### 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) |
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Title of host publication | 2012 IEEE Statistical Signal Processing Workshop, SSP 2012 |

Pages | 728-731 |

Number of pages | 4 |

DOIs | |

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 |
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### Other

Other | 2012 IEEE Statistical Signal Processing Workshop, SSP 2012 |
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Country | 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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## Cite this

*2012 IEEE Statistical Signal Processing Workshop, SSP 2012*(pp. 728-731). [6319806] (2012 IEEE Statistical Signal Processing Workshop, SSP 2012). https://doi.org/10.1109/SSP.2012.6319806