Tractable Graph Matching via Soft Seeding

Fang, F.; Sussman, Daniel L.; Lyzinski, Vince

doi:10.48550/arxiv.1807.09299

Cited by 3 publications

(5 citation statements)

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“…Of course, the error for a local optimum will not necessarily be close to the error for a random permutation, so this result is not directly applicable. Another related result is in [27], which shows that under certain conditions there will be no local optimum which correctly align more than θ( √ n) vertices. This gap in the number of aligned vertices presumably leads to a gap in the objective function.…”

Section: Discussionmentioning

confidence: 99%

“…In this work, we employ a soft seeding approach [27] which uses the prior information to initialize the algorithm but does not insist that the correspondence for seeds be fixed; i.e., if the seeded vertices are [m] in both networks, we initialize the gradient descent graph matching algorithm at I m ⊕ D for suitable D in D n−m . Conversely, in the hard-seeded setting we optimize over Π n−m with the global solution being then of the form I m ⊕P for P ∈ Π n−m .…”

Section: Graph Matching Algorithmsmentioning

confidence: 99%

See 1 more Smart Citation

Matched Filters for Noisy Induced Subgraph Detection

Sussman

Park

Priebe

et al. 2019

IEEE Trans. Pattern Anal. Mach. Intell.

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The problem of finding the vertex correspondence between two noisy graphs with different number of vertices where the smaller graph is still large has many applications in social networks, neuroscience, and computer vision. We propose a solution to this problem via a graph matching matched filter: centering and padding the smaller adjacency matrix and applying graph matching methods to align it to the larger network. The centering and padding schemes can be incorporated into any algorithm that matches using adjacency matrices. Under a statistical model for correlated pairs of graphs, which yields a noisy copy of the small graph within the larger graph, the resulting optimization problem can be guaranteed to recover the true vertex correspondence between the networks. However, there are currently no efficient algorithms for solving this problem. To illustrate the possibilities and challenges of such problems, we use an algorithm that can exploit a partially known correspondence and show via varied simulations and applications to Drosophila and human connectomes that this approach can achieve good performance.

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Section: Discussionmentioning

confidence: 99%

Section: Graph Matching Algorithmsmentioning

confidence: 99%

Matched Filters for Noisy Induced Subgraph Detection

Sussman

Park

Priebe

et al. 2019

IEEE Trans. Pattern Anal. Mach. Intell.

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“…The above results range from providing theoretic phase transitions on matchability [14,15,38] to providing nearly efficient methods for achieving matchability from an algorithmic perspective [18,4,16,22]. While they have served to establish a novel theoretical understanding of the matchability problem, in each case the transition from matchable to unmatchable graphs is defined in terms of decreasing across-graph correlation and within-graph sparsity.…”

Section: Introductionmentioning

confidence: 99%

Matchability of heterogeneous networks pairs

Lyzinski

Sussman

2020

Information and Inference: A Journal of the IMA

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We consider the problem of graph matchability in non-identically distributed networks. In a general class of edge-independent networks, we demonstrate that graph matchability can be lost with high probability when matching the networks directly.We further demonstrate that under mild model assumptions, matchability is almost perfectly recovered by centering the networks using Universal Singular Value Thresholding before matching. These theoretical results are then demonstrated in both real and synthetic simulation settings. We also recover analogous core-matchability results in a very general core-junk network model, wherein some vertices do not correspond between the graph pair.of the recent applications and approaches to the graph matching problem, see the sequence of survey papers [13,24,20]. While recent results [3] have whittled away at the complexity of the related graph isomorphism problem-determining whether a permutation matrix P exists satisfying A = P BP T -at its most general, where A and B are allowed to be weighted and directed, the graph matching problem is known to be NP-hard. Indeed, in this case, the graph matching problem is equivalent to the notoriously difficult quadratic assignment problem [37,8,7]. However, recent approaches that leverage efficient representation/learning methodologies (see, for example, [5,59,26]) have shown excellent empirical performance matching networks with up to millions of nodes.In addition to algorithmic advancements in graph matching, there has been a flurry of activity studying the closely related problem of graph matchability: Given a latent alignment between the vertex sets of two graphs, can graph matching uncover this alignment in the presence of shuffled vertex labels? This problem arises in a variety of contexts, from network de-anonymization and privatization to multi-network hypothesis testing [38] to multimodality graph embedding methodologies [11]. Many existing results are concerned with recovering a latent alignment present across random graph models where each of A and B have identical marginal distributions, and exciting advancements on the threshold of matchable versus unmatchable graphs have been made across many random graph settings, including: the homogeneous correlated Erdős-Renyi model (see, for example, [46,40,4,18]), the correlated stochastic blockmodel setting (see, for example, [45,38]), the ρ-correlated heterogeneous Erdős-Renyi model (see, for example, [39,41]), and in the correlated heterogeneous Erdős-Renyi model with varying edge correlations (see, for example, [49,42]).In the non-identically distributed model setting, the work in [14,15,16] provide theoretic phase transitions on matchability in the (A, B) ∼Erdős-Rényi(p,q, ) model (i.e., A ∼Erdős-Rényi(n,p), B ∼Erdős-Rényi(n,q) and the edge correlation across graphs in provided by the constant ; see Definition 2).The above results range from providing theoretic phase transitions on matchability [14,15,38] to providing nearly efficient methods for achieving matchability from an algori...

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Section: Introductionmentioning

confidence: 99%

“…An arguably common perception, as suggested in Zhang (2018) and possibly implicitly in Fang et al (2018), too, is that learning even one seed in the unseeded setting may be hard. But in this paper, we will show that learning a set of "soft" seed nodes, where "soft" has similar meanings as that in (Fang et al, 2018), turns out to be easy. Then under smoothness assumption, the "soft" seeds we select would gradually become "hard" as the bias asymptotically diminishes, and even randomly sampled sets of nodes can serve as quite good seed sets, as long as their numbers slowly grows to infinity.…”

Section: Introductionmentioning

confidence: 99%

Consistent polynomial-time unseeded graph matching for Lipschitz graphons

Zhang¹

2018

Preprint

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We propose a consistent polynomial-time method for the unseeded node matching problem for networks with smooth underlying structures. Despite widely conjectured by the research community that the structured graph matching problem to be significantly easier than its worst case counterpart, well-known to be NP-hard, the statistical version of the problem has stood a challenge that resisted any solution both provable and polynomial-time. The closest existing work requires quasi-polynomial time. Our method is based on the latest advances in graphon estimation techniques and analysis on the concentration of empirical Wasserstein distances. Its core is a simple yet unconventional sampling-and-matching scheme that reduces the problem from unseeded to seeded. Our method allows flexible efficiencies, is convenient to analyze and potentially can be extended to more general settings. Our work enables a rich variety of subsequent estimations and inferences.

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Tractable Graph Matching via Soft Seeding

Cited by 3 publications

References 25 publications

Matched Filters for Noisy Induced Subgraph Detection

Matched Filters for Noisy Induced Subgraph Detection

Matchability of heterogeneous networks pairs

Consistent polynomial-time unseeded graph matching for Lipschitz graphons

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