In mathematics, projections onto convex sets (POCS), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets. It is a very simple algorithm and has been rediscovered many times. The simplest case, when the sets are affine spaces, was analyzed by John von Neumann. The case when the sets are affine spaces is special, since the iterates not only converge to a point in the intersection (assuming the intersection is non-empty) but to the orthogonal projection of the point onto the intersection. For general closed convex sets, the limit point need not be the projection. Classical work on the case of two closed convex sets shows that the rate of convergence of the iterates is linear.There are now extensions that consider cases when there are more than one set, or when the sets are not convex, or that give faster convergence rates. Analysis of POCS and related methods attempt to show that the algorithm converges (and if so, find the rate of convergence), and whether it converges to the projection of the original point. These questions are largely known for simple cases, but a topic of active research for the extensions. There are also variants of the algorithm, such as Dykstra's projection algorithm. See the references in the section for an overview of the variants, extensions and applications of the POCS method; a good historical background can be found in section III of. (Wikipedia).
Projections (video 5): Example N-dimensional Projections
Recordings of the corresponding course on Coursera. If you are interested in exercises and/or a certificate, have a look here: https://www.coursera.org/learn/pca-machine-learning
From playlist Projections
New Results on Projections - Guy Moshkovitz
Computer Science/Discrete Mathematics Seminar II Topic: New Results on Projections Speaker: Guy Moshkovitz Affiliation: Member, School of Mathematics Date: January 22, 2019 For more video please visit http://video.ias.edu
From playlist Mathematics
Projections (video 4): N-dimensional projections
Recordings of the corresponding course on Coursera. If you are interested in exercises and/or a certificate, have a look here: https://www.coursera.org/learn/pca-machine-learning
From playlist Projections
Projections (video 2): Projection onto 1D Subspaces
Recordings of the corresponding course on Coursera. If you are interested in exercises and/or a certificate, have a look here: https://www.coursera.org/learn/pca-machine-learning
From playlist Projections
Spectrahedral lifts of convex sets – Rekha Thomas – ICM2018
Control Theory and Optimization Invited Lecture 16.6 Spectrahedral lifts of convex sets Rekha Thomas Abstract: Efficient representations of convex sets are of crucial importance for many algorithms that work with them. It is well-known that sometimes, a complicated convex set can be expr
From playlist Control Theory and Optimization
Recordings of the corresponding course on Coursera. If you are interested in exercises and/or a certificate, have a look here: https://www.coursera.org/learn/pca-machine-learning
From playlist Projections
Lecture 3 | Convex Optimization II (Stanford)
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From playlist Lecture Collection | Convex Optimization
Karol Życzkowski : Geometry of Quantum Entanglement
Recording during the thematic meeting : "Geometrical and Topological Structures of Information" the August 31, 2017 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematician
From playlist Geometry
Lecture 1 | Random polytopes | Zakhar Kabluchko | EIMI
Online school "Randomness online" November 4 – 8, 2020 https://indico.eimi.ru/event/40/
From playlist Talks of Mathematics Münster's reseachers
Haotian Jiang: Minimizing Convex Functions with Integral Minimizers
Given a separation oracle SO for a convex function f that has an integral minimizer inside a box with radius R, we show how to find an exact minimizer of f using at most • O(n(n + log(R))) calls to SO and poly(n,log(R)) arithmetic operations, or • O(nlog(nR)) calls to SO and exp(O(n)) · po
From playlist Workshop: Continuous approaches to discrete optimization
Ramon van Handel: The mysterious extremals of the Alexandrov-Fenchel inequality
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From playlist Trimester Seminar Series on the Interplay between High-Dimensional Geometry and Probability
Priya Donti - Optimization-in-the-loop AI for energy and climate - IPAM at UCLA
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From playlist 2023 Artificial Intelligence and Discrete Optimization
Jelena Diakonikolas: Local Acceleration of Frank-Wolfe Methods
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From playlist Workshop: Continuous approaches to discrete optimization
John McCarthy: Norm-preserving extensions of bounded holomorphic functions
Recording during the meeting "Interpolation in Spaces of Analytic Functions" the November 18, 2019 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audio
From playlist Analysis and its Applications
Tropical Geometry - Lecture 9 - Tropical Convexity | Bernd Sturmfels
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From playlist Twelve Lectures on Tropical Geometry by Bernd Sturmfels
Suvrit Sra: Lecture series on Aspects of Convex, Nonconvex, and Geometric Optimization (Lecture 2)
The lecture was held within the framework of the Hausdorff Trimester Program "Mathematics of Signal Processing". (26.1.2016)
From playlist HIM Lectures: Trimester Program "Mathematics of Signal Processing"
Convex mirrors and ray diagrams, demonstrated and explained; from fizzics.org
Ray diagrams for convex mirrors are explained using demonstrations with laser light rays. You see what the image looks like and how to relate that to a ray diagram. Notes and many more video lessons available here https://www.fizzics.org/fizzics-guide/
From playlist Light, lenses and mirrors