Riemannian Manifolds: An Introduction to Curvature

By John M. Lee. 1997

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Citations in the corpus

Riemannian Walk for Incremental Learning: Understanding Forgetting and Intransigence (2018, positive): “Thus, when ∆θ → 0, computing KL-divergence 12‖∆θ‖(2)Fθ is equivalent to computing distance in a Riemannian manifold3 [13] induced by the Fisher information matrix at θ.”
Multi-Step Regression Learning for Compositional Distributional Semantics (2013, methodology): “This is done by ﬁrst running a type inference algorithm on the generative rules and obtaining types, then assigning to each basic type a vector space and to each function type a tensor space, and representing arguments by vectors and functions by tensors, ﬁnally, model function application by…”
On the Diffusion of Shape (2007, background): “We direct interested readers to [22] for complete computations and explanations.”
Stable group theory and approximate subgroups (2009, background): “l around pis the image under the exponential map expp of a ball bof radius raround 0 in the tangent space to p, where ris small enough that expp is a diﬀeomorphism. We have ρ(p,expp(v)) = |v| if v∈ b([24], Proposition 6.10, p. 105.) A subset U is called convex if for each p,q∈ U there is a unique ge”
A Geometric Approach to Maximum Likelihood Estimation of the Functional Principal Components From Sparse Longitudinal Data (2007, methodology): “The idea of maximizing the restricted likelihood is in the same framework as that studied by James, Hastie and Sugar (2000), who propose an EM algorithm to maximize the log-likelihood.”
A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank (2013, background): “Substituting these identities in Koszul’s formula (Lee, 1997, (5.1)), 2gA(∇νη, λ) = DgA(η, λ)[ν] + DgA(ν, λ)[η]−DgA(η, ν)[λ] + gA(λ, [ν, η]) + gA(η, [λ, ν])− gA(ν, [η, λ]), we obtain 2tr(A−TλT∇νηA−1) = 2tr(A−TλTDη[ν]A−1 −A−T ηTλA−1νA−1 −A−TλT ηA−1νA−1 −A−T νTλA−1ηA−1 −A−TλT νA−1ηA−1 +A−T ηT…”
Statistical topology via Morse theory, persistence and nonparametric estimation (2009, background): “Let (U , (x)) be any normal coordinate chart centered at xi, then the components of the metric at xi are gij = δij , so √ |gij(xi)| = 1, see [22].”
Geodesic finite elements on simplicial grids (2012, unknown): “11 from [18] we get dist(p, q)(2) = ∑m i=1 q 2 i , which is C .”
Intrinsic dimension estimation: Advances and open problems (2016, background): “2 A Riemannian manifold is a differentiable manifold M endowed with a smooth inner product (Riemannian metric) g(u, v) on each tangent space TpM [57].”