Riemannian Geometry Cheat sheet
Patrick Nicolas
Director Data Engineering @ aidéo technologies |software & data engineering, operations, and machine learning.
Facing challenges with high-dimensional, densely packed but limited data, and complex distributions? Riemannian geometry offers a solution by enabling data scientists to grasp the true shape and distribution of data.
M, N: Manifolds,
Tp M: Tangent space, plane,
U, V: Vectors
X,Y: Vector fields
?u. Directional derivative
φ: Smooth map,
g: Metric Tensor
<,> Dot product
[X, Y] Lie bracket for vector fields,
领英推荐
Γij^k: Christoffel symbol,
γ: Curve on manifold
Differential geometry is a branch of mathematics that uses techniques from calculus, algebra and topology to study the properties of curves, surfaces, and higher-dimensional objects in space. It focuses on concepts such as curvature, angles, and distances, examining how these properties vary as one moves along different paths on a geometric object.
Differential geometry is crucial in understanding the shapes and structures of objects that can be continuously altered, and it has applications in many fields including physics (I.e., general relativity and quantum mechanics), engineering, computer science, and data exploration and analysis. It ultimately provides data scientists with a mathematical framework facilitates the creation of models that are accurate and complex by leveraging geometric and topological insights.
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Patrick Nicolas has over 25 years of experience in software and data engineering, architecture design and end-to-end deployment and support with extensive knowledge in machine learning.?He has been director of data engineering at Aideo Technologies since 2017 and he is the?author of "Scala for Machine Learning", Packt Publishing ISBN 978-1-78712-238-3
Machine Learning/AI Leadership
8 个月I find the notion of "curved spaces" intuitive but pre-conceived and confusing. I think of Riemannian geometry as the study of differentiable manifolds (spaces that can be modeled locally on open sets in Euclidean space, but glued together in a differentiable way ), and different ways of measuring distances in these spaces (a.k.a Riemannian metrics). We then develop a general theory of Riemannian metrics, connections and curvature which aligns with our intuitions of curved spaces embedded in Euclidean space, where we use the Euclidean metric (technically pull back) metric onto the embedded manifold.