Auction dynamics: A volume constrained MBO scheme

Jacobs, Matt; Merkurjev, Ekaterina; Esedoḡlu, Selim

doi:10.1016/j.jcp.2017.10.036

Cited by 49 publications

(70 citation statements)

References 24 publications

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“…To numerically implement (2), a straightforward scheme is to incorporate volume constraints with Lagrange multipliers, however, a more efficient approach employs auction algorithms (Jacobs et al, 2018).…”

Section: Level Set-based Approachmentioning

confidence: 99%

“…, but implementing a scaling in this parameter improves both computational time and accuracy (Jacobs et al, 2018).…”

Section: Level Set-based Approachmentioning

confidence: 99%

“…It is possible to modify the auction algorithm to extend it to this type of constraint. The implementation of the upper bound does not significantly change the algorithm but the lower bound requires running also a reverse auction where cell regions bid on points (see (Jacobs et al, 2018), Section 3.3). One can also incorporate random effects in the algorithm -either by randomly changing cell volumes (see (Jacobs et al, 2018), Section 3.4) or by adding a suitable noise to the weights σ ij .…”

Section: Parameters Of the Algorithmmentioning

confidence: 99%

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A level set-based approach for modeling cellular rearrangements in tissue morphogenesis

Mohammad

Murakawa

Svadlenka

et al. 2020

Preprint

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Mathematical models and numerical simulations can provide an essential insight into the mechanisms through which local cell-cell interactions affect tissue-level cell morphology. Among such morphological phenomena, cellular patterns observed in developing sensory epithelia have gained keen attention of researchers in recent years, because they are thought to be of utmost importance for accurate sensory functions. However, most of current computational approaches to cellular rearrangements lack solid mathematical background and involve experimentally unreachable parameters, whereby only weak and ambiguous conclusions can be made based on simulation results. Here we present a simple mathematical model for tissue morphogenesis together with a level set-based numerical scheme for its solution as a tool to rigorously investigate evolving cellular patterns. This combined framework of a model and a numerical method features minimum possible number of physical parameters and guarantees reliability of simulation results, including correct handling of topology changes, such as cell intercalations. In this framework, we adopt the viewpoint of free energy minimization principle, and take cellular rearrangement as a gradient flow of a weighted surface energy associated with cell membrane, where the weights are related to physical parameters of the cells, for example, cell-cell adhesion and cell contractility. We present the applicability of this model to a wide range of tissue morphological phenomena, such as cell sorting, engulfment or internalization. In particular, we stress that this method is the first one to be successful in computationally reproducing the experimentally observed development of cellular mosaic patterns in sensory epithelia. Thanks to its simplicity and reliability, the model is able to capture the essence of biological phenomena, and may give a strong helping hand in deciphering unsolved questions of morphology.

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Section: Level Set-based Approachmentioning

confidence: 99%

“…, but implementing a scaling in this parameter improves both computational time and accuracy (Jacobs et al, 2018).…”

Section: Level Set-based Approachmentioning

confidence: 99%

Section: Parameters Of the Algorithmmentioning

confidence: 99%

See 1 more Smart Citation

A level set-based approach for modeling cellular rearrangements in tissue morphogenesis

Mohammad

Murakawa

Svadlenka

et al. 2020

Preprint

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“…The main idea is to approximate the cut term using the GL functional and then use PDE-based methods for gradient descent in a spectral approach. Jacobs, Merkurjev, and Esedoğlu [2018] have a very efficient MBO-based method for solving the volume-constrained classification problem with different phases and with prescribed volume constraints and volume inequalities for the different phases. This work combines some of the best features of the MBO scheme in both Euclidean space and on graphs with a highly efficient algorithm of Bertsekas [1979] for the auction problem with volume constraints.…”

Section: Volume Penaltiesmentioning

confidence: 99%

Graphical Models in Machine Learning, Networks and Uncertainty Quantification

Bertozzi

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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This paper is a review article on semi-supervised and unsupervised graph models for classification using similarity graphs and for community detection in networks. The paper reviews graph-based variational models built on graph cut metrics. The equivalence between the graph mincut problem and total variation minimization on the graph for an assignment function allows one to cast graph-cut variational problems in the language of total variation minimization, thus creating a parallel between low dimensional data science problems in Euclidean space (e.g. image segmentation) and high dimensional clustering. The connection paves the way for new algorithms for data science that have a similar structure to well-known computational methods for nonlinear partial differential equations. This paper focuses on a class of methods build around diffuse interface models (e.g. the Ginzburg-Landau functional and the Allen-Cahn equation) and threshold dynamics, developed by the Author and collaborators. Semi-supervised learning with a small amount of training data can be carried out in this framework with diverse applications ranging from hyperspectral pixel classification to identifying activity in police body worn video. It can also be extended to the context of uncertainty quantification with Gaussian noise models. The problem of community detection in networks also has a graph-cut structure and algorithms are presented for the use of threshold dynamics for modularity optimization. With efficient methods, this allows for the use of network modularity for unsupervised machine learning problems with unknown number of classes.

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“…Recently, Jacobs showed how to apply techniques from models of crystal growth to graph-cut problems from semisupervised learning [43]. (See [44] for additional related work.) Several other recent papers, which do not directly involve surface tension, have used ideas from perimeter minimization and/or TV minimization for graph cuts and clustering in machine learning [8].…”

mentioning

confidence: 99%

Stochastic Block Models are a Discrete Surface Tension

Boyd

Porter²,

Bertozzi³

2019

J Nonlinear Sci

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Networks, which represent agents and interactions between them, arise in myriad applications throughout the sciences, engineering, and even the humanities. To understand large-scale structure in a network, a common task is to cluster a network's nodes into sets called "communities", such that there are dense connections within communities but sparse connections between them. A popular and statistically principled method to perform such clustering is to use a family of generative models known as stochastic block models (SBMs). In this paper, we show that maximum likelihood estimation in an SBM is a network analog of a well-known continuum surface-tension problem that arises from an application in metallurgy.To illustrate the utility of this relationship, we implement network analogs of three surface-tension algorithms, with which we successfully recover planted community structure in synthetic networks and which yield fascinating insights on empirical networks that we construct from hyperspectral videos.(MBO) scheme, geometric partial differential equations AMS subject classifications. 65K10, 49M20, 35Q56, 62H30, 91C20, 91D30, 94C151. Introduction. The study of networks, in which nodes represent entities and edges encode interactions between entities [66], can provide useful insights into a wide variety of complex systems in myriad fields, such as granular materials [73], disease spreading [74], criminology [37], and more. In the study of such applications, the analysis of large data sets -from diverse sources and applications -continues to grow ever more important.The simplest type of network is a graph, and empirical networks often appear to exhibit a complicated mixture of regular and seemingly random features [66]. Additionally, it is increasingly important to study networks with more complicated features, such as time-dependence [39], multiplexity [50], annotations [67], and connections that go beyond a pairwise paradigm [72]. One also has to worry about "features" such as missing information and false positives [48]. Nevertheless, it is convenient in the present paper to restrict our attention to undirected, unweighted graphs for simplicity.To try to understand the large-scale structure of a network, it can be very insightful to coarse-grain it in various ways [27,79,81,83,84]. The most popular type of clustering is the detection of assortative "communities," in which dense sets of nodes are connected sparsely to other dense sets of nodes [27,81]. A statistically principled approach is to treat community detection as a statistical inference problem using a model such * Submitted to the editors DATE.

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Auction dynamics: A volume constrained MBO scheme

Cited by 49 publications

References 24 publications

A level set-based approach for modeling cellular rearrangements in tissue morphogenesis

A level set-based approach for modeling cellular rearrangements in tissue morphogenesis

Graphical Models in Machine Learning, Networks and Uncertainty Quantification

Stochastic Block Models are a Discrete Surface Tension

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