Riemannian center of mass and mollifier smoothing

By Hermann Karcher. 1977

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Citations in the corpus

Optimization Techniques on Riemannian Manifolds (2014, unknown): “Nevertheless, the amount by which exp fails to preserve inner products can be quantiﬁed via the Gauss Lemma and Jacobi’s equation; see, e.g., Cheeger and Ebin [ 12 ], or the appendices of Karcher [ 28 ].”
Coding Kendall's Shape Trajectories for 3D Action Recognition (2018, methodology): “In summary, we have analytical expressions for computing exponential map, logarithm map, and intrinsic mean [24] of shapes on S.”
INTRINSIC SHAPE ANALYSIS: GEODESIC PCA FOR RIEMANNIAN MANIFOLDS MODULO ISOMETRIC LIE GROUP ACTIONS (2007, background): “…of a random variable Y is contained in a geodesic ball Br(p) for some p ∈ M , and if B4r(p) is contained in a normal neighborhood U on which positive sectional curvatures are bounded by κ > 0, then the condition r (π/(2κ)) ensures that Y has a unique IM, see Karcher (1977) and Le (2001).”
Geodesic finite elements on simplicial grids (2012, background): “The precise conditions have been given by Karcher [16].”
Geodesic Finite Elements of Higher Order (2015, methodology): “For the first-order case this question was answered by recognizing the interpolation rule (1) as a special case of the Riemannian center of mass with a positive measure of unit weight, and using the now classic result of Karcher [14].”
Geodesic Finite Mixture Models (2014, background): “For instance, the mean of N points xi on a manifold can be calculated as μ = argmin p ∑i=1 d (xi, p) 2 [17].”
Weighted deterministic walks for the least squares mean on Hadamard spaces (2014, background): “This concept was first introduced by Élie Cartan a long time ago, then came of interest again in [9].”
Probability measures on the space of persistence diagrams (2011, background): “There are several results on the existence (and uniqueness) of the Fréchet mean for particular manifolds and distributions [27, 28].”
Statistical shape analysis: clustering, learning, and testing (2005, methodology): “algorithm for computing the Karcher mean is presented in [16, 13 ] and is particularized to S in [15].”
Shape Analysis of Elastic Curves in Euclidean Spaces (2011, methodology): “Using ideas presented in earlier papers [6], [3 0 we demonstrate a simple model where we: (i) first compute the sample Karcher mean [10] of the given shapes, (ii) learn a probability model on the tangent space (at the mean) by mapping the observations to that tangent space, and (iii) wrap the probab”
Human gesture recognition on product manifolds (2012, methodology): “One possible way to realize the composition map, G ◦H , is to employ the tangent space and modify the Karcher mean (Karcher, 1977).”
Nonparametric statistics on manifolds with applications to shape spaces (2008, methodology): “Recall that (Karchar [19]; see also Theorem 2.”
Human Detection via Classification on Riemannian Manifolds (2007, background): “Similar to Euclidean spaces, the Karcher mean [10] of points on Riemannian manifold, is the point on M which minimizes the sum of squared distances”
Fréchet means and Procrustes analysis in Wasserstein space (2017, background): “A closely related notion is that of a Karcher mean (Karcher [45]; Le [50]), a term that describes stationary points of the sum of squares functional, when the latter is differentiable.”