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This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds.
[author] Fréchet, MauriceM. (1948). Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l’Institut Henri Poincaré 10 215–310
1948
Earlier work this paper cites.
[author] Busemann, HerbertH. (1955). The Geometry of Geodesics. Academic Press
1955
Earlier work this paper cites.
[author] Bishop, R. L.R. L. and O’Neill, B.B. (1969). Manifolds of negative curvature. Transactions of the American Mathematical Society 145 1–49. 10.1090/S0002-9947-1969-0251664-4
1969
Earlier work this paper cites.
[author] Bishop, R. L.R. L. and O’Neill, B.B. (1969). Manifolds of negative curvature. Transactions of the American Mathematical Society 145 1–49. 10.1090/S0002-9947-1969-0251664-4
1969
Earlier work this paper cites.
[author] Karcher, HermannH. (1977). Riemannian center of mass and mollifier smoothing. Communications in Pure and Applied Mathematics 30 509–541
1977
Earlier work this paper cites.
[author] Buser, PeterP. and Karcher, HermannH. (1981). Gromov’s almost flat manifolds. Société mathématique de France
1981
Earlier work this paper cites.
[author] Kendall, Wilfrid S.W. S. (1990). Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proc. London Math. Soc. 61 371–406
1990
Earlier work this paper cites.
[author] Cutler, AdeleA. and Breiman, LeoL. (1994). Archetypal Analysis. Technometrics 36 338–347. http://dx.doi.org/10.2307/1269949
1994
Earlier work this paper cites.
[author] Crouch, PeterP. and Leite, F. SilvaF. S. (1995). The dynamic interpolation problem: on Riemannian manifolds, Lie groups, and symmetric spaces. J. of Dynamical and Control Systems 1 177–202
1995
Earlier work this paper cites.
[author] Small, Christopher G.C. G. (1996). The Statistical Theory of Shapes. Springer series in statistics. Springer
1996
Earlier work this paper cites.
[author] Lee, John M.J. M. (1997). Riemannian Manifolds: An Introduction to Curvature. Springer
1997
Earlier work this paper cites.
[author] Tipping, Michael E.M. E. and Bishop, Chris M.C. M. (1999). Probabilistic Principal Component Analysis. Journal of the Royal Statistical Society, Series B 61 611–622
1999
Earlier work this paper cites.
[author] Buss, Samuel R.S. R. and Fillmore, Jay P.J. P. (2001). Spherical Averages and Applications to Spherical Splines and Interpolation. ACM Trans. Graph. 20 95–126. 10.1145/502122.502124
2001
Earlier work this paper cites.
[author] Olver, Peter J.P. J. (2001). Geometric Foundations of Numerical Algorithms and Symmetry. Applicable Algebra in Engineering, Communication and Computing 11 417–436. 10.1007/s002000000053
2001
Earlier work this paper cites.
[author] Buss, Samuel R.S. R. and Fillmore, Jay P.J. P. (2001). Spherical Averages and Applications to Spherical Splines and Interpolation. ACM Trans. Graph. 20 95–126. 10.1145/502122.502124
2001
Earlier work this paper cites.
[author] Bhattacharya, RabiR. and Patrangenaru, VicV. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds, I. Annals of Statistics 31 1-29
2003
Earlier work this paper cites.
[author] Fletcher, P. ThomasP. T., Lu, ConglinC., Pizer, Stephen M.S. M. and Joshi, SarangS. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging 23 995–1005. 10.1109/TMI.2004.831793
2004
Earlier work this paper cites.
[author] Groisser, DavidD. (2004). Newton’s method, zeroes of vector fields, and the Riemannian center of mass. Adv. in Applied Math 33 95-135
2004
Earlier work this paper cites.
[author] Le, HuilingH. (2004). Estimation of Riemannian barycenters. LMS J. Comput. Math. 7 193-200
2004
Cited alongside, same era.
[author] Bhattacharya, RabiR. and Patrangenaru, VicV. (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds, II. Annals of Statistics 33 1225-1259
2005
Cited alongside, same era.
[author] Dryden, Ian L.I. L. (2005). Statistical analysis on high-dimensional spheres and shape spaces. Ann. Statist. 33 1643–1665. 10.1214/009053605000000264
2005
Cited alongside, same era.
[author] Huckemann, StephanS. and Ziezold, HerbertH. (2006). Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces. Advances in Applied Probability 38 299–319. 10.1239/aap/1151337073
2006
Cited alongside, same era.
[author] Gay-Balmaz, FF., Holm, DDD., Meier, DMD., Ratiu, TST. and Vialard, F-XF.-X. (2012). Invariant Higher-Order Variational Problems. Comm. in Mathematical Physics 309 413–458. 10.1007/s00220-011-1313-y
2012
Later among the works it cites.
[author] Jung, SungkyuS., Dryden, Ian L.I. L. and Marron, J. SteveJ. S. (2012). Analysis of principal nested spheres. Biometrika 99 551–568. 10.1093/biomet/ass022
2012
Later among the works it cites.
[author] Pennec, XavierX. and Arsigny, VincentV. (2012). Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups. In Matrix Information Geometry (FredericF. Barbaresco, AmitA. Mishra and FrankF. Nielsen, eds.) 123-168. Springer. 10.1007/978-3-642-30232-9_7
2012
Later among the works it cites.
[author] Damon, JamesJ. and Marron, J. S.J. S. (2013). Backwards Principal Component Analysis and Principal Nested Relations. Journal of Mathematical Imaging and Vision 50 107–114. 10.1007/s10851-013-0463-2
2013
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2006
Cited alongside, same era.
[author] Pennec, XavierX., Fillard, PierreP. and Ayache, NicholasN. (2006). A Riemannian Framework for Tensor Computing. International Journal of Computer Vision 66 41–66. 10.1007/s11263-005-3222-z
2006
Cited alongside, same era.
[author] Pennec, XavierX. (2006). Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. Journal of Mathematical Imaging and Vision 25 127-154. 10.1007/s10851-006-6228-4
2006
Cited alongside, same era.
[author] Pennec, XavierX., Fillard, PierreP. and Ayache, NicholasN. (2006). A Riemannian Framework for Tensor Computing. International Journal of Computer Vision 66 41–66. 10.1007/s11263-005-3222-z
2006
Cited alongside, same era.
[author] Absil, P. A.P. A., Mahony, R.R. and Sepulchre, R.R. (2008). Optimization Algorithms on Matrix Manifolds. Princeton University Press
2008
Cited alongside, same era.
Leporé, N
2008
Cited alongside, same era.
[author] Wang, XiaohuiX. and Marron, J. S.J. S. (2008). A scale-based approach to finding effective dimensionality in manifold learning. Electronic Journal of Statistics 2 127–148. 10.1214/07-EJS137
2008
Cited alongside, same era.
[author] Brewin, LeoL. (2009). Riemann normal coordinate expansions using Cadabra. Classical and Quantum Gravity 26 175017
2009
Cited alongside, same era.
Later among the works it cites.
[author] Feragen, AasaA., Owen, MeganM., Petersen, JensJ., Wille, Mathilde M. W.M. M. W., Thomsen, Laura H.L. H., Dirksen, AsgerA. and de Bruijne, MarleenM. (2013). Tree-Space Statistics and Approximations for Large-Scale Analysis of Anatomical Trees In Proc of Inf. Proc. in Medical Imaging (IPMI 2013), Asilomar, CA, USA LNCS 7917 74–85. Springer. 10.1007/978-3-642-38868-2_7
2013
Later among the works it cites.
[author] Lorenzi, MarcoM. and Pennec, XavierX. (2013). Geodesics, Parallel Transport & One-parameter Subgroups for Diffeomorphic Image Registration. International Journal of Computer Vision 105 111-127. 10.1007/s11263-012-0598-4
2013
Later among the works it cites.
[author] Sommer, StefanS. (2013). Horizontal Dimensionality Reduction and Iterated Frame Bundle Development. In Proc. of Geometric Science of Information (GSI 2013), (FrankF. Nielsen and FrédéricF. Barbaresco, eds.). LNCS 8085 76–83. Springer Berlin Heidelberg
2013
Later among the works it cites.
[author] Sommer, StefanS., Lauze, FrançoisF. and Nielsen, MadsM. (2013). Optimization over geodesics for exact principal geodesic analysis. Advances in Computational Mathematics 40 283–313. 10.1007/s10444-013-9308-1
2013
Later among the works it cites.
Zhang, M
2013
Later among the works it cites.
[author] Costa, Sueli I. R.S. I. R., Santos, Sandra A.S. A. and Strapasson, João E.J. E. (2015). Fisher information distance: A geometrical reading. Discrete Applied Mathematics 197 59–69. 10.1016/j.dam.2014.10.004
2014
Later among the works it cites.
[author] Wilson, R. C.R. C., Hancock, E. R.E. R., Pekalska, E.E. and Duin, R. P. W.R. P. W. (2014). Spherical and Hyperbolic Embeddings of Data. IEEE Transactions on Pattern Analysis and Machine Intelligence 36 2255–2269. 10.1109/TPAMI.2014.2316836
2014
Later among the works it cites.
Eltzner, B
2015
Later among the works it cites.
[author] Lorenzi, MarcoM., Ayache, NicholasN. and Pennec, XavierX. (2015). Regional flux analysis for discovering and quantifying anatomical changes: An application to the brain morphometry in Alzheimer’s disease. NeuroImage 115 224–234. 10.1016/j.neuroimage.2015.04.051
2015
Later among the works it cites.
Pennec, X
2015
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[author] Weyenberg, Grady S.G. S. (2015). Statistics in the Billera-Holmes-Vogtmann treespace PhD thesis, University of Kentucky
2015
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[author] Zhai, HaojinH. (2016). Principal component analysis in phylogenetic tree space PhD thesis, University of North Carolina at Chapel Hill
2016
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