Dirichlet Graph Densifiers

2018 24th International Conference on Pattern Recognition (ICPR)

Escolano

Lozano

et al. 2018

Self Cite

Section: Methodsmentioning

confidence: 99%

“…This motivated the development of an unsupervised and scalable method for graph densification. Dirichlet graph densifiers [13] rely on two elements: a) return random walks and b) a Dirichlet process. Return random walks are designed to retain the diffusion process inside each cluster (i.e.…”

Section: Introductionmentioning

confidence: 99%

Semi-supervised Graph Rewiring with the Dirichlet Principle

2018 24th International Conference on Pattern Recognition (ICPR)

Escolano

Lozano

et al. 2018

Self Cite

“…In [13] we develop a novel densifier, which infers new intra-class edges while minimizing the number of new inter-class edges. To this end, we proceed to design a structural filter, using Return Random Walks (RRW), and then we build the line graph and run a Dirichlet process on it.…”

Section: Dirichlet Densifiersmentioning

confidence: 99%

“…In [12] we highlighted the fact that densification leads to a shrinkage of the inter-cluster distances, thus making Commute Times meaningful in large graphs. Later on, in [13], we highlighted the fact that state-of-the-art densifiers rely on semi-definite programming and motivate a novel algorithm, which is more scalable and robust. The core of this algorithm is harmonic analysis.…”

Section: Introductionmentioning

confidence: 99%

Dirichlet densifier bounds: Densifying beyond the spectral gap constraint

Pattern Recognition Letters

Lozano

Escolano

et al. 2019

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In this paper, we characterize the universal bounds of our recently reported Dirichlet Densifier. In particular we aim to study the impact of densification on the bounding of intra-class node similarities. To this end we derive a new bound for commute time estimation. This bound does not rely on the spectral gap, but on graph densification (or graph rewiring). Firstly, we explain how our densifier works and we motivate the bound by showing that implicitly constraining the spectral gap through graph densification cannot fully explain the cluster structure in real-world datasets. Then, we pose our hypothesis about densification: a graph densifier can only deal with a moderate degradation of the spectral gap if the inter-cluster commute distances are significantly shrunk. This points to a more detailed bound which explicitly accounts for the shrinking effect of densification. Finally, we formally develop this bound, thus revealing the deeper implications of graph densification in commute time estimation.

“…5. In particular, Escolano et al [5] showed that densifying G significantly decreases the spectral gap which in turn enlarges the von Luxburg bound. As a result, effective resistances do not depend only on local properties and become meaningful for large graphs provided that these graphs have been properly densified.…”

Section: Motivation Of the Experimental Setupmentioning

confidence: 99%

On the Interplay Between Strong Regularity and Graph Densification

Fiorucci

Torcinovich

Graph-Based Representations in Pattern Recognition

et al. 2017

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In this paper we analyze the practical implications of Szemerédi's regularity lemma in the preservation of metric information contained in large graphs. To this end, we present a heuristic algorithm to find regular partitions. Our experiments show that this method is quite robust to the natural sparsification of proximity graphs. In addition, this robustness can be enforced by graph densification.The aim of this work is to analyze the ideal density regime where the regularity lemma can find useful applications. In particular, we use the regularity lemma to reduce an input graph and we then exploit the key lemma to obtain an expanded version which preserves some topological properties of the original graph. If we are out of the ideal density regime, we have to densify the graph before applying the regularity lemma. Among the many topological measures we test the effective resistance (or equivalently the scaled commute time), one of the most important metrics between the vertices in the graph, which has been very recently questioned. In [12] it is argued that this measure is meaningless for large graphs. However, recent experimental results show that the graph can be pre-processed (densified) to provide some informative estimation of this metric [5] [4]. Therefore, in this paper, we analyze the practical implications of the key lemma in the estimation of commute time in large graphs.