Smooth Manifolds

By Fabian Immler, Bohua Zhan. Arch. Formal Proofs, 2018

454 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/b37775b5240d5f5ef6b60811a865c3eea89ba000

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Citations in the corpus

• Moser Flow: Divergence-based Generative Modeling on Manifolds (2021, methodology): “…Lie derivative L v ω of a smooth vector field v and smooth differential form ω can be computed using Cartan’s "magic formula" (see e.g., Theorem 14.35 in Lee (2013)): where i v ω is the interior multiplication of a vector field and a differential form defined by In case ω is an n -form (as ˆ α t in…”
• On Lazy Training in Differentiable Programming (2018, background): “…, the smallest singular value of Dh ( w 0 ) is positive and that the initialization satisfies (cid:107) h ( w The proof of this result relies on the fact that αh ( w α ( t )) follows the gradient flow of R in a time-dependent and non degenerate metric: the pushforward metric [21] induced by h on F .”
• Linear growth of quantum circuit complexity (2021, background): “DxF A has the same rank as D0(exp FA(x) ◦F A ◦ exp×R x ), because exp×R x and expFA(x) are local charts [31].”
• Riemannian Score-Based Generative Modeling (2022, methodology): “Then, we define div : The following Stokes formula (also called divergence theorem, see Lee (2018, p.51)) holds for any f ∈ C ∞ ( M ) and X ∈ X ( M ), Using the Stokes formula and the definition of the gradient we get that in local coordinates The Laplace-Beltrami operator is given by ∆ M : C ∞ ( M )…”
• Unsupervised Transformation Learning via Convex Relaxations (2017, methodology): “…that G is a one-dimensional Lie group, we have more: for each R near I , there is a t ∈ R satisfying The matrix tA = log R in the exponential map is the derivative of our transformation (as A ≈ ( R − I ) /t for R − I small) and is analogous to locally linear neighborhoods in manifold learning [10].”
• SE(3) diffusion model with application to protein backbone generation (2023, methodology): “We refer the reader to Lee (2013) and Hsu (2002) for background on differential geometry and stochastic analysis on manifolds.”