Sampling-based algorithms for dimension reduction

Can one compute a low-dimensional representation of any given data by looking only at its small sample, chosen cleverly on the fly? Motivated by the above question, we consider the problem of low-rank matrix approximation: given a matrix A..., one wants to compute a rank-k matrix (where k << min{m, n}) nearest to A in the Frobenius norm (also known as the Hilbert-Schmidt norm). We prove that using a sample of roughly O(k/[epsilon]) rows of A one can compute, with high probability, a (1 + [epsilon])-approximation to the nearest rank-k matrix. This gives an algorithm for low-rank approximation with an improved error guarantee (compared to the additive [epsilon]... guarantee known earlier from the work of Frieze, Kannan, and Vempala) and running time O(Mk/[epsilon]), where M is the number of non-zero entries of A. The proof is based on two sampling techniques called adaptive sampling and volume sampling, and some linear algebraic tools. Low-rank matrix approximation under the Frobenius norm is equivalent to the problem of finding a low-dimensional subspace that minimizes the sum of squared distances to given points. The general subspace approximation problem asks one to find a low-dimensional subspace that minimizes the sum of p-th powers of distances (for p > 1) to given points. We generalize our sampling techniques and prove similar sampling-based dimension reduction results for subspace approximation. However, the proof is geometric.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2007

Author dc:creator

• Deshpande, Amit Jayant

Advisor dc:contributor.advisor

• Santosh S. Vempala and Daniel A. Spielman.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/38935

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/38935