Tangent space estimation for smooth embeddings of Riemannian manifolds

Download

index.pdf

Date

2013-06-01

Author

Tyagi, Hemant
Vural, Elif
Frossard, Pascal

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

49
views

0
downloads

Numerous dimensionality reduction problems in data analysis involve the recovery of low-dimensional models or the learning of manifolds underlying sets of data. Many manifold learning methods require the estimation of the tangent space of the manifold at a point from locally available data samples. Local sampling conditions such as (i) the size of the neighborhood (sampling width) and (ii) the number of samples in the neighborhood (sampling density) affect the performance of learning algorithms. In this work, we propose a theoretical analysis of local sampling conditions for the estimation of the tangent space at a point P lying on an m-dimensional Riemannian manifold S in ℝn. Assuming a smooth embedding of S in ℝn, we estimate the tangent space TPS by performing a principal component analysis (PCA) on points sampled from the neighborhood of P on S. Our analysis explicitly takes into account the second-order properties of the manifold at P, namely the principal curvatures as well as the higher-order terms. We consider a random sampling framework and leverage recent results from random matrix theory to derive conditions on the sampling width and the local sampling density for an accurate estimation of tangent subspaces. We measure the estimation accuracy by the angle between the estimated tangent space T̂PS and the true tangent space TPS and we give conditions for this angle to be bounded with high probability. In particular, we observe that the local sampling conditions are highly dependent on the correlation between the components in the second-order local approximation of the manifold. We finally provide numerical simulations to validate our theoretical findings.

URI

https://hdl.handle.net/11511/42184

Journal

Information and Inference: A Journal of the IMA

DOI

https://doi.org/10.1093/imaiai/iat003

Collections

Department of Electrical and Electronics Engineering, Article

Suggestions

OpenMETU
Core

Distance-based discretization of parametric signal manifolds Vural, Elif (2010-06-28) The characterization of signals and images in manifolds often lead to efficient dimensionality reduction algorithms based on manifold distance computation for analysis or classification tasks. We propose in this paper a method for the discretization of signal manifolds given in a parametric form. We present an iterative algorithm for the selection of samples on the manifold that permits to minimize the average error in the manifold distance computation. Experimental results with image appearance manifolds d...
Nonlinear supervised dimensionality reduction via smooth regular embeddings Ornek, Cem; Vural, Elif (2019-03-01) The recovery of the intrinsic geometric structures of data collections is an important problem in data analysis. Supervised extensions of several manifold learning approaches have been proposed in the recent years. Meanwhile, existing methods primarily focus on the embedding of the training data, and the generalization of the embedding to initially unseen test data is rather ignored. In this work, we build on recent theoretical results on the generalization performance of supervised manifold learning algori...
Nonlinear supervised dimensionality reduction via smooth regular embeddings Örnek, Cem; Schmidt, Şenan Ece; Department of Electrical and Electronics Engineering (2018) The recovery of the intrinsic geometric structures of data collections is an important problem in data analysis. Supervised extensions of several manifold learning approaches have been proposed in the recent years. Meanwhile, existing methods primarily focus on the embedding of the training data, and the generalization of the embedding to initially unseen test data is rather ignored. In this work, we build on recent theoretical results on the generalization performance of supervised manifold learning algori...
Derivative free algorithms for large scale non-smooth optimization and their applications Tor, Ali Hakan; Karasözen, Bülent; Department of Mathematics (2013) In this thesis, various numerical methods are developed to solve nonsmooth and in particular, nonconvex optimization problems. More speciﬁcally, three numerical algorithms are developed for solving nonsmooth convex optimization problems and one algorithm is proposed to solve nonsmooth nonconvex optimization problems. In general, main diﬀerences between algorithms of smooth optimization are in the calculation of search directions, line searches for ﬁnding step-sizes and stopping criteria. However, in nonsmoo...
Clustering of manifold-modeled data based on tangent space variations Gökdoğan, Gökhan; Vural, Elif; Department of Electrical and Electronics Engineering (2017) An important research topic of the recent years has been to understand and analyze data collections for clustering and classification applications. In many data analysis problems, the data sets at hand have an intrinsically low-dimensional structure and admit a manifold model. Most state-of-the-art clustering methods developed for data of non-linear and low-dimensional structure are based on local linearity assumptions. However, clustering algorithms based on locally linear representations can tolerate diff...

Citation Formats

H. Tyagi, E. Vural, and P. Frossard, “Tangent space estimation for smooth embeddings of Riemannian manifolds,” Information and Inference: A Journal of the IMA, pp. 69–114, 2013, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/42184.