Correspondence edit distance to obtain a set of weighted means of graph correspondences.

Given a pair of data structures, such as strings, trees, graphs or sets of points, several correspondences (also referred in literature as labellings, matchings or assignments) can be defined between their local parts. The Hamming distance has been largely used to define the dissimilarity of a pair of correspondences between two data structures. Although it has the advantage of being simple in computation, it does not consider the data structures themselves, which the correspondences relate to. In this paper, we extend the definitions of a recently presented distance between correspondences based on the concept of the edit distance, which we called Correspondence edit distance. Moreover, we present an algorithm to compute the set of weighted means between a pair of graph correspondences. Both the Correspondence edit distance and the computation of the set of weighted means are necessary for the calculation of a more representative prototype between a set of correspondences. In the validation section, we show how the use of the Correspondence edit distance increases the quality of the set of weighted means compared to using the Hamming distance

Structural graph matching using the EM algorithm and singular value decomposition

This paper describes an efficient algorithm for inexact graph matching. The method is purely structural, that is, it uses only the edge or connectivity structure of the graph and does not draw on node or edge attributes. We make two contributions: 1) commencing from a probability distribution for matching errors, we show how the problem of graph matching can be posed as maximum-likelihood estimation using the apparatus of the EM algorithm; and 2) we cast the recovery of correspondence matches between the graph nodes in a matrix framework. This allows one to efficiently recover correspondence matches using the singular value decomposition. We experiment with the method on both real-world and synthetic data. Here, we demonstrate that the method offers comparable performance to more computationally demanding method

Structural Data Recognition with Graph Model Boosting

This paper presents a novel method for structural data recognition using a large number of graph models. In general, prevalent methods for structural data recognition have two shortcomings: 1) Only a single model is used to capture structural variation. 2) Naive recognition methods are used, such as the nearest neighbor method. In this paper, we propose strengthening the recognition performance of these models as well as their ability to capture structural variation. The proposed method constructs a large number of graph models and trains decision trees using the models. This paper makes two main contributions. The first is a novel graph model that can quickly perform calculations, which allows us to construct several models in a feasible amount of time. The second contribution is a novel approach to structural data recognition: graph model boosting. Comprehensive structural variations can be captured with a large number of graph models constructed in a boosting framework, and a sophisticated classifier can be formed by aggregating the decision trees. Consequently, we can carry out structural data recognition with powerful recognition capability in the face of comprehensive structural variation. The experiments shows that the proposed method achieves impressive results and outperforms existing methods on datasets of IAM graph database repository.Comment: 8 page

Pattern vectors from algebraic graph theory

Graphstructures have proven computationally cumbersome for pattern analysis. The reason for this is that, before graphs can be converted to pattern vectors, correspondences must be established between the nodes of structures which are potentially of different size. To overcome this problem, in this paper, we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner. We extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a Hermitian property matrix that can be viewed as a complex analogue of the Laplacian. To embed the graphs in a pattern space, we explore whether the vectors of invariants can be embedded in a low- dimensional space using a number of alternative strategies, including principal components analysis ( PCA), multidimensional scaling ( MDS), and locality preserving projection ( LPP). Experimentally, we demonstrate that the embeddings result in well- defined graph clusters. Our experiments with the spectral representation involve both synthetic and real- world data. The experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure. The real- world experiments show that the method can be used to locate clusters of graphs

Learning the Consensus of Multiple Correspondences between Data Structures

En aquesta tesi presentem un marc de treball per aprendre el consens donades múltiples correspondències. S'assumeix que les diferents parts involucrades han generat aquestes correspondències per separat, i el nostre sistema actua com un mecanisme que calibra diferents característiques i considera diferents paràmetres per aprendre les millors assignacions i així, conformar una correspondència amb la major precisió possible a costa d'un cost computacional raonable. Aquest marc de treball de consens és presentat en una forma gradual, començant pels desenvolupaments més bàsics que utilitzaven exclusivament conceptes ben definits o únicament un parell de correspondències, fins al model final que és capaç de considerar múltiples correspondències, amb la capacitat d'aprendre automàticament alguns paràmetres de ponderació. Cada pas d'aquest marc de treball és avaluat fent servir bases de dades de naturalesa variada per demostrar efectivament que és possible tractar diferents escenaris de matching. Addicionalment, dos avanços suplementaris relacionats amb correspondències es presenten en aquest treball. En primer lloc, una nova mètrica de distància per correspondències s'ha desenvolupat, la qual va derivar en una nova estratègia per a la cerca de mitjanes ponderades. En segon lloc, un marc de treball específicament dissenyat per a generar correspondències al camp del registre d'imatges s'ha modelat, on es considera que una de les imatges és una imatge completa, i l'altra és una mostra petita d'aquesta. La conclusió presenta noves percepcions de com el nostre marc de treball de consens pot ser millorada, i com els dos desenvolupaments paral·lels poden convergir amb el marc de treball de consens.En esta tesis presentamos un marco de trabajo para aprender el consenso dadas múltiples correspondencias. Se asume que las distintas partes involucradas han generado dichas correspondencias por separado, y nuestro sistema actúa como un mecanismo que calibra distintas características y considera diferentes parámetros para aprender las mejores asignaciones y así, conformar una correspondencia con la mayor precisión posible a expensas de un costo computacional razonable. El marco de trabajo de consenso es presentado en una forma gradual, comenzando por los acercamientos más básicos que utilizaban exclusivamente conceptos bien definidos o únicamente un par de correspondencias, hasta el modelo final que es capaz de considerar múltiples correspondencias, con la capacidad de aprender automáticamente algunos parámetros de ponderación. Cada paso de este marco de trabajo es evaluado usando bases de datos de naturaleza variada para demostrar efectivamente que es posible tratar diferentes escenarios de matching. Adicionalmente, dos avances suplementarios relacionados con correspondencias son presentados en este trabajo. En primer lugar, una nueva métrica de distancia para correspondencias ha sido desarrollada, la cual derivó en una nueva estrategia para la búsqueda de medias ponderadas. En segundo lugar, un marco de trabajo específicamente diseñado para generar correspondencias en el campo del registro de imágenes ha sido establecida, donde se considera que una de las imágenes es una imagen completa, y la otra es una muestra pequeña de ésta. La conclusión presenta nuevas percepciones de cómo nuestro marco de trabajo de consenso puede ser mejorada, y cómo los dos desarrollos paralelos pueden converger con éste.In this work, we present a framework to learn the consensus given multiple correspondences. It is assumed that the several parties involved have generated separately these correspondences, and our system acts as a mechanism that gauges several characteristics and considers different parameters to learn the best mappings and thus, conform a correspondence with the highest possible accuracy at the expense of a reasonable computational cost. The consensus framework is presented in a gradual form, starting from the most basic approaches that used exclusively well-known concepts or only two correspondences, until the final model which is able to consider multiple correspondences, with the capability of automatically learning some weighting parameters. Each step of the framework is evaluated using databases of varied nature to effectively demonstrate that it is capable to address different matching scenarios. In addition, two supplementary advances related on correspondences are presented in this work. Firstly, a new distance metric for correspondences has been developed, which lead to a new strategy for the weighted mean correspondence search. Secondly, a framework specifically designed for correspondence generation in the image registration field has been established, where it is considered that one of the images is a full image, and the other one is a small sample of it. The conclusion presents insights of how our consensus framework can be enhanced, and how these two parallel developments can converge with it

Generalised median of graph correspondences.

A graph correspondence is defined as a function that maps the elements of two attributed graphs. Due to the increasing availability of methods to perform graph matching, numerous graph correspondences can be deducted for a pair of attributed graphs. To obtain a representative prototype for a set of data structures, the concept of the median has been largely employed, as it has proven to deliver a robust sample. Nonetheless, the calculation of the exact (or generalised) median is known to be an NP-complete problem for most domains. In this paper, we present a method based on an optimisation function to calculate the generalised median graph correspondence. This method makes use of the Correspondence Edit Distance, which is a metric that considers the attributes and the local structures of the graphs to obtain more interesting and meaningful results. Experimental validation shows that this approach is capable of obtaining the generalised median in a comparable runtime with respect to state-of-the-art methods on artificial data, while maintaining the success rate for a real-application case

Obtaining the consensus of multiple correspondences between graphs through online learning.

In structural pattern recognition, it is usual to compare a pair of objects through the generation of a correspondence between the elements of each of their local parts. To do so, one of the most natural ways to represent these objects is through attributed graphs. Several existing graph extraction methods could be implemented and thus, numerous graphs, which may not only differ in their nodes and edge structure but also in their attribute domains, could be created from the same object. Afterwards, a matching process is implemented to generate the correspondence between two attributed graphs, and depending on the selected graph matching method, a unique correspondence is generated from a given pair of attributed graphs. The combination of these factors leads to the possibility of a large quantity of correspondences between the two original objects. This paper presents a method that tackles this problem by considering multiple correspondences to conform a single one called a consensus correspondence, eliminating both the incongruences introduced by the graph extraction and the graph matching processes. Additionally, through the application of an online learning algorithm, it is possible to deduce some weights that influence on the generation of the consensus correspondence. This means that the algorithm automatically learns the quality of both the attribute domain and the correspondence for every initial correspondence proposal to be considered in the consensus, and defines a set of weights based on this quality. It is shown that the method automatically tends to assign larger values to high quality initial proposals, and therefore is capable to deduce better consensus correspondences

Generalised median of a set of correspondences based on the hamming distance.

A correspondence is a set of mappings that establishes a relation between the elements of two data structures (i.e. sets of points, strings, trees or graphs). If we consider several correspondences between the same two structures, one option to define a representative of them is through the generalised median correspondence. In general, the computation of the generalised median is an NP-complete task. In this paper, we present two methods to calculate the generalised median correspondence of multiple correspondences. The first one obtains the optimal solution in cubic time, but it is restricted to the Hamming distance. The second one obtains a sub-optimal solution through an iterative approach, but does not have any restrictions with respect to the used distance. We compare both proposals in terms of the distance to the true generalised median and runtime

Correspondence consensus of two sets of correspondences through optimisation functions.

We present a consensus method which, given the two correspondences between sets of elements generated by separate entities, enounces a final correspondence consensus considering the existence of outliers. Our method is based on an optimisation technique that minimises the cost of the correspondence while forcing (to the most) to be the mean correspondence of the two original correspondences. The method decides the mapping of the elements that the original correspondences disagree and returns the same element mapping when both correspondences agree. We first show the validity of the method through an experiment in ideal conditions based on palmprint identification, and subsequently present two practical experiments based on image retrieval

Distances and Isomorphism between Networks and the Stability of Network Invariants

We develop the theoretical foundations of a network distance that has recently been applied to various subfields of topological data analysis, namely persistent homology and hierarchical clustering. While this network distance has previously appeared in the context of finite networks, we extend the setting to that of compact networks. The main challenge in this new setting is the lack of an easy notion of sampling from compact networks; we solve this problem in the process of obtaining our results. The generality of our setting means that we automatically establish results for exotic objects such as directed metric spaces and Finsler manifolds. We identify readily computable network invariants and establish their quantitative stability under this network distance. We also discuss the computational complexity involved in precisely computing this distance, and develop easily-computable lower bounds by using the identified invariants. By constructing a wide range of explicit examples, we show that these lower bounds are effective in distinguishing between networks. Finally, we provide a simple algorithm that computes a lower bound on the distance between two networks in polynomial time and illustrate our metric and invariant constructions on a database of random networks and a database of simulated hippocampal networks