computing
Linear-time approximation schemes for clustering problems in any dimensions
Journal of the ACM (JACM)
Universal ε-approximators for integrals
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Coresets and sketches for high dimensional subspace approximation problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Geometric optimization and sums of algebraic functions
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
Coresets for discrete integration and clustering
FSTTCS'06 Proceedings of the 26th international conference on Foundations of Software Technology and Theoretical Computer Science
FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
Proceedings of the twenty-ninth annual symposium on Computational geometry
Learning Big (Image) Data via Coresets for Dictionaries
Journal of Mathematical Imaging and Vision
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We consider the problem of projective clustering in Euclidean spaces of non-fixed dimension. Here, we are given a set P of n points in Rm and integers j ≥ 1, k ≥ 0, and the goal is to find j k-subspaces so that the sum of the distances of each point in P to the nearest subspace is minimized. Observe that this is a shape fitting problem where we wish to find the best fit in the L1 sense. Here we will treat the number j of subspaces we want to fit and the dimension k of each of them as constants. We consider instances of projective clustering where the point coordinates are integers of magnitude polynomial in m and n. Our main result is a randomized algorithm that for any ε 0 runs in time O(mn polylog(mn)) and outputs a solution that with high probability is within (1 + ε) of the optimal solution. To obtain this result, we show that the fixed dimensional version of the above projective clustering problem has a small coreset. We do that by observing that in a fairly general sense, shape fitting problems that have small coresets in the L∞ setting also have small coresets in the L1 setting, and then exploiting an existing construction for the L∞ setting. This observation seems to be quite useful for other shape fitting problems as well, as we demonstrate by constructing the first "regular" coreset for the circle fitting problem in the plane.