Nearly optimal linear embeddings into very low dimensions
Name
Nearly optimal.pdf
Size
246.49 KB
Format
Adobe PDF
Checksum (MD5)
a2ac454696df3ce2cb9725ce506efcb2
Author(s)
Grant, Elyot
Hegde, Chinmay
Indyk, Piotr
Date Issued
February 2014
Journal
2013 IEEE Global Conference on Signal and Information Processing
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Grant, Elyot, Chinmay Hegde, and Piotr Indyk. "Nearly Optimal Linear Embeddings into Very Low Dimensions." 3-5 Dec. 2013, Austin, Texas, IEEE, 2013, pp. 973–76.
Version
Author's final manuscript
Abstract
We propose algorithms for constructing linear embeddings of a finite dataset V ⊂ ℝ[superscript d] into a k-dimensional subspace with provable, nearly optimal distortions. First, we propose an exhaustive-search-based algorithm that yields a k-dimensional linear embedding with distortion at most ε[subscript opt](k)+δ, for any δ > 0 where ε[subscript opt](k) is the smallest achievable distortion over all possible orthonormal embeddings. This algorithm is space-efficient and can be achieved by a single pass over the data V. However, the runtime of this algorithm is exponential in k. Second, we propose a convex-programming-based algorithm that yields an O(k/δ)-dimensional orthonormal embedding with distortion at most (1 + δ)ε[subscript opt](k). The runtime of this algorithm is polynomial in d and independent of k. Several experiments demonstrate the benefits of our approach over conventional linear embedding techniques, such as principal components analysis (PCA) or random projections.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1109/GlobalSIP.2013.6737055
