Near-isometric linear embeddings of manifolds

Chinmay Hegde, Aswin C. Sankaranarayanan, Richard G. Baraniuk

Research output: Chapter in Book/Report/Conference proceedingConference contribution

5 Scopus citations

Abstract

We propose a new method for linear dimensionality reduction of manifold-modeled data. Given a training set X of Q points belonging to a manifold M ⊂ ℝ N, we construct a linear operator P : ℝ N → ℝ M that approximately preserves the norms of all (Q2) - pairwise difference vectors (or secants) of X. We design the matrix P via a trace-norm minimization that can be efficiently solved as a semi-definite program (SDP). When X comprises a sufficiently dense sampling of M, we prove that the optimal matrix P preserves all pairs of secants over M. We numerically demonstrate the considerable gains using our SDP-based approach over existing linear dimensionality reduction methods, such as principal components analysis (PCA) and random projections.

Original languageEnglish (US)
Title of host publication2012 IEEE Statistical Signal Processing Workshop, SSP 2012
Pages728-731
Number of pages4
DOIs
StatePublished - Nov 6 2012
Event2012 IEEE Statistical Signal Processing Workshop, SSP 2012 - Ann Arbor, MI, United States
Duration: Aug 5 2012Aug 8 2012

Other

Other2012 IEEE Statistical Signal Processing Workshop, SSP 2012
CountryUnited States
CityAnn Arbor, MI
Period8/5/128/8/12

Keywords

  • Adaptive sampling
  • Linear Dimensionality Reduction
  • Whitney's Theorem

ASJC Scopus subject areas

  • Signal Processing

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