A nonlinear dimensionality reduction framework using smooth geodesics.

Herein, we propose a framework for nonlinear dimensionality reduction that generates a manifold in terms of smooth geodesics that is designed to treat problems in which manifold measurements are either ... Then, we fit points in each geodesic by a smoothing spline to emphasize the smoothness. ... With this idea in mind, herein we introduced a novel nonlinear dimensionality reduction framework using smooth geodesics that emphasizes the underlying smoothness of the manifold. ...

arXiv:1707.06757v2 fatcat:xghxz4mrtngspjigvcev7o3spm

Multiple Versions

Dimensionality reduction techniques are designed to exploit the fact that most high-dimensional datasets from the real world do not uniformly fill the hyperspaces in which they are represented but instead ... their distributions usually concentrate to nonlinear manifolds of lower intrinsic dimensions. ... CONCLUSIONS We have introduced a framework for an efficient intrinsic dimensionality estimation and dimensionality reduction. ...

doi:10.1109/icdsp.2009.5201196 fatcat:7ksgzjaq3bhrpo7he7c2io3pxm

The idea of manifold learning using geodesic distance can fully consider the nonlinear structure of the original dataset. In this paper, comparative learning is the main framework. ... Classical dimensionality reduction algorithms are mainly divided into linear dimensionality reduction algorithms and nonlinear dimensionality reduction algorithms. ... With the development of the machine learning, more data dimensionality reduction algorithms will be applied to these fields to optimise the learning process and improve the performance of intelligent systems ...

doi:10.56028/aetr.9.1.522.2024 fatcat:7rqh32xgubgadnt7eopelv6vai

In addition to being efficient for the smoothing of curves, it is a novel scheme for computing geodesics in shape space. ... The scheme can be used for smoothing and noise removal, e.g., for reducing jittering artifacts in motion capture data. We introduce a reduced-order method for the computation of the flow. ... Dimensional reduction For the dimensional reduction, we are restricting the variations of every shape to a low-dimensional affine subspace of R n . ...

doi:10.1111/cgf.12832 fatcat:xmzdr2fb2za7pftsb57xa2rcwq

We describe a new approach for elucidating the nonlinear degrees of freedom in a distribution of shapes depicted in digital images. ... The novel approach takes into account the nonlinear nature of shape manifolds and is related to the ISOMAP algorithm. ... [9] who propose a method for principal geodesic analysis for the study of nonlinear statistics of shape. ...

doi:10.1109/isbi.2008.4541042 dblp:conf/isbi/RohdeWPM08 fatcat:exempm7dlje5dgnlnkbm2jimke

We present this method as an application to signal classification in a dimensionality reduction framework. ... A computational experiment is presented indicating the potential and shortcomings of this framework. ... embedding: Use the geodesic distance in a MDS algorithm for computing a d-dimensional embedding. ...

doi:10.1007/s00006-015-0608-z fatcat:bng3jble45fkjo4zsfiaqogoaa

Szczepanski

It is based on extending the classical multidimensional scaling method for dimension reduction, replacing pairwise Euclidean distances by the geodesic distances on the manifold. ... A continuous version of Isomap called continuum Isomap is proposed. Manifold learning in the continuous framework is then reduced to an eigenvalue problem of an integral operator. ... Coifman for a introduction and discussion of bi-Lipschitz mappings. ...

doi:10.1016/j.csda.2006.11.027 fatcat:havlb3yvfjcz5axf46yi7h4bvq

In particular, we establish that the optical flow forms a low-dimensional smooth manifold. ... The core premise is that we can view a collection of images, each of which is indexed by a small number of degrees of freedom (3D camera pose, motion/deformation, etc.), as a low-dimensional nonlinear ... . , e M } ∈ E ≡ R K using a dimensionality reduction tool such as (LLE [12] , ISOMAP [18] , etc. ...

doi:10.1109/allerton.2011.6120390 dblp:conf/allerton/SankaranarayananHNB11 fatcat:7yndldtnbfea7orpytxzdghm4a

C., [2000], “A global geometric framework for nonlinear dimensionality reduction,” Science, 260:2319 - 2323. • Tikhonov, A. N., Vasileva, A. B., Sveshnikov, A. ... uniform regridding using a stiff bivariate cubic smoothing spline. ...

doi:10.1142/s021812740701777x fatcat:pov4iyu3jzem7ku4dzhtrdlcl4

The shape filter is realized using a square-root velocity to project the shapes on a shape manifold in which distances between shapes are based on elastic changes. ... Two data-driven weighting methods are proposed to achieve a trade-off between shape smoothness and consistency with the data. ... Fig. 4 : Two-dimensional visualization of the path in shape space using isomap [27] dimensionality reduction to project the path of shapes after shape filter on the Riemannian manifold. ...

arXiv:1802.05318v2 fatcat:6snpu4ujorcdzezar4tzt5237m

Open Access Multiple Versions

In recent years, nonlinear dimensionality reduction (NLDR) techniques have attracted much attention in visual perception and many other areas of science. ... A Riemannian manifold can be constructed in the form of a simplicial complex, and thus its intrinsic dimension can be reliably estimated. ... Conclusion We presented a RNC-based manifold learning method for nonlinear dimensionality reduction, which can learn intrinsic geometry of the underlying manifold with metric-preserving properties. ...

doi:10.1007/11744023_4 fatcat:tnopubylqzg37ex7gtd7o7dpcq

To achieve this, we develop a latent variable model for principal geodesic analysis (PGA) that provides a probabilistic framework for factor analysis on diffeomorphisms. ... In this paper, we present a generative Bayesian approach for automatic dimensionality reduction of shape variability represented through diffeomorphic mappings. ... We propose instead to treat the dimensionality reduction step as a probabilistic inference problem on discrete images, in a model called Bayesian principal geodesic analysis (BPGA), which jointly estimates ...

doi:10.1007/978-3-319-10443-0_16 fatcat:k7fl3bolrzf3pkyvoa3ebjz3om

In many cases, there is enough structure in the data (CT, MRI, ultrasound) so a lower dimensional object can describe the degrees of freedom, such as in a manifold structure. ... Manifold learning theory has seen a surge of interest in the modeling of large and extensive datasets in medical imaging since they capture the essence of data in a way that fundamentally outperforms linear ... In order to minimize E γ ð Þ, a nonlinear conjugate gradient technique defined in the low-dimensional space R d is used, thus avoiding convergence and speed issues. ...

doi:10.5772/intechopen.79989 fatcat:jmbsgxgojbc3rekjhdsthzst4e

Open Access

Geodesic distance, as an essential measurement for data dissimilarity, has been successfully used in manifold learning. ... However, most geodesic distance-based manifold learning algorithms have two limitations when applied to classification: 1) class information is rarely used in computing the geodesic distances between data ... In [27] , Nie et al. proposed a flexible manifold embedding framework, which unifies a lot of graph embedding related, linear, and kernelized nonlinear semi-supervised dimension reduction methods. ...

doi:10.1109/tip.2014.2312643 pmid:24723575 fatcat:pzblshuvifetrdg5cerfldv6lm

This paper proposes a framework to construct such input stochastic models for the topology and thermal diffusivity variations in heterogeneous media using a data-driven strategy. ... Given a set of microstructure realizations (input samples) generated from given statistical information about the medium topology, the framework constructs a reduced-order stochastic representation of ... The computing was conducted in part using the resources of the Cornell University Center for Advanced Computing. ...

doi:10.1016/j.jcp.2008.03.023 fatcat:ue4fmvgtuzdubmqsn2qtoxmzhm

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