Sketching Earth-Mover Distance on Graph Metrics

McGregor, Andrew; Stubbs, Daniel M.

doi:10.1007/978-3-642-40328-6_20

Cited by 13 publications

(8 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…That method was used to compare microbial communities by measuring the phylogenetic distance between sets of taxa in a phylogenetic tree as the fraction of the branch length of the tree that leads to descendants from either one environment or the other, but not both [44]. In this section, we follow [15,21,46] to leverage the geometric structure of tree metrics, and recall their main result. Let T be a tree rooted at r with non-negative edge lengths, and d T be the tree metric on T .…”

Section: Reminders On Optimal Transport and Tree Metricsmentioning

confidence: 99%

“…In this work, we follow the first direction to provide a fast computation for OT. To do so, we consider tree metrics as ground costs for OT, which results in the so-called tree-Wasserstein (TW) distance [15,21,46]. We consider two practical procedures to sample tree metrics based on spatial information for both low-dimensional and high-dimensional spaces of supports.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Tree-Sliced Variants of Wasserstein Distances

Le¹,

Yamada²,

Fukumizu³

et al. 2019

Preprint

View full text Add to dashboard Cite

Optimal transport (OT) theory defines a powerful set of tools to compare probability distributions. OT suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants of OT in the recent literature, one of the most notable being the sliced formulation, which exploits the closed-form formula between univariate distributions by projecting high-dimensional measures onto random lines. We consider in this work a more general family of ground metrics, namely tree metrics, which also yield fast closed-form computations and negative definite, and of which the sliced-Wasserstein distance is a particular case (the tree is a chain). We propose the tree-sliced Wasserstein distance, computed by averaging the Wasserstein distance between these measures using random tree metrics, built adaptively in either low or high-dimensional spaces. Exploiting the negative definiteness of that distance, we also propose a positive definite kernel, and test it against other baselines on a few benchmark tasks.

show abstract

Section: Reminders On Optimal Transport and Tree Metricsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tree-Sliced Variants of Wasserstein Distances

Le¹,

Yamada²,

Fukumizu³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The time complexity of both algorithms, however, is supercubic in n. Thus, a precise computation of SND using existing techniques is prohibitively expensive at the scale of real-world online social networks. Furthermore, the existing approximations of EMD are either not applicable to the comparison of histograms derived from the states of a social network, since they drastically simplify the ground distance [26,17], or are effective only for some graphs, such as trees, structurally not characteristic of social networks [20].…”

Section: Efficient Computation Of Sndmentioning

confidence: 99%

A Distance Measure for the Analysis of Polar Opinion Dynamics in Social Networks

Amelkin

Bogdanov

Singh

2019

ACM Trans. Knowl. Discov. Data

View full text Add to dashboard Cite

Analysis of opinion dynamics in social networks plays an important role in today's life. For applications such as predicting users' political preference, it is particularly important to be able to analyze the dynamics of competing opinions. While observing the evolution of polar opinions of a social network's users over time, can we tell when the network "behaved" abnormally? Furthermore, can we predict how the opinions of the users will change in the future? Do opinions evolve according to existing network opinion dynamics models? To answer such questions, it is not sufficient to study individual user behavior, since opinions can spread far beyond users' egonets. We need a method to analyze opinion dynamics of all network users simultaneously and capture the effect of individuals' behavior on the global evolution pattern of the social network.In this work, we introduce Social Network Distance (SND) -a distance measure that quantifies the "cost" of evolution of one snapshot of a social network into another snapshot under various models of polar opinion propagation. SND has a rich semantics of a transportation problem, yet, is computable in time linear in the number of users, which makes SND applicable to the analysis of large-scale online social networks. In our experiments with synthetic and realworld Twitter data, we demonstrate the utility of our distance measure for anomalous event detection. It achieves a true positive rate of 0.83, twice as high as that of alternatives. When employed for opinion prediction in Twitter, our method's accuracy is 75.63%, which is 7.5% higher than that of the next best method.otherwise.

show abstract

“…In the above categorization, this is an optimization-based spectral distance measure, but is implicit, since it does not produce a map from vertices of G 1 to those of G 2 (informally, this is because the EMD is not translating one set of eigenvalues into the other, but instead transforming their respective histograms). Recent work applying the EMD to graph classification includes [9] and [26].…”

Section: Graph-edit Distance (Structural Explicit Disc-opt)mentioning

confidence: 99%

Novel diffusion-derived distance measures for graphs

Scott,

Mjolsness

2019

Preprint

View full text Add to dashboard Cite

We define a new family of similarity and distance measures on graphs, and explore their theoretical properties in comparison to conventional distance metrics. These measures are defined by the solution(s) to an optimization problem which attempts find a map minimizing the discrepancy between two graph Laplacian exponential matrices, under norm-preserving and sparsity constraints. Variants of the distance metric are introduced to consider such optimized maps under sparsity constraints as well as fixed time-scaling between the two Laplacians. The objective function of this optimization is multimodal and has discontinuous slope, and is hence difficult for univariate optimizers to solve. We demonstrate a novel procedure for efficiently calculating these optima for two of our distance measure variants. We present numerical experiments demonstrating that (a) upper bounds of our distance metrics can be used to distinguish between lineages of related graphs; (b) our procedure is faster at finding the required optima, by as much as a factor of 10 3 ; and (c) the upper bounds satisfy the triangle inequality exactly under some assumptions and approximately under others. We also derive an upper bound for the distance between two graph products, in terms of the distance between the two pairs of factors. Additionally, we present several possible applications, including the construction of infinite "graph limits" by means of Cauchy sequences of graphs related to one another by our distance measure.

show abstract

Sketching Earth-Mover Distance on Graph Metrics

Cited by 13 publications

References 13 publications

Tree-Sliced Variants of Wasserstein Distances

Tree-Sliced Variants of Wasserstein Distances

A Distance Measure for the Analysis of Polar Opinion Dynamics in Social Networks

Novel diffusion-derived distance measures for graphs

Contact Info

Product

Resources

About