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Functions Are Vectors (2023)
are Vectors Conceptualizing functions as infinite-dimensional vectors lets us apply the tools of linear algebra to a vast landscape of new problems, from image and geometry processing to curve fitting, light transport, and machine learning. Prerequisites: introductory linear algebra, introductory calculus, introductory differential equations.
Conceptualizing functions as infinite-dimensional vectors lets us apply the tools of linear algebra to a vast landscape of new problems, from image and geometry processing to curve fitting, light transport, and machine learning. In this post, we won’t attempt to prove our results in infinite dimensions: we will focus on building intuition via analogies to finite-dimensional linear algebra. Geometry: Distances, Parallel Transport, Flattening, Non-manifold Meshes, and Polygonal Meshes Simulation: the Finite Element Method, Monte Carlo PDEs, Minimal Surfaces, and Fluid Cohomology Light Transport: Radiosity, Operator Formulation (Ch.4), Low-Rank Approximation, and Inverse Rendering Machine Learning: DiffusionNet, MeshCNN, Kinematics, Fourier Features, and Inverse Geometry Splines: C2 Interpolation, Quadratic Approximation, and Simplification
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