On spectra of Hermitian Randic matrix of second kind

We propose the Hermitian Randi\'c matrix $R^\omega(X)=(R^\omega_{ij})$, where $\omega=\frac{1+i \sqrt{3}}{2}$ and $R^\omega_{ij}={1}/{\sqrt{d_id_j}}$ if $v_iv_j$ is an unoriented edge, ${\omega}/{\sqrt{d_id_j}}$ if $v_i\rightarrow v_j$, ${\overline{\omega}}/{\sqrt{d_id_j}}$ if $v_i\leftarrow v_j$, and 0 otherwise. This appears to be more natural because of $\omega+\overline{\omega}=1$ and $|\omega|=1$. In this paper, we investigate some features of this novel Hermitian matrix and study a few properties like positiveness, bipartiteness, edge-interlacing etc. We also compute the characteristic polynomial for this new matrix and obtain some upper and lower bounds for the eigenvalues and the energy of this matrix

More on minors of Hermitian (quasi-)Laplacian matrix of the second kind for mixed graphs

A mixed graph $M_{G}$ is the graph obtained from an unoriented simple graph $G$ by giving directions to some edges of $G$, where $G$ is often called the underlying graph of $M_{G}$. In this paper, we introduce two classes of incidence matrices of the second kind of $M_{G}$, and discuss the determinants of these two matrices for rootless mixed trees and unicyclic mixed graphs. Applying these results, we characterize the explicit expressions of various minors for Hermitian (quasi-)Laplacian matrix of the second kind of $M_{G}$. Moreover, we give two sufficient conditions that the absolute values of all the cofactors of Hermitian (quasi-)Laplacian matrix of the second kind are equal to the number of spanning trees of the underlying graph $G$.Comment: 16 pages,7 figure

A New Surrogating Algorithm by the Complex Graph Fourier Transform (CGFT)

[EN] The essential step of surrogating algorithms is phase randomizing the Fourier transform while preserving the original spectrum amplitude before computing the inverse Fourier transform. In this paper, we propose a new method which considers the graph Fourier transform. In this manner, much more flexibility is gained to define properties of the original graph signal which are to be preserved in the surrogates. The complex case is considered to allow unconstrained phase randomization in the transformed domain, hence we define a Hermitian Laplacian matrix that models the graph topology, whose eigenvectors form the basis of a complex graph Fourier transform. We have shown that the Hermitian Laplacian matrix may have negative eigenvalues. We also show in the paper that preserving the graph spectrum amplitude implies several invariances that can be controlled by the selected Hermitian Laplacian matrix. The interest of surrogating graph signals has been illustrated in the context of scarcity of instances in classifier training.This research was funded by the Spanish Administration and the European Union under grant TEC2017-84743-P.Belda, J.; Vergara Domínguez, L.; Safont Armero, G.; Salazar Afanador, A.; Parcheta, Z. (2019). A New Surrogating Algorithm by the Complex Graph Fourier Transform (CGFT). Entropy. 21(8):1-18. https://doi.org/10.3390/e21080759S118218Schreiber, T., & Schmitz, A. (2000). Surrogate time series. Physica D: Nonlinear Phenomena, 142(3-4), 346-382. doi:10.1016/s0167-2789(00)00043-9Miralles, R., Vergara, L., Salazar, A., & Igual, J. (2008). Blind detection of nonlinearities in multiple-echo ultrasonic signals. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 55(3), 637-647. doi:10.1109/tuffc.2008.688Mandic, D. ., Chen, M., Gautama, T., Van Hulle, M. ., & Constantinides, A. (2008). 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A survey of canonical forms and invariants for unitary similarity. Linear Algebra and its Applications, 147, 101-167. doi:10.1016/0024-3795(91)90232-lFutorny, V., Horn, R. A., & Sergeichuk, V. V. (2017). Specht’s criterion for systems of linear mappings. Linear Algebra and its Applications, 519, 278-295. doi:10.1016/j.laa.2017.01.006Mazumder, R., & Hastie, T. (2012). The graphical lasso: New insights and alternatives. Electronic Journal of Statistics, 6(0), 2125-2149. doi:10.1214/12-ejs740Baba, K., Shibata, R., & Sibuya, M. (2004). PARTIAL CORRELATION AND CONDITIONAL CORRELATION AS MEASURES OF CONDITIONAL INDEPENDENCE. Australian New Zealand Journal of Statistics, 46(4), 657-664. doi:10.1111/j.1467-842x.2004.00360.xChen, X., Xu, M., & Wu, W. B. (2013). Covariance and precision matrix estimation for high-dimensional time series. The Annals of Statistics, 41(6), 2994-3021. doi:10.1214/13-aos1182Theiler, J., Eubank, S., Longtin, A., Galdrikian, B., & Doyne Farmer, J. (1992). 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A Magnetic Framelet-Based Convolutional Neural Network for Directed Graphs

Recent years have witnessed the surging popularity among studies on directed graphs (digraphs) and digraph neural networks. With the unique capability of encoding directional relationships, digraphs have shown their superiority in modelling many real-life applications, such as citation analysis and website hyperlinks. Spectral Graph Convolutional Neural Networks (spectral GCNNs), a powerful tool for processing and analyzing undirected graph data, have been recently introduced to digraphs. Although spectral GCNNs typically apply frequency filtering via Fourier transform to obtain representations with selective information, research shows that model performance can be enhanced by framelet transform-based filtering. However, the massive majority of such research only considers spectral GCNNs for undirected graphs. In this thesis, we introduce Framelet-MagNet, a magnetic framelet-based spectral GCNN for digraphs. The model adopts magnetic framelet transform which decomposes the input digraph data to low-pass and high-pass frequency components in the spectral domain, forming a more sophisticated digraph representation for filtering. Digraph framelets are constructed with the complex-valued magnetic Laplacian, simultaneously leading to signal processing in both real and complex domains. To our best knowledge, this approach is the first attempt to conduct framelet-based convolution on digraph data in both real and complex domains. We empirically validate the predictive power of Framelet-MagNet via various tasks, including node classification, link prediction, and denoising. Besides, we show through experiment results that Framelet-MagNet can outperform the state-of-the-art approaches across several benchmark datasets

Nuevas contribuciones a la teoría y aplicación del procesado de señal sobre grafos

[ES] El procesado de señal sobre grafos es un campo emergente de técnicas que combinan conceptos de dos áreas muy consolidadas: el procesado de señal y la teoría de grafos. Desde la perspectiva del procesado de señal puede obtenerse una definición de la señal mucho más general asignando cada valor de la misma a un vértice de un grafo. Las señales convencionales pueden considerarse casos particulares en los que los valores de cada muestra se asignan a una cuadrícula uniforme (temporal o espacial). Desde la perspectiva de la teoría de grafos, se pueden definir nuevas transformaciones del grafo de forma que se extiendan los conceptos clásicos del procesado de la señal como el filtrado, la predicción y el análisis espectral. Además, el procesado de señales sobre grafos está encontrando nuevas aplicaciones en las áreas de detección y clasificación debido a su flexibilidad para modelar dependencias generales entre variables. En esta tesis se realizan nuevas contribuciones al procesado de señales sobre grafos. En primer lugar, se plantea el problema de estimación de la matriz Laplaciana asociada a un grafo, que determina la relación entre nodos. Los métodos convencionales se basan en la matriz de precisión, donde se asume implícitamente Gaussianidad. En esta tesis se proponen nuevos métodos para estimar la matriz Laplaciana a partir de las correlaciones parciales asumiendo respectivamente dos modelos no Gaussianos diferentes en el espacio de las observaciones: mezclas gaussianas y análisis de componentes independientes. Los métodos propuestos han sido probados con datos simulados y con datos reales en algunas aplicaciones biomédicas seleccionadas. Se demuestra que pueden obtenerse mejores estimaciones de la matriz Laplaciana con los nuevos métodos propuestos en los casos en que la Gaussianidad no es una suposición correcta. También se ha considerado la generación de señales sintéticas en escenarios donde la escasez de señales reales puede ser un problema. Los modelos sobre grafos permiten modelos de dependencia por pares más generales entre muestras de señal. Así, se propone un nuevo método basado en la Transformada de Fourier Compleja sobre Grafos y en el concepto de subrogación. Se ha aplicado en el desafiante problema del reconocimiento de gestos con las manos. Se ha demostrado que la extensión del conjunto de entrenamiento original con réplicas sustitutas generadas con los métodos sobre grafos, mejora significativamente la precisión del clasificador de gestos con las manos.[CAT] El processament de senyal sobre grafs és un camp emergent de tècniques que combinen conceptes de dues àrees molt consolidades: el processament de senyal i la teoria de grafs. Des de la perspectiva del processament de senyal pot obtindre's una definició del senyal molt més general assignant cada valor de la mateixa a un vèrtex d'un graf. Els senyals convencionals poden considerar-se casos particulars en els quals els valors de la mostra s'assignen a una quadrícula uniforme (temporal o espacial). Des de la perspectiva de la teoria de grafs, es poden definir noves transformacions del graf de manera que s'estenguen els conceptes clàssics del processament del senyal com el filtrat, la predicció i l'anàlisi espectral. A més, el processament de senyals sobre grafs està trobant noves aplicacions en les àrees de detecció i classificació a causa de la seua flexibilitat per a modelar dependències generals entre variables. En aquesta tesi es donen noves contribucions al processament de senyals sobre grafs. En primer lloc, es planteja el problema d'estimació de la matriu Laplaciana associada a un graf, que determina la relació entre nodes. Els mètodes convencionals es basen en la matriu de precisió, on s'assumeix implícitament la gaussianitat. En aquesta tesi es proposen nous mètodes per a estimar la matriu Laplaciana a partir de les correlacions parcials assumint respectivament dos models no gaussians diferents en l'espai d'observació: mescles gaussianes i anàlisis de components independents. Els mètodes proposats han sigut provats amb dades simulades i amb dades reals en algunes aplicacions biomèdiques seleccionades. Es demostra que poden obtindre's millors estimacions de la matriu Laplaciana amb els nous mètodes proposats en els casos en què la gaussianitat no és una suposició correcta. També s'ha considerat el problema de generar senyals sintètics en escenaris on l'escassetat de senyals reals pot ser un problema. Els models sobre grafs permeten models de dependència per parells més generals entre mostres de senyal. Així, es proposa un nou mètode basat en la Transformada de Fourier Complexa sobre Grafs i en el concepte de subrogació. S'ha aplicat en el desafiador problema del reconeixement de gestos amb les mans. S'ha demostrat que l'extensió del conjunt d'entrenament original amb rèpliques substitutes generades amb mètodes sobre grafs, millora significativament la precisió del classificador de gestos amb les mans.[EN] Graph signal processing appears as an emerging field of techniques that combine concepts from two highly consolidated areas: signal processing and graph theory. From the perspective of signal processing, it is possible to achieve a more general signal definition by assigning each value of the signal to a vertex of a graph. Conventional signals can be considered particular cases where the sample values are assigned to a uniform (temporal or spatial) grid. From the perspective of graph theory, new transformations of the graph can be defined in such a way that they extend the classical concepts of signal processing such as filtering, prediction and spectral analysis. Furthermore, graph signal processing is finding new applications in detection and classification areas due to its flexibility to model general dependencies between variables. In this thesis, new contributions are given to graph signal processing. Firstly, it is considered the problem of estimating the Laplacian matrix associated with a graph, which determines the relationship between nodes. Conventional methods are based on the precision matrix, where Gaussianity is implicitly assumed. In this thesis, new methods to estimate the Laplacian matrix from the partial correlations are proposed respectively assuming two different non-Gaussian models in the observation space: Gaussian Mixtures and Independent Component Analysis. The proposed methods have been tested with simulated data and with real data in some selected biomedical applications. It is demonstrate that better estimates of the Laplacian matrix can be obtained with the new proposed methods in cases where Gaussianity is not a correct assumption. The problem of generating synthetic signal in scenarios where real signals scarcity can be an issue has also been considered. Graph models allow more general pairwise dependence models between signal samples. Thus a new method based on the Complex Graph Fourier Transform and on the concept of subrogation is proposed. It has been applied in the challenging problem of hand gesture recognition. It has been demonstrated that extending the original training set with graph surrogate replicas, significantly improves the accuracy of the hand gesture classifier.Belda Valls, J. (2022). Nuevas contribuciones a la teoría y aplicación del procesado de señal sobre grafos [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/19133