Asymptotic Lipschitz Regularity for Tug-of-War Games with Varying Probabilities

Arroyo, Ángel; Luiro, Hannes; Parviainen, Mikko; Ruosteenoja, Eero

doi:10.1007/s11118-019-09778-8

Cited by 12 publications

(4 citation statements)

References 13 publications

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“…The method has been significantly generalized and applied to parabolic and elliptic equations [39,40,50,64,65]. Coupling methods in the discrete setting have also been used to establish Hölder and Lipschitz regularity in nonlinear potential theory, and in particular, for the p-Laplacian via the connection to stochastic two player tug-of-war games [2,3,34,44,47]. There are also recent application to Hölder regularity for the Robin problem [41,42].…”

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confidence: 99%

Lipschitz regularity of graph Laplacians on random data clouds

Calder¹,

Trillos²,

Lewicka³

2020

Preprint

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In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study arises in data analysis in the context of graph-based learning and contains, as important examples, the equations satisfied by graph Laplacian eigenvectors. In particular, we prove high probability interior and global Lipschitz estimates for solutions of graph Poisson equations. Our results can be used to show that graph Laplacian eigenvectors are, with high probability, essentially Lipschitz regular with constants depending explicitly on their corresponding eigenvalues. Our analysis relies on a probabilistic coupling argument of suitable random walks at the continuum level, and an interpolation method for extending functions on random point clouds to the continuum manifold. As a byproduct of our general regularity results, we obtain high probability L ∞ and approximate C 0,1 convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators. The convergence rates we obtain scale like the L 2 -convergence rates established in [13].

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confidence: 99%

Lipschitz regularity of graph Laplacians on random data clouds

Calder¹,

Trillos²,

Lewicka³

2020

Preprint

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“…This argument plays an important role in obtaining the desired estimate. Several regularity results for functions satisfying various time-independent DPPs were proved by calculations based on this argument (see [LP18,AHP17,ALPR]). It was proved in [PR16] that functions satisfying another time-dependent DPP have Hölder regularity.…”

Section: Hölder Regularitymentioning

confidence: 97%

Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle

Han¹

2020

Communications on Pure &Amp; Applied Analysis

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“…and the other estimate we obtain in Theorem 3.4 are and need to be necessarily asymptotic (notice the extra term ε 1+γ that appears in the right hand side). To the best of our knowledge, only Hölder and Lipschitz estimates, for example in [PS08, LPS13, LP18] as well as in [ALPR20] (under some additional assumptions, and different operators), have been so far available in this context.…”

Section: Introductionmentioning

confidence: 99%