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Chentsov's Theorem
The Fisher information is often the default choice of the Riemannian metric for manifolds of probability distributions. In this post, ...
The Curvature of the Manifold of Gaussian Distributions
The Gaussian probability distribution is central in statistics and machine learning. As it turns out, by equipping the set of ...
Hessian and Curvatures in Machine Learning: A Differential-Geometric View
In machine learning, especially in neural networks, the Hessian matrix is often treated synonymously with curvatures. But, from calculus alone, ...
Optimization and Gradient Descent on Riemannian Manifolds
One of the most ubiquitous applications in the field of differential geometry is the optimization problem. In this article we ...
Minkowski's, Dirichlet's, and Two Squares Theorem
Application of Minkowski's Theorem in geometry problems, Dirichlet's Approximation Theorem, and Two Squares Theorem.
Reduced Betti number of sphere: Mayer-Vietoris Theorem
A proof of reduced homology of sphere with Mayer-Vietoris sequence.
Brouwer's Fixed Point Theorem: A Proof with Reduced Homology
A proof of special case (ball) of Brouwer's Fixed Point Theorem with Reduced Homology.