Near-isometric linear embeddings of manifolds

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

doi:10.1109/SSP.2012.6319806

Near-isometric linear embeddings of manifolds

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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 language	English (US)
Title of host publication	2012 IEEE Statistical Signal Processing Workshop, SSP 2012
Pages	728-731
Number of pages	4
DOIs	https://doi.org/10.1109/SSP.2012.6319806
State	Published - 2012
Event	2012 IEEE Statistical Signal Processing Workshop, SSP 2012 - Ann Arbor, MI, United States Duration: Aug 5 2012 → Aug 8 2012

Publication series

Name	2012 IEEE Statistical Signal Processing Workshop, SSP 2012

Other

Other	2012 IEEE Statistical Signal Processing Workshop, SSP 2012
Country/Territory	United States
City	Ann Arbor, MI
Period	8/5/12 → 8/8/12

Keywords

Adaptive sampling
Linear Dimensionality Reduction
Whitney's Theorem

ASJC Scopus subject areas

Signal Processing

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10.1109/SSP.2012.6319806

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Hegde, C, Sankaranarayanan, AC & Baraniuk, RG 2012, Near-isometric linear embeddings of manifolds. in 2012 IEEE Statistical Signal Processing Workshop, SSP 2012., 6319806, 2012 IEEE Statistical Signal Processing Workshop, SSP 2012, pp. 728-731, 2012 IEEE Statistical Signal Processing Workshop, SSP 2012, Ann Arbor, MI, United States, 8/5/12. https://doi.org/10.1109/SSP.2012.6319806

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keywords = "Adaptive sampling, Linear Dimensionality Reduction, Whitney's Theorem",

author = "Chinmay Hegde and Sankaranarayanan, {Aswin C.} and Baraniuk, {Richard G.}",

year = "2012",

doi = "10.1109/SSP.2012.6319806",

language = "English (US)",

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series = "2012 IEEE Statistical Signal Processing Workshop, SSP 2012",

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AU - Hegde, Chinmay

AU - Sankaranarayanan, Aswin C.

AU - Baraniuk, Richard G.

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N2 - 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.

AB - 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.

KW - Adaptive sampling

KW - Linear Dimensionality Reduction

KW - Whitney's Theorem

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BT - 2012 IEEE Statistical Signal Processing Workshop, SSP 2012

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ER -

Near-isometric linear embeddings of manifolds

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Keywords

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