The Moreau-Yosida Regularization: Difference between revisions
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Let <math>(X,d)</math> be a metric space. A function <math>g : X \to (-\infty,+\infty]</math> is said to be '''proper''' if it is not identically equal to <math>+\infty</math>, that is, if there exists <math>x \in X</math> such that <math>g(x) < +\infty</math>. | Let <math>(X,d)</math> be a metric space. A function <math>g : X \to (-\infty,+\infty]</math> is said to be '''proper''' if it is not identically equal to <math>+\infty</math>, that is, if there exists <math>x \in X</math> such that <math>g(x) < +\infty</math>. | ||
For a given function <math>g : X \to (-\infty,+\infty]</math> and <math>k \geq 0</math>, its '''Moreau-Yosida regularization''' <math>g_k : X \to (-\infty,+\infty]</math> is given by | |||
:<math>g_k(x) := \inf\limits_{y \in X} \left[ g(y) + k d(x,y) \right] | :<math>g_k(x) := \inf\limits_{y \in X} \left[ g(y) + k d(x,y) \right].</math> | ||
==Examples== | ==Examples== | ||
Revision as of 21:18, 21 January 2022
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Motivation
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Definitions
Let be a metric space. A function is said to be proper if it is not identically equal to , that is, if there exists such that .
![{\displaystyle g:X\to (-\infty ,+\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb30feface8bba531b0a2ad5b5dffc066ed4acbd)
For a given function and , its Moreau-Yosida regularization is given by
![{\displaystyle g_{k}:X\to (-\infty ,+\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b718f4b10f7cd1014c845ca702eafc9eab1aa223)
![{\displaystyle g_{k}(x):=\inf \limits _{y\in X}\left[g(y)+kd(x,y)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22a6e02a5f403ff1a143b2dce1719ad675b26942)
Examples
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References
Possible list of references, will fix accordingly
Bauschke-Combette Ch 12.[1]; Santambrogio (6)[2]; Ambrosio-Gigli-Savare (59-61)[3]
- ↑ Bauschke, Heinz H. and Patrick L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd Ed. Ch. 12. Springer, 2017.
- ↑ Santambrogio, Filippo. Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling Ch. 1.1. Birkhäuser, 2015.
- ↑ Ambrosio, Luigi, Nicola Gigli, and Giuseppe Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Ch. 3.1. Birkhäuser, 2005.