# Nonlinear Dimension Reduction via Local Tangent Space Alignment

@inproceedings{Zhang2003NonlinearDR, title={Nonlinear Dimension Reduction via Local Tangent Space Alignment}, author={Zhenyue Zhang and Hongyuan Zha}, booktitle={IDEAL}, year={2003} }

In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction. Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry of the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood of each data point. Those tangent spaces are aligned to give the internal global coordinates of the data points with respect to the underlying manifold by way of a partial eigendecomposition of…

## 656 Citations

Tangent Bundle Manifold Learning via Grassmann&Stiefel Eigenmaps

- Computer Science, MathematicsArXiv
- 2012

This work proposes an amplification of the ML, called Tangent Bundle ML, in which the proximity not only between the original manifold and its estimator but also between their tangent spaces is required.

Robust Local Tangent Space Alignment

- Computer ScienceICONIP
- 2009

A robust version of LTSA called RLTSA is proposed, which makes LTSA more robust from three aspects: robust PCA algorithm is used instead of the standard SVD to reduce influence of noise on local tangent space coordinates, and R LTSA chooses neighborhoods that are approximated well by the local tangENT space coordinates to align with the global coordinates.

Active Neighborhood Selection for Locally Linear Embedding

- Computer Science2009 Second International Symposium on Knowledge Acquisition and Modeling
- 2009

Experimental results demonstrate that metric LLE usually performs better than LLE in feature extraction, and the strategy of active neighborhood selection to extend LLE is made use.

Neighborhood smoothing embedding for noisy manifold learning

- Computer Science2008 IEEE International Conference on Granular Computing
- 2008

This paper proposes neighbor smoothing embedding (NSE) for noisy points sampled from a nonlinear manifold based on LLE and local linear surface estimator, which smoothes the neighbors of each sample data point and then computes the reconstruction matrix of the projections on the estimated surface.

LOCAL LINEAR SMOOTHING FOR NONLINEAR MANIFOLD LEARNING

- Mathematics
- 2003

In this paper, we develop methods for outlier removal and noise reduction based on weighted local linear smoothing for a set of noisy points sampled from a nonlinear manifold. The methods can be used…

Matrix perturbation analysis of local tangent space alignment

- Computer Science
- 2009

Analysis of an alignment algorithm for nonlinear dimensionality reduction

- Computer Science
- 2007

An analysis of the alignment process is presented, giving conditions under which the null space of the aligned matrix recovers the global coordinate system up to an affine transformation, and it is shown that Local Tangent Space Alignment method (LTSA) can recover a locally isometric embedding up to a rigid motion.

Data-based Manifold Reconstruction via Tangent Bundle Manifold Learning

- Mathematics, Computer Science
- 2014

A new geometrically motivated method for the TBML problem in which the manifold, its tangent spaces and lowdimensional representation accurately reconstructed from a sample is presented.

Iterative Hyperplane Merging: A Framework for Manifold Learning

- Computer ScienceBMVC
- 2010

A Minimum Spanning Tree provides the skeleton needed to traverse the manifold so that the local hyperplanes can be used to build a global, locally stable, embedding.

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