Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

By Xavier Pennec. Journal of Mathematical Imaging and Vision, 2006

874 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fd579d0231979aeaf80cda8ca5cfa6d4e3a929d9

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In Applied Mathematics

In Computer Vision and Pattern Recognition

In Geometry and Topology

Citations in the corpus

Differential Geometry and Lie Groups (2020, background): “Fréchet means are not unique, but if X is a complete Riemannian manifold, certain sufficient conditions on the dispersion of the data are known that ensure the existence and uniqueness of the Fréchet mean (see Pennec [92]).”
Geodesic Finite Mixture Models (2014, background): “However, it is possible to approximate this normalization factor based on maximum entropy criterion computed directly on the manifold [23].”
Weighted deterministic walks for the least squares mean on Hadamard spaces (2014, methodology): “In the case when the NPC space of interest is in fact a non-positively curved Riemannian manifold, then one can use gradient or Newton like optimization methods [6, 14, 16, 17, 19] for approximating the mean.”
Matching Normalizing Flows and Probability Paths on Manifolds (2022, background): “Solutions to the heat equation are known in very few cases (Pennec, 2006), and even for the sphere the solution is only known as an infinite series of Legendre polynomials (Tulovsky & Papiez, 2001).”
Barycentric subspace analysis on manifolds (2016, methodology): “…of the exponential and logarithmic maps (from now on exp and log) is indeed the basis of programming on Riemannian manifolds, and we can express using them practically all the geometric operations needed for statistics (Pennec, 2006) or image processing (Pennec, Fillard and Ayache, 2006).”
Sampling From A Manifold (2012, background): “The development of mean and variance estimators appears in Pennec (2006) and Bhattacharya and Patrangenaru (2003).”
Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach (2012, methodology): “[4] introduced the Affine Invariant Riemannian Metric (AIRM) and”
Dreaming More Data: Class-dependent Distributions over Diffeomorphisms for Learned Data Augmentation (2015, methodology): “Here Σ y = 1 N y nm ( v θ nm )( v θ nm ) (cid:62) denotes the Riemannian covariance of the transformations [39], where N y is the number of sample pairs used for this class.”
Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices (2007, methodology): “The framework of Riemannian geometry [8] is particularly adapted to this task, since many statistical tools [18] and PDEs can be generalized to this framework.”
Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups (2013, neutral): “Thus, exactly as in Riemannian manifolds [55], we can locally linearize the Lie group around the mean using the map Logx and define the 2-covariant tensor:”
Riemannian Score-Based Generative Modeling (2022, background): “Problem 2: The scores are intractable so the dynamics (2) cannot be implemented.”
Population Shape Regression from Random Design Data (2007, methodology): “In [9, 14, 30], statistical concepts such as averaging and principal components analysis were extended to manifolds representing anatomical shape variability.”
Latent Space Non-Linear Statistics (2018, methodology): “We aim at using the constructions, tools and methods from nonlinear statistics [15] to perform statistical analysis of data in the latent representation.”