Theory and Applications of Differential Equation Methods for Graph-based Learning

J.M. Budd

doi:10.4233/uuid:b8e0648c-d38e-4f95-bcd7-b99a943cb2d1

Theory and Applications of Differential Equation Methods for Graph-based Learning

J.M. Budd

Mathematical Physics

Research output: Thesis › Dissertation (TU Delft)

278 Downloads (Pure)

Abstract

A large number of modern learning problems involve working with highly interrelated and interconnected data. Graph-based learning is an emerging technique for approaching such problems, by representing this data as a graph (a.k.a. a network). That is, the points of data are represented by the vertices of the graph, and then the edges linking these vertices represent the relationships between the points of data. This provides a unified perspective for thinking about all sorts of interrelated data: the vertices could represent pixels in an image or people in a social network, and the underlying framework would be the same...

Original language	English
Qualification	Doctor of Philosophy
Awarding Institution	Delft University of Technology
Supervisors/Advisors	Dubbeldam, J.L.A., Supervisor van Gennip, Y., Advisor
Award date	25 Jan 2022
DOIs	https://doi.org/10.4233/uuid:b8e0648c-d38e-4f95-bcd7-b99a943cb2d1
Publication status	Published - 2022

Keywords

Graph dynamics
Allen-Cahn equation
Ginzburg–Landau functional
Merriman—Bence—Osher scheme
double-obstacle potential
Mass conservation
fidelity constraint
mean curvature flow
Image segmentation
joint reconstruction-segmentation

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10.4233/uuid:b8e0648c-d38e-4f95-bcd7-b99a943cb2d1

Thesis_final_Jeremy_buddFinal published version, 3.29 MB

3 Article

Classification and image processing with a semi-discrete scheme for fidelity forced Allen–Cahn on graphs
Budd, J., van Gennip, Y. & Latz, J., 2021, In: GAMM Mitteilungen. 44, 1, p. 1-43 43 p., e202100004.
Research output: Contribution to journal › Article › Scientific › peer-review

Open Access
File
6 Citations (Scopus)

53 Downloads (Pure)
Mass-conserving diffusion-based dynamics on graphs
Budd, J. M. & Van Gennip, Y., 2021, In: European Journal of Applied Mathematics. 33 (2022), 3, p. 423–471 49 p.
Research output: Contribution to journal › Article › Scientific › peer-review

Open Access
File
40 Downloads (Pure)
Graph Merriman-Bence-Osher as a semidiscrete implicit Euler scheme for graph Allen-Cahn flow
Budd, J. & van Gennip, Y., 2020, In: SIAM Journal on Mathematical Analysis. 52, 5, p. 4101–4139 39 p.
Research output: Contribution to journal › Article › Scientific › peer-review

Joint reconstruction segmentation code
Budd, J. M. (Creator), Latz, J. (Creator) & Parisotto, S. (Creator), TU Delft - 4TU.ResearchData, 15 Sept 2021
DOI: 10.17632/bgjcwk3v3x
Dataset/Software: Software
Image segmentation code
Budd, J. M. (Creator) & Latz, J. (Creator), TU Delft - 4TU.ResearchData, 15 Sept 2021
DOI: 10.17632/drkpzht233
Dataset/Software: Software

Cite this

@phdthesis{b8e0648cd38e4f95bcd7b99a943cb2d1,

title = "Theory and Applications of Differential Equation Methods for Graph-based Learning",

abstract = "A large number of modern learning problems involve working with highly interrelated and interconnected data. Graph-based learning is an emerging technique for approaching such problems, by representing this data as a graph (a.k.a. a network). That is, the points of data are represented by the vertices of the graph, and then the edges linking these vertices represent the relationships between the points of data. This provides a unified perspective for thinking about all sorts of interrelated data: the vertices could represent pixels in an image or people in a social network, and the underlying framework would be the same...",

keywords = "Graph dynamics, Allen-Cahn equation, Ginzburg–Landau functional, Merriman—Bence—Osher scheme, double-obstacle potential, Mass conservation, fidelity constraint, mean curvature flow, Image segmentation, joint reconstruction-segmentation",

author = "J.M. Budd",

year = "2022",

doi = "10.4233/uuid:b8e0648c-d38e-4f95-bcd7-b99a943cb2d1",

language = "English",

type = "Dissertation (TU Delft)",

school = "Delft University of Technology",

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TY - THES

T1 - Theory and Applications of Differential Equation Methods for Graph-based Learning

AU - Budd, J.M.

PY - 2022

Y1 - 2022

N2 - A large number of modern learning problems involve working with highly interrelated and interconnected data. Graph-based learning is an emerging technique for approaching such problems, by representing this data as a graph (a.k.a. a network). That is, the points of data are represented by the vertices of the graph, and then the edges linking these vertices represent the relationships between the points of data. This provides a unified perspective for thinking about all sorts of interrelated data: the vertices could represent pixels in an image or people in a social network, and the underlying framework would be the same...

AB - A large number of modern learning problems involve working with highly interrelated and interconnected data. Graph-based learning is an emerging technique for approaching such problems, by representing this data as a graph (a.k.a. a network). That is, the points of data are represented by the vertices of the graph, and then the edges linking these vertices represent the relationships between the points of data. This provides a unified perspective for thinking about all sorts of interrelated data: the vertices could represent pixels in an image or people in a social network, and the underlying framework would be the same...

KW - Graph dynamics

KW - Allen-Cahn equation

KW - Ginzburg–Landau functional

KW - Merriman—Bence—Osher scheme

KW - double-obstacle potential

KW - Mass conservation

KW - fidelity constraint

KW - mean curvature flow

KW - Image segmentation

KW - joint reconstruction-segmentation

U2 - 10.4233/uuid:b8e0648c-d38e-4f95-bcd7-b99a943cb2d1

DO - 10.4233/uuid:b8e0648c-d38e-4f95-bcd7-b99a943cb2d1

M3 - Dissertation (TU Delft)

ER -

Theory and Applications of Differential Equation Methods for Graph-based Learning

Abstract

Keywords

Access to Document

Fingerprint

Research output

Classification and image processing with a semi-discrete scheme for fidelity forced Allen–Cahn on graphs

Mass-conserving diffusion-based dynamics on graphs

Graph Merriman-Bence-Osher as a semidiscrete implicit Euler scheme for graph Allen-Cahn flow

Datasets

Joint reconstruction segmentation code

Image segmentation code

Cite this