Timeline for $L^2$-projection onto monotone functions
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 6 at 2:50 | comment | added | ViktorStein | In A Wasserstein approach to the one-dimensional sticky particle system, Natile and Savaré prove in Thm 3.1 that the projection of $f \in L_2(0, 1)$ to the set of nondecreasing functions (identified with their right-continuous representative) is given by $D^+ F^{**}$, where $F^{**}$ is the convex envelope of the primitive of $f$ and $D^+$ denotes the derivative from the right. | |
| Mar 24, 2021 at 14:19 | comment | added | Denis Serre | @WillieWong. Sketch of proof: on the one hand $\phi'\in K$ (obvious). Then because $K$ is convex and the distance is euclidian, it is enough to prove $\langle f-\phi',\phi'-g\rangle\ge0$ for every $g\in K$. This is the sum of integrals over disjoint integrals of term $\int\theta'hdx$, where $\theta=\psi-\phi\ge0$ vanishes on the boundaries, and $h=g-\phi'$ (here $\phi'$ is constant) is non-decreasing. Integration by parts give the correct sign. | |
| Mar 24, 2021 at 13:16 | comment | added | Willie Wong | Yes, the hard thing for me was proving that the formula above has the right properties that you would expect of $\phi'$ (specifically that when $\pi f \neq f$ almost surely there exists an interval on which $\pi f$ is constant). After that the proof that it is the optimizer is short. Since the formula is the min-max of the difference quotient of the primitive, there's probably a shorter proof just by directly connecting to $\phi'$. | |
| Mar 24, 2021 at 7:01 | comment | added | Denis Serre | @WillieWong. Thank you so much for this close formula. I'll check soon that it coincides with $\phi'$. As to the fact that $\phi'$ is the projection of $f$, I have written up a one-page proof. | |
| Mar 23, 2021 at 15:34 | comment | added | Willie Wong | I can confirm your guess is true. Another formula for $\pi f$ is $$ \pi f(x) = \inf_{z > x} \sup_{y \leq x} \frac{1}{z-y} \int_y^z f(s) ~ds $$ I would also love a reference for it, especially if there is a short convex analysis proof. It took me a few days to work out all the details of a brute force proof and my proof is 12 pages long. | |
| Mar 17, 2021 at 21:48 | history | edited | Denis Serre | CC BY-SA 4.0 |
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| Mar 17, 2021 at 12:51 | history | asked | Denis Serre | CC BY-SA 4.0 |