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Multiple Manifolds Clustering via Local Linear Analysis

Wei Zheng, Shuo Chen
2016 Cybernetics and Information Technologies  
Then, Multiple Manifolds Clustering (LMMC) is proposed on the base of the Local Linear Analysis (LLA) via this definition and neighbor-growing algorithm, which are especially effective under the condition  ...  The smoothness and local linearity of data samples are utilized to define the local linear degree which is motivated by Principal Component Analysis (PCA) and Depth First Search (DFS).  ...  The principal component theory provides a basis for this algorithm to define linearity.  ... 
doi:10.1515/cait-2016-0088 fatcat:huchmpjbtbfvnimx3vmvrvfiny

Error analysis of algorithms for computing the projection of a point onto a linear manifold

M. Arioli, A. Laratta
1986 Linear Algebra and its Applications  
The accumulation of rounding errors in methods used to compute the projection of a point onto a linear manifold is studied.  ...  LINEAR ALGEBRA AND ITS APPLICATIONS 82:1-26 (1986) 1 0  ...  BASIC RESULTS FOR THE ERROR ANALYSIS Let us produce some perturbation theory results which will be used in the next section, where the computational errors of algorithms (3.4) and (3.6) will be studied  ... 
doi:10.1016/0024-3795(86)90141-2 fatcat:f4d3bue6bbgnhnpwhudh4lse3q

Center manifold and multivariable approximants applied to non-linear stability analysis

J.-J. Sinou, F. Thouverez, L. Jezequel
2003 International Journal of Non-Linear Mechanics  
The center manifold approach, the multivariable approximants theory, and the alternate frequency/time domain (AFT) method are applied.  ...  This paper outlines the stability analysis and a procedure to reduce and simplify the non-linear system, in order to obtain limit cycle amplitudes.  ...  The center manifold theorem (Marsden and McCracken [17] ) characterises the local bifurcation analysis in the vicinity of a fixed point of the non-linear system.  ... 
doi:10.1016/s0020-7462(02)00080-x fatcat:m4h3adsa35bzpbyau5g4awu6ce

Think globally, fit locally under the Manifold Setup: Asymptotic Analysis of Locally Linear Embedding [article]

Hau-Tieng Wu, Nan Wu
2017 arXiv   pre-print
Since its introduction in 2000, the locally linear embedding (LLE) has been widely applied in data science. We provide an asymptotical analysis of the LLE under the manifold setup.  ...  A comparison with the other commonly applied nonlinear algorithms, particularly the diffusion map, is provided, and its relationship with the locally linear regression is also discussed.  ...  The resulting algorithms could be roughly classified into two types, linear and nonlinear. Linear methods include principal component analysis (PCA), multidimensional scaling, and others.  ... 
arXiv:1703.04058v2 fatcat:gnpwp4krwvgv7n3i7vsne7j3xu

Compressive independent component analysis: theory and algorithms

Michael P Sheehan, Mike E Davies
2022 Information and Inference A Journal of the IMA  
We provide analysis of the CICA algorithms including the effects of finite samples.  ...  By considering synthetic and real datasets, we show the substantial memory gains achieved over well-known ICA algorithms by using one of the proposed CICA algorithms.  ...  In this paper, we develop a CL framework, including theory and practical algorithms, for independent component analysis (ICA).  ... 
doi:10.1093/imaiai/iaac016 fatcat:rh2l4hgytfaxxnyvpazm5qx4fq

Compressive Independent Component Analysis: Theory and Algorithms [article]

Michael P. Sheehan, Mike E. Davies
2021 arXiv   pre-print
We provide analysis of the CICA algorithms including the effects of finite samples.  ...  By considering synthetic and real datasets, we show the substantial memory gains achieved over well-known ICA algorithms by using one of the proposed CICA algorithms.  ...  Code Availability A MATLAB implementation of the proposed CICA algorithms are available at the repository https://gitlab.com/mpsheehan1995/CICA.  ... 
arXiv:2110.08045v1 fatcat:cybamhq77fdsbhgmls2u65ofpy

On non-parametric density estimation on linear and non-linear manifolds using generalized Radon transforms [article]

James Webber, Erika Hussey, Eric Miller, Shuchin Aeron
2019 arXiv   pre-print
Here we present a new non-parametric approach to density estimation and classification derived from theory in Radon transforms and image reconstruction.  ...  We subsequently extend the ideas to address problems in manifold learning and density estimation on manifolds.  ...  Our analysis was consisitent with the theory of [5] , and the higher curvature manifolds proved more troublesome for an accurate density estimation.  ... 
arXiv:1901.03780v3 fatcat:a54ov2vmk5cahe6hrq54xnhtdu

Sparse Subspace Clustering: Algorithm, Theory, and Applications

E. Elhamifar, R. Vidal
2013 IEEE Transactions on Pattern Analysis and Machine Intelligence  
The proposed algorithm is efficient and can handle data points near the intersections of subspaces.  ...  In this paper, we propose and study an algorithm, called sparse subspace clustering, to cluster data points that lie in a union of low-dimensional subspaces.  ...  ACKNOWLEDGMENTS This work was partially supported by US National Science Foundation grants NSF-ISS 0447739 and NSF-CSN 0931805.  ... 
doi:10.1109/tpami.2013.57 pmid:24051734 fatcat:34st7xdfw5gadp2ud5r3f7kzhm

Local Linear Regression on Manifolds and its Geometric Interpretation [article]

Ming-Yen Cheng, Hau-tieng Wu
2012 arXiv   pre-print
High-dimensional data analysis has been an active area, and the main focuses have been variable selection and dimension reduction.  ...  To the first aim, we suggest directly reducing the dimensionality to the intrinsic dimension d of the manifold, and performing the popular local linear regression (LLR) on a tangent plane estimate.  ...  Implications to Manifold Learning Another branch of approaches to high-dimensional, massive data analysis are the graph based algorithms such as locally linear embedding (LLE) [32] , ISOMAP [37] , Hessian  ... 
arXiv:1201.0327v3 fatcat:i2pbacutynfdfbzgdapswzn2gq

Center manifold analysis of a point vortex model of vortex shedding with control

Bartosz Protas
2007 Physica D : Non-linear phenomena  
This manifold is shown to coincide with the uncontrollable subspace of the linearized system.  ...  In this paper we use methods of dynamical systems theory to provide a precise mathematical characterization of the behavior of the point vortex Föppl system with linear feedback control.  ...  The research was supported by NSERC-Discovery (Canada) and CNRS (France).  ... 
doi:10.1016/j.physd.2007.03.008 fatcat:4h75tuz54fd3rksktnlploqoe4

Tangent-based manifold approximation with locally linear models

Sofia Karygianni, Pascal Frossard
2014 Signal Processing  
We start by considering each manifold sample as a different group and we use the difference of tangents to determine appropriate group mergings.  ...  Our experiments verify the effectiveness of the proposed scheme and show its superior performance compared to state-of-the-art methods for manifold approximation.  ...  Our approach, however, specifically addresses the problem of linear manifold approximation as it effectively combines the tangent distances with the theory of constrained clustering towards the design  ... 
doi:10.1016/j.sigpro.2014.03.047 fatcat:t2qqcud4brdofg2lu22nhxueim

Think globally, fit locally under the manifold setup: Asymptotic analysis of locally linear embedding

Hau-Tieng Wu, Nan Wu
2018 Annals of Statistics  
Since its introduction in 2000, Locally Linear Embedding (LLE) has been widely applied in data science. We provide an asymptotical analysis of LLE under the manifold setup.  ...  A comparison with other commonly applied nonlinear algorithms, particularly diffusion map, is provided, and its relationship with locally linear regression is also discussed.  ...  The resulting algorithms can be roughly classified into two types: linear and nonlinear. Linear methods include Principal Component Analysis (PCA), multidimensional scaling, and others.  ... 
doi:10.1214/17-aos1676 fatcat:lsjcfx7jdfeahm6kl4kmumguda

Efficient computation of linear response of chaotic attractors with one-dimensional unstable manifolds [article]

Nisha Chandramoorthy, Qiqi Wang
2022 arXiv   pre-print
In this process, we develop new algorithms, which may be useful beyond linear response, to compute i) a fundamental statistical quantity we introduce called the density gradient, and ii) the unstable derivatives  ...  This paper presents the space-split sensitivity or the S3 algorithm to transform Ruelle's linear response formula into a well-conditioned ergodic-averaging computation.  ...  The authors would like to thank Angxiu Ni, AdamŚliwiak, Malo Jézéquel and Semyon Dyatlov for pointing out errors in the manuscript.  ... 
arXiv:2103.08816v5 fatcat:b7tur6mpyjcy3k3ql53kpd37uq

Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds [article]

Ronny Bergmann and Roland Herzog and Maurício Silva Louzeiro and Daniel Tenbrinck and José Vidal-Núñez
2020 arXiv   pre-print
These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions  ...  Furthermore, we show numerically that our novel algorithm even converges on manifolds of positive curvature.  ...  RB would like to thank Fjedor Gaede and Leon Bungert for fruitful discussions concerning the Chambolle-Pock algorithm in R , especially concerning the choice of parameters as well as DT for hospitality  ... 
arXiv:1908.02022v4 fatcat:hqf7c5dkzzea3mws4zyd6wiufq

A Non-linear Manifold Strategy for SHM Approaches

N. Dervilis, I. Antoniadou, E. J. Cross, K. Worden
2015 Strain  
Acknowledgements The support of the UK Engineering and Physical Sciences Research Council (EPSRC) through grant reference number EP/J016942/1 and EP/K003836/2 is gratefully acknowledged.  ...  First, the whole data set of the four natural frequencies is reduced to two dimensions using a nonlinear manifold technique, in this case locally linear embedding (LLE) (nonlinear principal component analysis  ...  Algorithm theory The initial and basic step in order to apply Gaussian process regression is to obtain a mean m({x}) and covariance function k({x}, {x ′ }) as GPs are completely specified by them, {x}  ... 
doi:10.1111/str.12143 fatcat:25xyzn347rd7jhivbta3jmikru
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