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Dimensionality reduction of collective motion by principal manifolds

Kelum Gajamannage, Sachit Butail, Maurizio Porfiri, Erik M. Bollt
2015 Physica D : Non-linear phenomena  
While the existence of low-dimensional embedding manifolds has been shown in patterns of collective motion, the current battery of nonlinear dimensionality reduction methods are not amenable to the analysis  ...  Through representative examples, we show that compared to existing nonlinear dimensionality reduction methods, the principal manifold retains the original structure even in noisy and sparse datasets.  ...  Algorithms Here, we state the algorithms of dimensionality reduction by principal manifold by three components as, clustering, smoothing, and embedding.  ... 
doi:10.1016/j.physd.2014.09.009 fatcat:7y72cgagrvawzimj5ufglqwmba

Modeling the product manifold of posture and motion

Ankur Datta, Yaser Sheikh, Takeo Kanade
2009 2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops  
A collection of local linear models, thus, avoids the limitation of global models that require a uniform dimensionality for the latent motion manifold.  ...  We propose to model the nonlinear motion manifold as a collection of local linear models, noting that given a particular posture, the variation in motion for that posture can be well-approximated by a  ...  Acknowledgements The research described in this paper was supported by the DENSO Corporation, Japan.  ... 
doi:10.1109/iccvw.2009.5457588 dblp:conf/iccvw/DattaSK09 fatcat:52bue7sbozhavcocidtzfbdtne

Gauge theory of collective modes

G Rosensteel
2012 Journal of Physics, Conference Series  
The classical theory of Riemann ellipsoids is formulated naturally as a gauge theory based on a principal G-bundle P.  ...  The base manifold is the space of positive-definite real 3 × 3 symmetric matrices, identified geometrically with the space of inertia ellipsoids.  ...  Acknowledgments My thanks to my mentor and friend, David Rowe, who identified many of the key group theoretic ideas hidden in the original formulation of the Bohr collective model.  ... 
doi:10.1088/1742-6596/403/1/012010 fatcat:b3wdfmm4tzbd3hs7soaieezsjq

Motion Key-Frame Extraction by Using Optimized t-Stochastic Neighbor Embedding

Qiang Zhang, Yi Yao, Dongsheng Zhou, Rui Liu
2015 Symmetry  
Key-frame extracting technology has been widely used in the field of human motion synthesis. Efficient and accurate key frames extraction methods can improve the accuracy of motion synthesis.  ...  The experimental results show that the validity of this method is better than the existing methods under the same experimental data.  ...  [11] clustered the motion data of N frame to K collection and took first frame of each collection as a key frame. Park [12] took motion data with parameters and represented it by the quaternion.  ... 
doi:10.3390/sym7020395 fatcat:xwbqhmtig5eo3oqiyyrkr372om

Global Optimization of Robotic Grasps

Carlos Rosales Gallegos, Josep Porta, Lluis Ros
2011 Robotics: Science and Systems VII  
The main difficulties of the problem include that the set of feasible grasps is a manifold implicitly defined by a system of non-linear equations, the high dimension of this manifold, and the multi-modal  ...  The proposed procedure finds a way around these difficulties by focussing the exploration on a relevant subset of grasps of lower dimension, which is traced out exhaustively using higher-dimensional continuation  ...  ACKNOWLEDGMENTS This work has been partially supported by the Spanish Ministry of Science and Innovation under contracts DPI2010-18449 and DPI2010-15446.  ... 
doi:10.15607/rss.2011.vii.012 dblp:conf/rss/GallegosPR11 fatcat:5dbx7wpyr5hsteny5xzvbb4era

Page 4095 of Mathematical Reviews Vol. , Issue 95g [page]

1995 Mathematical Reviews  
Let E be a vector bundle over a 1-dimensional manifold M, with local coordinate f, and let e),---,e, be a local basis of sections of E, so that any section s of E can be written as s = }*) _; Vala, Where  ...  In the first principal case one must distinguish four subcases, resulting in a total of six normal forms of six types of motion.  ... 

Nonlinear dimensionality reduction in molecular simulation: The diffusion map approach

Andrew L. Ferguson, Athanassios Z. Panagiotopoulos, Ioannis G. Kevrekidis, Pablo G. Debenedetti
2011 Chemical Physics Letters  
The diffusion map is a nonlinear dimensionality reduction technique with the capacity to systematically extract the essential dynamical modes of high-dimensional simulation trajectories, furnishing a kinetically  ...  While the atomic-level resolution provides unparalleled detail, it can be non-trivial to extract the important motions underlying simulations of complex systems containing many degrees of freedom.  ...  The existence of low-dimensional effective descriptions is supported, for example, by studies of proteins demonstrating the important dynamics to be confined within a handful of collective motions [6]  ... 
doi:10.1016/j.cplett.2011.04.066 fatcat:e5ldnrjncngdphkm2jnjlgr3nq

Detecting phase transitions in collective behavior using manifold's curvature

Kelum Gajamannage, Erik M. Bollt
2016 Mathematical Biosciences and Engineering  
We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists.  ...  Here, we propose a method of detecting phase transitions and splitting the manifold into phase transitions free sub-manifolds.  ...  Approximating the curvature of a manifold. We extend the two dimensional assertion into higher dimensions by intersecting principal sections, made by the shape operator, with the manifold.  ... 
doi:10.3934/mbe.2017027 pmid:27879108 fatcat:53lu67pixrazbe7zcbrnrxwmuy

Generalized PCA via the Backward Stepwise Approach in Image Analysis [chapter]

Sungkyu Jung, Xiaoxiao Liu, J. S. Marron, Stephen M. Pizer
2010 Advances in Intelligent and Soft Computing  
In an example describing the motion of the lung based on CT images, we show that composite Principal Nested Spheres captures landmark data more succinctly than forward PCA methods.  ...  We see that for manifold data the backward view gives much more natural and accessible generalizations of PCA.  ...  In particular, the quadratic motion in the PC 1-2 plane is efficiently captured by the 1-dimensional principal arc.  ... 
doi:10.1007/978-3-642-16259-6_9 fatcat:kbx6tkkdrngcth5x7pzcdt5ep4

Page 4777 of Mathematical Reviews Vol. , Issue 88i [page]

1988 Mathematical Reviews  
In more sophisticated approaches the space-time manifold is the base space of a principal fiber bundle.  ...  Dimensional reduction [P. Forgacs and N. S. Manton, Comm. Math.  ... 

Page 6530 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews  
Let M be an n-dimensional smooth manifold and let 7? M be the vector bundle of the n-dimensional vector densities of weight p. T?  ...  The author finds all 2-dimensional spaces % having the t-dimensional motion groups G, for all t > 0. Some interesting results are obtained in the case of the spaces %, having motion group G3.  ... 

Invariant manifolds and collective motion in many-body systems

T. Papenbrock
2001 AIP Conference Proceedings  
The importance of collective configurations depends on the stability of the manifold. We present an example of quantum collective motion on the manifold  ...  Collective modes of interacting many-body systems can be related to the motion on classically invariant manifolds. We introduce suitable coordinate systems.  ...  TP acknowledges support as a Wigner Fellow and staff member at Oak Ridge National Laboratory, managed by UT-Battelle, LLC for the U.S. Department of Energy under contract DE-AC05-00OR22725.  ... 
doi:10.1063/1.1427476 fatcat:ozazsklkgfbwzjh74th25iyuj4

Dimension Reduction of Dynamical Systems: Methods, Models, Applications

Giuseppe Rega, Hans Troger
2005 Nonlinear dynamics  
After presenting some basic introductory ideas concerning dimension reduction and reduced order modelling, an overview of the contents of the papers collected in this Special Issue of Nonlinear Dynamics  ...  If the motion is exactly represented by the motion on the manifold, this motion is called in [15] a Nonlinear Normal Mode.  ...  This type of analysis has been done in [13] for the spring pendulum. On the resulting invariant manifold a two-dimensional motion in the four dimensional space is obtained.  ... 
doi:10.1007/s11071-005-2790-3 fatcat:pv6oe5j4p5cftcxk6spwqsetee

EMG-Based Control of a Robot Arm Using Low-Dimensional Embeddings

P.K. Artemiadis, K.J. Kyriakopoulos
2010 IEEE Transactions on robotics  
A mathematical model is trained to decode upper limb motion from EMG recordings, using a dimensionality-reduction technique that represents muscle synergies and motion primitives.  ...  The accuracy of the method is assessed through real-time experiments, including random arm motions.  ...  Raw EMG signals are collected, preprocessed, and then represented by the low-dimensional manifolds using (3) . Then, the fitted model (5) is used.  ... 
doi:10.1109/tro.2009.2039378 fatcat:5vrmovytuvc3ni5uyms4aerydm

Nonlinear reconstruction of single-molecule free-energy surfaces from univariate time series

Jiang Wang, Andrew L. Ferguson
2016 Physical review. E  
The dynamics of the n-tetracosane polymer chain in water considered in this work are contained in a two-dimensional manifold parametrized by the collective variables [ϒ 1 ,ϒ 2 ] that are nonlinear combinations  ...  The diffusion map discovers the latent two-dimensional manifold, and extracts it into the two collective variables [ 2 , 3 ] quantifying, respectively, the location of the points along and perpendicular  ...  two principal components, confirming that the manifold is effectively two-dimensional and can be projected into [ ϒ * 1 , ϒ * 2 ] ∈ R 2 with essentially no loss of information. forcing by solvent motion  ... 
doi:10.1103/physreve.93.032412 pmid:27078395 fatcat:szwxoycaifc5pjpzuxxbaey5xu
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