Coresets and sketches for high dimensional subspace approximation problems

Dan Feldman, Morteza Monemizadeh, Christian Sohler, David P. Woodruff

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We consider the problem of approximating a set P of n points in ℝd by a j-dimensional subspace under the ℓp, measure, in which we wish to minimize the sum of ℓp, distances from each point of P to this subspace. More generally, the Fq (ℓp)-subspace approximation problem asks for a j-subspace that minimizes the sum of qth powers of ℓp-distances to this subspace, up to a multiplicative factor of (1 + ∈e). We develop techniques for subspace approximation, regression, and matrix approximation that can be used to deal with massive data sets in high dimensional spaces. In particular, we develop coresets and sketches, i.e. small space representations that approximate the input point set P with respect to the subspace approximation problem. Our results are: • A dimensionality reduction method that can be applied to Fq (ℓp)-clustering and shape fitting problems, such as those in [8, 15]. • The first strong coreset for F1 (ℓ2)- subspace approximation in high-dimensional spaces, i.e. of size polynomial in the dimension of the space. This coreset approximates the distances to any j-subspace (not just the optimal one). • A (1 + ∈)-approximation algorithm for the j-dimensional F1 (ℓ2)-subspace approximation problem with running time nd(j/∈)O(1) + (n + d)2poly(j/∈). • A streaming algorithm that maintains a coreset for the F1 (ℓ2)-subspace approximation problem and uses a space of d (2√log n/∈2)poly(j) (weighted) points. • Streaming algorithms for the above problems with bounded precision in the turnstile model, i.e, when coordinates appear in an arbitrary order and undergo multiple updates. We show that bounded precision can lead to further improvements. We extend results of [7] for approximate linear regression, distances to subspace approximation, and optimal rank-j approximation, to error measures other than the Frobenius norm.

Original languageEnglish
Title of host publicationProceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms
PublisherAssociation for Computing Machinery (ACM)
Number of pages20
ISBN (Print)9780898717013
StatePublished - 2010
Externally publishedYes
Event21st Annual ACM-SIAM Symposium on Discrete Algorithms - Austin, TX, United States
Duration: 17 Jan 201019 Jan 2010

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference21st Annual ACM-SIAM Symposium on Discrete Algorithms
Country/TerritoryUnited States
CityAustin, TX

ASJC Scopus subject areas

  • Software
  • Mathematics (all)


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