Motivations, ideas and applications of ramified optimal transportation

Xia, Qinglan

doi:10.1051/m2an/2015028

Cited by 27 publications

(29 citation statements)

References 28 publications

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“…Note that the usual 2 -distance does not work since φ i − φ j 2 = 2δ i j where δ i j is the Kronecker delta. Here, we propose to derive a natural distance between eigenvectors using the ideas gained from the ramified optimal transport theory [8], i.e., we view the cost to "transport" one eigenvector to another eigenvector as the natural distance between two such eigenvectors.…”

Section: Problems With Ordering Eigenvectors According To the Correspmentioning

confidence: 99%

“…The Ramified Optimal Transport (ROT) theory [8] is a branch of more general optimal transport theory [9]: it studies transporting "mass" from one probability measure µ + to another µ − along ramified transport paths with some specific transport cost functional, and has been used to analyze various branching structures, e.g., trees; veins on a leaf; cardiovascular systems; river channel networks; electrical grids; communication networks, to name a few. For simplicity, we only consider the discrete case: two probability mass functions (pmfs) in R d .…”

Section: A Brief Review Of Ramified Optimal Transport Theorymentioning

confidence: 99%

“…is a metric on the space of pmfs and of homogeneous of degree α; and moreover he derived numerical algorithms to generate the α-optimal path for a given pair ( f , g ); see [8] and the refereces therein for further information.…”

Section: A Brief Review Of Ramified Optimal Transport Theorymentioning

confidence: 99%

“…Hence, a fundamental question is how to order and organize Laplacian eigenvectors without using the eigenvalues, i.e., how to create a dual domain of a given graph. In this paper, we investigate this important problem further and propose a new method to order and organize those eigenvectors using the idea from the Ramified Optimal Transport (ROT) theory [8] to measure "natural" distances between eigenvectors followed by embedding them into a low-dimensional Euclidean domain.…”

Section: Introductionmentioning

confidence: 99%

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How Can We Naturally Order and Organize Graph Laplacian Eigenvectors?

Saito

2018

2018 IEEE Statistical Signal Processing Workshop (SSP)

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When attempting to develop wavelet transforms for graphs and networks, some researchers have used graph Laplacian eigenvalues and eigenvectors in place of the frequencies and complex exponentials in the Fourier theory for regular lattices in the Euclidean domains. This viewpoint, however, has a fundamental flaw: on a general graph, the Laplacian eigenvalues cannot be interpreted as the frequencies of the corresponding eigenvectors. In this paper, we discuss this important problem further and propose a new method to organize those eigenvectors by defining and measuring "natural" distances between eigenvectors using the Ramified Optimal Transport Theory followed by embedding them into a low-dimensional Euclidean domain. We demonstrate its effectiveness using a synthetic graph as well as a dendritic tree of a retinal ganglion cell of a mouse.

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Section: Problems With Ordering Eigenvectors According To the Correspmentioning

confidence: 99%

Section: A Brief Review Of Ramified Optimal Transport Theorymentioning

confidence: 99%

Section: A Brief Review Of Ramified Optimal Transport Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

How Can We Naturally Order and Organize Graph Laplacian Eigenvectors?

Saito

2018

2018 IEEE Statistical Signal Processing Workshop (SSP)

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“…The theory of ramified transport for general measures was developed independently in [12] and [19]. See also [1] for a comprehensive introduction, and [20] for a survey of the field. Further results on optimal ramified transport can be found in [2,5,13,14,17].…”

Section: Introductionmentioning

confidence: 99%

Variational problems for tree roots and branches

2019

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This paper studies two classes of variational problems introduced in [7], related to the optimal shapes of tree roots and branches. Given a measure µ describing the distribution of leaves, a sunlight functional S(µ) computes the total amount of light captured by the leaves. For a measure µ describing the distribution of root hair cells, a harvest functional H(µ) computes the total amount of water and nutrients gathered by the roots. In both cases, we seek a measure µ that maximizes these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk or from the trunk to the leaves.Compared with [7], here we do not impose any a priori bound on the total mass of the optimal measure µ, and more careful a priori estimates are thus required. In the unconstrained optimization problem for branches, we prove that an optimal measure exists, with bounded support and bounded total mass. In the unconstrained problem for tree roots, we prove that an optimal measure exists, with bounded support but possibly unbounded total mass. The last section of the paper analyzes how the size of the optimal tree depends on the parameters defining the various functionals.MSC: 35R06, 49Q10, 92C80.

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Partial Plateau’s problem with H-mass

Alvarado,

Xia

2024

Calc. Var.

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Classically, Plateau’s problem asks to find a surface of the least area with a given boundary B. In this article, we investigate a version of Plateau’s problem, where the boundary of an admissible surface is only required to partially span B. Our boundary data is given by a flat $$(m-1)$$ ( m - 1 ) -chain B and a smooth compactly supported differential $$(m-1)$$ ( m - 1 ) -form $$\Phi $$ Φ . We are interested in minimizing $$ \textbf{M}(T) - \int _{\partial T} \Phi $$ M ( T ) - ∫ ∂ T Φ over all m-dimensional rectifiable currents T in $$\mathbb {R}^n$$ R n such that $$\partial T$$ ∂ T is a subcurrent of the given boundary B. The existence of a rectifiable minimizer is proven with Federer and Fleming’s compactness theorem. We generalize this problem by replacing the mass $$\textbf{M}$$ M with the H-mass of rectifiable currents. By minimizing over a larger class of objects, called scans with boundary, and by defining their H-mass as a type of lower-semicontinuous envelope over the H-mass of rectifiable currents, we prove an existence result for this problem by using Hardt and De Pauw’s BV compactness theorem.

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Motivations, ideas and applications of ramified optimal transportation

Abstract: In this survey article, the author summarizes the motivations, key ideas and main applications of ramified optimal transportation that the author has studied in recent years.

Cited by 27 publications

References 28 publications

How Can We Naturally Order and Organize Graph Laplacian Eigenvectors?

How Can We Naturally Order and Organize Graph Laplacian Eigenvectors?

Variational problems for tree roots and branches

Partial Plateau’s problem with H-mass

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