Wasserstein distance, Fourier series and applications

Steinerberger, Stefan

doi:10.1007/s00605-020-01497-2

Cited by 25 publications

(19 citation statements)

References 23 publications

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“…with Neumann boundary conditions; note that to simplify notation we dispense with the tilde notation ψ and ũ from §3.4 and just write ψ and u. In order to precondition (27) we define U and Ψ by…”

Section: 2mentioning

confidence: 99%

“…Several authors have considered the connection between negative homogeneous Sobolev norms and the W 2 metric in several different contexts. In analysis, this connection has been used to establish estimates by many authors, see for example [13,15,23,26,27]. Moreover, this connection also arises in the study of how measures change under heat diffusion, see [5,21,30].…”

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confidence: 99%

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On a linearization of quadratic Wasserstein distance

Greengard¹,

Hoskins²,

Marshall³

et al. 2022

Preprint

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This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance W 2 . In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving a elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form H = −∆ + V , and describe an associated embedding. Third, for the case of probability distributions on the unit square [0, 1] 2 represented by n × n arrays we present a fast code demonstrating our approach. Several numerical examples are presented.

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Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

On a linearization of quadratic Wasserstein distance

Greengard¹,

Hoskins²,

Marshall³

et al. 2022

Preprint

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“…and {kα}, the decimal part of kα, is approximately 2 −ℓ -separated. Using these facts we mimic the computation in [Ste21] to obtain…”

Section: Introductionmentioning

confidence: 99%

A study in quantitative equidistribution on the unit square

Goering¹,

Weiß²

2022

Preprint

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The distributional properties of the translation flow on the unit square have been considered in different fields of mathematics, including algebraic geometry and discrepancy theory. One method to quantify equidistribution is to compare the error between the actual time the translation flow spent in specific sets E ⊂ [0, 1] 2 to the expected time. In this article, we prove that when E is in the algebra generated by convex sets the error is of order at most log(T ) 1+ε for all but countably many directions. Whenever the direction is badly approximable the bound can be sharpened to log(T ) 1/2+ε . The error estimates we produce are smaller than for general measurable sets as proved by Beck, while our class of examples is larger than in the work of Grepstad-Larcher who obtained the bounded remainder property for their sets. Our proof relies on the duality between local convexity of the boundary and regularity of sections of the flow.

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“…In other words, the work done to "move" the positive mass to the negative mass should be large. This has been subsequently used in [M] to prove a conjectured sharp lower bound on the Wasserstein distance between the measures (see [St1]). • Lastly, one can ask the natural question as to whether versions of Theorem 1.1 could hold for p > 1.…”

Section: Introductionmentioning

confidence: 99%

Some applications of heat flow to Laplace eigenfunctions

Georgiev¹,

Mukherjee²

2021

Preprint

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We consider mass concentration properties of Laplace eigenfunctions ϕ λ , that is, smooth functions satisfying the equation −∆ϕ λ = λϕ λ , on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central "thick" part and "thin" elongated branches representing tunnels of sub-wavelength opening. Finally, in an Appendix, we record some new observations regarding sub-level sets of the eigenfunctions and interactions of different level sets.

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Wasserstein distance, Fourier series and applications

Cited by 25 publications

References 23 publications

On a linearization of quadratic Wasserstein distance

On a linearization of quadratic Wasserstein distance

A study in quantitative equidistribution on the unit square

Some applications of heat flow to Laplace eigenfunctions

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