Introduction to Smooth Manifolds

By John M. Lee. 2002

3,885 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

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Citations in the corpus

Variational Autoencoders and Nonlinear ICA: A Unifying Framework (2019, background): “14 in Lee (2003), we conclude that Ci is a smooth (C1) submanifold of co-dimension 1 in R, and so is C̃i by equality.”
Meta-Learning with Warped Gradient Descent (2019, background): “The metric tensor deﬁnes the steepest direction of descent by − G − 1 ∇ L (Lee, 2003).”
Bounded cohomology of discrete groups (2016, methodology): “[Lee03] J. M. Lee, Introduction to smooth manifolds, Graduate Texts in Mathematics, no. 218, Springer-Verlag, New York, 2003.”
Is Voter Competence Good for Voters?: Information, Rationality, and Democratic Performance (2014, background): “Specializing Theorem 8.8 of Lee (2003, 182) yields Theorem 1 (Constant-Rank Level Set Theorem) Let M be an open subset of R n and let f : M → R be a smooth (i.e., inﬁnitely many times continuously differentiable) map with the rank of Df ( a ) constant at k .”
From Frenet to Cartan: The Method of Moving Frames (2017, background): “[Lee13] John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathe- matics, vol. 218, Springer, New York, 2013.”
Graph Laplacians and their Convergence on Random Neighborhood Graphs (2006, methodology): “For more technical details behind the definition of a manifold with boundary we refer to Lee (2003). Note that the boundary ∂M of M is a (d − 1)-dimensional manifold without boundary.”
Grassmannian Sparse Representations and Motion Depth Surfaces for 3D Action Recognition (2013, background): “The geometrical characteristics and structuring of Grassmann manifolds are discussed in [21], [22].”
Differentiating through the Fréchet Mean (2020, methodology): “For a more in-depth introduction, we refer the interested reader to our Appendix C or texts such as Lee (2003) and Lee (1997).”
Convergence of Learning Dynamics in Stackelberg Games (2019, background): “This set can be characterized in terms of an implicit map r, defined by the leader’s belief that the follower is playing a best response to its choice at each iteration, which would imply D2f2(x1,k, x2,k) = 0, and under sufficient regularity conditions the implicit mapping theorem [23] gives rise to”
Proto-value Functions: A Laplacian Framework for Learning Representation and Control in Markov Decision Processes (2007, background): “There are technical conditions that additionally need to be satisfied for a manifold to be smooth, as explained in Lee (2003). there have been several attempts to fix the shortcomings of traditional function approximators to address the inherent nonlinear nature of value functions, these approaches…”
Alternating Minimization Algorithms for Hybrid Precoding in Millimeter Wave MIMO Systems (2016, background): “More background on manifolds and manifold optimization can be found in [41]–[43], and there are some recent applications in wireless communications [44].”
Metric Learning and Manifolds: Preserving the Intrinsic Geometry (2017, methodology): “Following (Lee, 2003), we will identify local coordinates of an open set U ⊂ M by the image coordinate chart homeomorphism.”
The Geometric Foundations of Hamiltonian Monte Carlo (2014, methodology): “Although we will relate each result to probabilistic theory and introduce intuition where we can, the formal details in the following require a working knowledge of diﬀerential geometry up to Lee (2013).”