Nearly optimal linear embeddings into very low dimensions
Author(s)Grant, Elyot; Hegde, Chinmay; Indyk, Piotr
MetadataShow full item record
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.
DepartmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
2013 IEEE Global Conference on Signal and Information Processing
Institute of Electrical and Electronics Engineers (IEEE)
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.
Author's final manuscript