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A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics [article]

Kelum Gajamannage, Randy Paffenroth, Erik M. Bollt
2018 arXiv   pre-print
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

Intrinsic dimensionality estimation and dimensionality reduction through scale space filtering

Konstantinos Karantzalos
2009 2009 16th International Conference on Digital Signal Processing  
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

Deformation-based nonlinear dimension reduction: Applications to nuclear morphometry

Gustavo K. Rohde, Wei Wang, Tao Peng, Robert F. Murphy
2008 2008 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro  
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

Geometric Flows of Curves in Shape Space for Processing Motion of Deformable Objects

Christopher Brandt, Christoph von Tycowicz, Klaus Hildebrandt
2016 Computer graphics forum (Print)  
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

Geometric Data Manipulation with Clifford Algebras and Möbius Transforms

Mijail Guillemard, Armin Iske, Udo Zölzer
2015 Advances in Applied Clifford Algebras  
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

Go with the flow: Optical flow-based transport operators for image manifolds

Aswin C. Sankaranarayanan, Chinmay Hegde, Sriram Nagaraj, Richard G. Baraniuk
2011 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton)  
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

Continuum Isomap for manifold learnings

Hongyuan Zha, Zhenyue Zhang
2007 Computational Statistics & Data Analysis  
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

ATTRACTOR MODELING AND EMPIRICAL NONLINEAR MODEL REDUCTION OF DISSIPATIVE DYNAMICAL SYSTEMS

ERIK BOLLT
2007 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering  
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

Nonlinear Shape Regression For Filtering Segmentation Results From Calcium Imaging [article]

Jie Wang, Zhongxiao Fu, Nasrin Sadeghzadehyazdi, Jonathan Kipnis, Scott T. Acton
2018 arXiv   pre-print
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

Riemannian Manifold Learning for Nonlinear Dimensionality Reduction [chapter]

Tony Lin, Hongbin Zha, Sang Uk Lee
2006 Lecture Notes in Computer Science  
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

Bayesian Principal Geodesic Analysis in Diffeomorphic Image Registration [chapter]

Miaomiao Zhang, P. Thomas Fletcher
2014 Lecture Notes in Computer Science  
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

Manifold Learning in Medical Imaging [chapter]

Samuel Kadoury
2018 Manifolds II-Theoretical and Applicable [Working Title]  
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

A non-linear dimension reduction methodology for generating data-driven stochastic input models

Baskar Ganapathysubramanian, Nicholas Zabaras
2008 Journal of Computational Physics  
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

A Regularized Approach for Geodesic-Based Semisupervised Multimanifold Learning

Mingyu Fan, Xiaoqin Zhang, Zhouchen Lin, Zhongfei Zhang, Hujun Bao
2014 IEEE Transactions on Image Processing  
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

Geometric hydrodynamics via Madelung transform

Boris Khesin, Gerard Misiolek, Klas Modin
2018 Proceedings of the National Academy of Sciences of the United States of America  
We introduce a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities.  ...  It turns out that several important PDEs of hydrodynamical origin can be described in this framework in a natural way.  ...  A part of this work was completed while B.K. held a Weston Visiting Professorship at the Weizmann Institute of Science.  ... 
doi:10.1073/pnas.1719346115 pmid:29844194 fatcat:n2tmxdfub5bz3dq4vjy5kcakfq
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