Compact Graph Research Articles

Graph-based multiple rank regression for image classification

Abstract Image classification is one important task in image processing and pattern recognition. Traditional image classification methods commonly transform the image into a vector. However, in essence, image is a matrix data and using vector instead of image loses the correlations of the matrix data. To address this problem, we propose a graph-based multiple rank regression model (GMRR), which employs multiple-rank left and right projecting vectors to regress each matrix data to its label for each category. To exploit the discriminating structure of the data space, a class compactness graph is constructed to constrain these left and right projecting vectors. The extensive experimental results on image classification have demonstrated the effectiveness of our proposed method.

Quantifying Topological Uncertainty in Fractured Systems using Graph Theory and Machine Learning

Fractured systems are ubiquitous in natural and engineered applications as diverse as hydraulic fracturing, underground nuclear test detection, corrosive damage in materials and brittle failure of metals and ceramics. Microstructural information (fracture size, orientation, etc.) plays a key role in governing the dominant physics for these systems but can only be known statistically. Current models either ignore or idealize microscale information at these larger scales because we lack a framework that efficiently utilizes it in its entirety to predict macroscale behavior in brittle materials. We propose a method that integrates computational physics, machine learning and graph theory to make a paradigm shift from computationally intensive high-fidelity models to coarse-scale graphs without loss of critical structural information. We exploit the underlying discrete structure of fracture networks in systems considering flow through fractures and fracture propagation. We demonstrate that compact graph representations require significantly fewer degrees of freedom (dof) to capture micro-fracture information and further accelerate these models with Machine Learning. Our method has been shown to improve accuracy of predictions with up to four orders of magnitude speedup.

Practical dynamic de Bruijn graphs.

The de Bruijn graph is fundamental to the analysis of next generation sequencing data and so, as datasets of DNA reads grow rapidly, it becomes more important to represent de Bruijn graphs compactly while still supporting fast assembly. Previous implementations of compact de Bruijn graphs have not supported node or edge deletion, however, which is important for pruning spurious elements from the graph. Belazzougui et al. (2016b) recently proposed a compact and fully dynamic representation, which supports exact membership queries and insertions and deletions of both nodes and edges. In this paper, we give a practical implementation of their data structure, supporting exact membership queries and fully dynamic edge operations, as well as limited support for dynamic node operations. We demonstrate experimentally that its performance is comparable to that of state-of-the-art implementations based on Bloom filters. Our source-code is publicly available at https://github.com/csirac/dynamicDBG under an open-source license.

Mandible and skull segmentation in cone beam computed tomography using super-voxels and graph clustering

Cone beam computed tomography (CBCT) is a medical imaging technique employed for diagnosis and treatment of patients with cranio-maxillofacial deformities. CBCT 3D reconstruction and segmentation of bones such as mandible or maxilla are essential procedures in surgical and orthodontic treatments. However, CBCT image processing may be impaired by features such as low contrast, inhomogeneity, noise and artifacts. Besides, values assigned to voxels are relative Hounsfield units unlike traditional computed tomography (CT). Such drawbacks render CBCT segmentation a difficult and time-consuming task, usually performed manually with tools designed for medical image processing. We present an interactive two-stage method for the segmentation of CBCT: (i) we first perform an automatic segmentation of bone structures with super-voxels, allowing a compact graph representation of the 3D data; (ii) next, a user-placed seed process guides a graph partitioning algorithm, splitting the extracted bones into mandible and skull. We have evaluated our segmentation method in three different scenarios and compared the results with ground truth data of the mandible and the skull. Results show that our method produces accurate segmentation and is robust to changes in parameters. We also compared our method with two similar segmentation strategy and showed that it produces more accurate segmentation. Finally, we evaluated our method for CT data of patients with deformed or missing bones and the segmentation was accurate for all data. The segmentation of a typical CBCT takes in average 5 min, which is faster than most techniques currently available.

On locally compact semitopological graph inverse semigroups

In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph $E$ is strongly connected and contains a finite amount of vertices then a locally compact semitopological graph inverse semigroup $G(E)$ is either compact or discrete. This result generalizes results of Gutik and Bardyla who proved the above dichotomy for locally compact semitopological polycyclic monoids $\mathcal{P}_1$ and $\mathcal{P}_{\lambda}$, respectively.

Time-Variant Graph Classification

Graphs are commonly used to represent objects, such as images and text, for pattern classification. In a dynamic world, an object may continuously evolve over time, and so does the graph extracted from the underlying object. These changes in graph structure with respect to the temporal order present a new representation of the graph, in which an object corresponds to a set of time-variant graphs. In this paper, we formulate a novel time-variant graph classification task and propose a new graph feature, called a graph-shapelet pattern , for learning and classifying time-variant graphs. Graph-shapelet patterns are compact and discriminative graph transformation subsequences . A graph-shapelet pattern can be regarded as a graphical extension of a shapelet —a class of discriminative features designed for vector-based temporal data classification. To discover graph-shapelet patterns, we propose to convert a time-variant graph sequence into time-series data and use the discovered shapelets to find graph transformation subsequences as graph-shapelet patterns. By converting each graph-shapelet pattern into a unique tokenized graph transformation sequence, we can measure the similarity between two graph-shapelet patterns and therefore classify time-variant graphs. Experiments on both synthetic and real-world data demonstrate the superior performance of the proposed algorithms.

Existence of infinitely many stationary solutions of the L2-subcritical and critical NLSE on compact metric graphs

We investigate the existence of stationary solutions for the nonlinear Schrödinger equation on compact metric graphs. In the L2-subcritical setting, we prove the existence of an infinite number of such solutions, for every value of the mass. In the critical regime, the existence of infinitely many solutions is established if the mass is lower than a threshold value, while global minimizers of the NLS energy exist if and only if the mass is lower or equal to the threshold. Moreover, the relation between this threshold and the topology of the graph is characterized. The investigation is based on variational techniques and some new versions of Gagliardo–Nirenberg inequalities.

Compact CMC Graphs in $$M\times \mathbb {R}$$ M × R with Boundary in Two Horizontal Slices

We establish existence and uniqueness of compact graphs of cons tant mean curvature in $$M\times \mathbb {R}$$ , over bounded domains contained in $$M\times \left\{ 0\right\} $$ , with boundary lying in two horizontal slices of $$M\times \mathbb {R}$$ .

Approximating the cone of copositive kernels to estimate the stability number of infinite graphs

It has been shown that the stable set problem in an infinite compact graph, and particularly the kissing number problem, reduces to an optimization problem over the cone of copositive kernels. We propose two converging hierarchies approximating this cone. Both are extensions of existing inner hierarchies for the finite dimensional copositive cone. We implement the first two levels of the new hierarchies for the kissing number problem.

Stackings and the W-cycles Conjecture

AbstractWe prove Wise’s W-cycles conjecture. Consider a compact graph Γ' immersing into another graph Γ. For any immersed cycle Λ: S1 ⟶ Γ, we consider the map Λ' from the circular components 𝕊 of the pullback to Γ'. Unless Λ' is reducible, the degree of the covering map 𝕊 ⟶ S1 is bounded above by minus the Euler characteristic of Γ'. As a corollary, any finitely generated subgroup of a one-relator group has a finitely generated Schur multiplier.

Perfect non-commuting graphs of matrices over chains

In this paper, we study the non-commuting graph [Formula: see text] of strictly upper triangular [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We prove that [Formula: see text] is a compact graph. From [Formula: see text], we construct a poset [Formula: see text]. We further prove that [Formula: see text] is a dismantlable lattice and its zero-divisor graph is isomorphic to [Formula: see text]. Lastly, we prove that [Formula: see text] is a perfect graph.

Borel-Piecewise Continuous Reducibility for Uniformization Problems

We study a fine hierarchy of Borel-piecewise continuous functions, especially, between closed-piecewise continuity and $G_\delta$-piecewise continuity. Our aim is to understand how a priority argument in computability theory is connected to the notion of $G_\delta$-piecewise continuity, and then we utilize this connection to obtain separation results on subclasses of $G_\delta$-piecewise continuous reductions for uniformization problems on set-valued functions with compact graphs. This method is also applicable for separating various non-constructive principles in the Weihrauch lattice.

Edge Coloring and Stopping Sets Analysis in Product Codes With MDS Components

We consider non-binary product codes with MDS components and their iterative row-column algebraic decoding on the erasure channel. Both independent and block erasures are considered in this paper. A compact graph representation is introduced on which we define double-diversity edge colorings via the rootcheck concept. An upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size $M$ . Then, we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols. The complexity of this algorithm per iteration is $o(M^{\aleph })$ , for a given differential evolution parameter $\aleph $ , where $M^{\aleph }$ itself is small with respect to the huge cardinality of the coloring ensemble. Stopping sets of a product code are defined in the context of MDS components and a relationship is established with the graph representation. A full characterization of these stopping sets is given up to a size $(d+1)^{2}$ , where $d$ is the minimum Hamming distance of the MDS component code. The performance of MDS-based product codes with and without double-diversity coloring is analyzed in presence of both the block and the independent erasures. In the latter case, ML and iterative decoding are proven to coincide at small channel erasure probability. Furthermore, numerical results show excellent performance in presence of unequal erasure probability due to double-diversity colorings.

On the Weyl Law for Quantum Graphs

Roughly speaking, the Weyl law describes the asymptotic distribution of eigenvalues of the Laplacian that can be attached to different objects and can be analyzed in different settings. The form of the remainder term in the Weyl law is very significant in applications, and a power-saving exponent in the remainder term is appreciated. We are dealing with Laplacian defined on compact metric graph with general self-adjoint boundary conditions. The main purpose of this paper is to present application of the special form of the Tauberian theorem for the Laplace transform to the suitably transformed trace formula in the above-mentioned quantum graphs setting. The key feature of our method is that it produces a power-saving form of the reminder term and hence represents improvement in classical methods, which may be applied in other settings as well. The obtained form of the Weyl law is with the power saving of 1 / 3 in the remainder term.

Regularized Label Relaxation Linear Regression.

Linear regression (LR) and some of its variants have been widely used for classification problems. Most of these methods assume that during the learning phase, the training samples can be exactly transformed into a strict binary label matrix, which has too little freedom to fit the labels adequately. To address this problem, in this paper, we propose a novel regularized label relaxation LR method, which has the following notable characteristics. First, the proposed method relaxes the strict binary label matrix into a slack variable matrix by introducing a nonnegative label relaxation matrix into LR, which provides more freedom to fit the labels and simultaneously enlarges the margins between different classes as much as possible. Second, the proposed method constructs the class compactness graph based on manifold learning and uses it as the regularization item to avoid the problem of overfitting. The class compactness graph is used to ensure that the samples sharing the same labels can be kept close after they are transformed. Two different algorithms, which are, respectively, based on -norm and -norm loss functions are devised. These two algorithms have compact closed-form solutions in each iteration so that they are easily implemented. Extensive experiments show that these two algorithms outperform the state-of-the-art algorithms in terms of the classification accuracy and running time.

On the structure of compact graphs

A simple graph \(G\) is called a compact graph if \(G\) contains no isolated vertices and for each pair \(x\), \(y\) of non-adjacent vertices of \(G\), there is a vertex \(z\) with \(N(x)\cup N(y)\subseteq N(z)\), where \(N(v)\) is the neighborhood of \(v\), for every vertex \(v\) of \(G\). In this paper, compact graphs with sufficient number of edges are studied. Also, it is proved that every regular compact graph is strongly regular. Some results about cycles in compact graphs are proved, too. Among other results, it is proved that if the ascending chain condition holds for the set of neighbors of a compact graph \(G\), then the descending chain condition holds for the set of neighbors of \(G\).

Spectral Problems on Compact Graphs

Spectral Problems on Compact Graphs

An efficient and optimised frequent pattern mining using novel multipath-graph structure

The frequent pattern mining is one of the most focused areas in the data mining domain. The frequent pattern growth (FP-growth) algorithm introduced compact prefix-based data structure, frequent pattern tree (FP-Tree), to store frequent itemsets in compressed format. The FP-growth attempts to overcome drawback of candidate generation approach of multiple database scan. This work aims to propose a novel optimised data structure multipath-Graph (MP-Graph) for improving memory utilisation and efficiency of mining algorithms. The MP-Graph is a compact graph structure to store frequent patterns in memory. It generates graph nodes equal to number of frequent 1-itemsets of transaction database. Further, it stores multiple occurrences of prefix subpaths along the edges of the graph in the form of transaction bitmap instead of storing frequency of individual item node. The proposed structure helps to mine frequent patterns without constructing conditional FP-Trees. The performance of the MP-Graph mining algorithm is compared with FP-growth, CT-PRO and IFP-growth. The experimental results show order of magnitude improvement in memory consumption to store frequent patterns, nodes generated and time complexity.

An efficient and optimised frequent pattern mining using novel multipath-graph structure

The frequent pattern mining is one of the most focused areas in the data mining domain. The frequent pattern growth (FP-growth) algorithm introduced compact prefix-based data structure, frequent pattern tree (FP-Tree), to store frequent itemsets in compressed format. The FP-growth attempts to overcome drawback of candidate generation approach of multiple database scan. This work aims to propose a novel optimised data structure multipath-Graph (MP-Graph) for improving memory utilisation and efficiency of mining algorithms. The MP-Graph is a compact graph structure to store frequent patterns in memory. It generates graph nodes equal to number of frequent 1-itemsets of transaction database. Further, it stores multiple occurrences of prefix subpaths along the edges of the graph in the form of transaction bitmap instead of storing frequency of individual item node. The proposed structure helps to mine frequent patterns without constructing conditional FP-Trees. The performance of the MP-Graph mining algorithm is compared with FP-growth, CT-PRO and IFP-growth. The experimental results show order of magnitude improvement in memory consumption to store frequent patterns, nodes generated and time complexity.

Graph Isomorphism, Color Refinement, and Compactness

Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color refinement procedure succeeds in distinguishing G from any non-isomorphic graph H. Babai et al. (SIAM J Comput 9(3):628–635, 1980) have shown that random graphs are amenable with high probability. We determine the exact range of applicability of color refinement by showing that amenable graphs are recognizable in time $${O((n+m)\log n)}$$ , where n and m denote the number of vertices and the number of edges in the input graph. We use our characterization of amenable graphs to analyze the approach to Graph Isomorphism based on the notion of compact graphs. A graph is called compact if the polytope of its fractional automorphisms is integral. Tinhofer (Discrete Appl Math 30(2–3):253–264, 1991) noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing compact graphs and recognizing them in polynomial time remains an open question. Our results in this direction are summarized below: