arXiv subjects

Richard C. Wilson

Publications and source records attributed to Richard C. Wilson.

11 recordsLinked to original sources

arXiv2025

LBONet: Supervised Spectral Descriptors for Shape Analysis

The Laplace-Beltrami operator has established itself in the field of non-rigid shape analysis due to its many useful properties such as being invariant under isometric transformation, having a countable eigensystem forming an orthornormal basis, and fully characterizing geodesic distances of the manifold. However, this invariancy only applies under isometric deformations, which leads to a performance breakdown in many real-world applications. In recent years emphasis has been placed upon extracting optimal features using deep learning methods,however spectral signatures play a crucial role and still add value. In this paper we take a step back, revisiting the LBO and proposing a supervised way to learn several operators on a manifold. Depending on the task, by applying these functions, we can train the LBO eigenbasis to be more task-specific. The optimization of the LBO leads to enormous improvements to established descriptors such as the heat kernel signature in various tasks such as retrieval, classification, segmentation, and correspondence, proving the adaption of the LBO eigenbasis to both global and highly local learning settings.

cs.CV↗

arXiv2025

Towards Simple Machine Learning Baselines for GNSS RFI Detection

Machine learning research in GNSS radio frequency interference (RFI) detection often lacks a clear empirical justification for the choice of deep learning architectures over simpler machine learning approaches. In this work, we argue for a change in research direction-from developing ever more complex deep learning models to carefully assessing their real-world effectiveness in comparison to interpretable and lightweight machine learning baselines. Our findings reveal that state-of-the-art deep learning models frequently fail to outperform simple, well-engineered machine learning methods in the context of GNSS RFI detection. Leveraging a unique large-scale dataset collected by the Swiss Air Force and Swiss Air-Rescue (Rega), and preprocessed by Swiss Air Navigation Services Ltd. (Skyguide), we demonstrate that a simple baseline model achieves 91\% accuracy in detecting GNSS RFI, outperforming more complex deep learning counterparts. These results highlight the effectiveness of pragmatic solutions and offer valuable insights to guide future research in this critical application domain.

eess.SP↗

arXiv2019

Computing Optimal Assignments in Linear Time for Approximate Graph Matching

Finding an optimal assignment between two sets of objects is a fundamental problem arising in many applications, including the matching of `bag-of-words' representations in natural language processing and computer vision. Solving the assignment problem typically requires cubic time and its pairwise computation is expensive on large datasets. In this paper, we develop an algorithm which can find an optimal assignment in linear time when the cost function between objects is represented by a tree distance. We employ the method to approximate the edit distance between two graphs by matching their vertices in linear time. To this end, we propose two tree distances, the first of which reflects discrete and structural differences between vertices, and the second of which can be used to compare continuous labels. We verify the effectiveness and efficiency of our methods using synthetic and real-world datasets.

cs.LG↗

arXiv2017

Cycle-centrality in complex networks

Networks are versatile representations of the interactions between entities in complex systems. Cycles on such networks represent feedback processes which play a central role in system dynamics. In this work, we introduce a measure of the importance of any individual cycle, as the fraction of the total information flow of the network passing through the cycle. This measure is computationally cheap, numerically well-conditioned, induces a centrality measure on arbitrary subgraphs and reduces to the eigenvector centrality on vertices. We demonstrate that this measure accurately reflects the impact of events on strategic ensembles of economic sectors, notably in the US economy. As a second example, we show that in the protein-interaction network of the plant Arabidopsis thaliana, a model based on cycle-centrality better accounts for pathogen activity than the state-of-art one. This translates into pathogen-targeted-proteins being concentrated in a small number of triads with high cycle-centrality. Algorithms for computing the centrality of cycles and subgraphs are available for download.

cs.SI↗

arXiv2017

On Valid Optimal Assignment Kernels and Applications to Graph Classification

The success of kernel methods has initiated the design of novel positive semidefinite functions, in particular for structured data. A leading design paradigm for this is the convolution kernel, which decomposes structured objects into their parts and sums over all pairs of parts. Assignment kernels, in contrast, are obtained from an optimal bijection between parts, which can provide a more valid notion of similarity. In general however, optimal assignments yield indefinite functions, which complicates their use in kernel methods. We characterize a class of base kernels used to compare parts that guarantees positive semidefinite optimal assignment kernels. These base kernels give rise to hierarchies from which the optimal assignment kernels are computed in linear time by histogram intersection. We apply these results by developing the Weisfeiler-Lehman optimal assignment kernel for graphs. It provides high classification accuracy on widely-used benchmark data sets improving over the original Weisfeiler-Lehman kernel.

cs.LG↗

arXiv2016

A general purpose algorithm for counting simple cycles and simple paths of any length

We describe a general purpose algorithm for counting simple cycles and simple paths of any length $\ell$ on a (weighted di)graph on $N$ vertices and $M$ edges, achieving a time complexity of $O\left(N+M+\big(\ell^ω+\ellΔ\big) |S_\ell|\right)$. In this expression, $|S_\ell|$ is the number of (weakly) connected induced subgraphs of $G$ on at most $\ell$ vertices, $Δ$ is the maximum degree of any vertex and $ω$ is the exponent of matrix multiplication. We compare the algorithm complexity both theoretically and experimentally with most of the existing algorithms for the same task. These comparisons show that the algorithm described here is the best general purpose algorithm for the class of graphs where $(\ell^{ω-1}Δ^{-1}+1) |S_\ell|\leq |\text{Cycle}_\ell|$, with $|\text{Cycle}_\ell|$ the total number of simple cycles of length at most $\ell$, including backtracks and self-loops. On Erdős-Rényi random graphs, we find empirically that this happens when the edge probability is larger than circa $4/N$. In addition, we show that some real-world networks also belong to this class. Finally, the algorithm permits the enumeration of simple cycles and simple paths on networks where vertices are labeled from an alphabet on $n$ letters with a time complexity of $O\left(N+M+\big(n^\ell\ell^ω+\ellΔ\big) |S_\ell|\right)$. A Matlab implementation of the algorithm proposed here is available for download.

cs.DS↗

arXiv2015

Modular Dynamics of Financial Market Networks

The financial market is a complex dynamical system composed of a large variety of intricate relationships between several entities, such as banks, corporations and institutions. At the heart of the system lies the stock exchange mechanism, which establishes a time-evolving network of trades among companies and individuals. Such network can be inferred through correlations between time series of companies stock prices, allowing the overall system to be characterized by techniques borrowed from network science. Here we study the presence of communities in the inferred stock market network, and show that the knowledge about the communities alone can provide a nearly complete representation of the system topology. This is done by defining a simple null model, a randomized version of the studied network sharing only the sizes and interconnectivity between communities observed. We show that many topological characteristics of the inferred networks are carried over the networks generated by the null model. In particular, we find that in periods of instability, such as during a financial crisis, the network strays away from a state of well-defined community structure to a much more uniform topological organization. We show that the framework presented here provides a good null model representation of topological variations taking place in the market during crises. Also, the general approach used in this work can be extended to other systems.

physics.soc-ph↗

arXiv2014

Concentric Network Symmetry

Quantification of symmetries in complex networks is typically done globally in terms of automorphisms. Extending previous methods to locally assess the symmetry of nodes is not straightforward. Here we present a new framework to quantify the symmetries around nodes, which we call connectivity patterns. We develop two topological transformations that allow a concise characterization of the different types of symmetry appearing on networks and apply these concepts to six network models, namely the Erdős-Rényi, Barabási-Albert, random geometric graph, Waxman, Voronoi and rewired Voronoi. Real-world networks, namely the scientific areas of Wikipedia, the world-wide airport network and the street networks of Oldenburg and San Joaquin, are also analyzed in terms of the proposed symmetry measurements. Several interesting results emerge from this analysis, including the high symmetry exhibited by the Erdős-Rényi model. Additionally, we found that the proposed measurements present low correlation with other traditional metrics, such as node degree and betweenness centrality. Principal component analysis is used to combine all the results, revealing that the concepts presented here have substantial potential to also characterize networks at a global scale.

physics.soc-ph↗

arXiv2013

Eigenfunctions of the Edge-Based Laplacian on a Graph

In this paper, we analyze the eigenfunctions of the edge-based Laplacian on a graph and the relationship of these functions to random walks on the graph. We commence by discussing the set of eigenfunctions supported at the vertices, and demonstrate the relationship of these eigenfunctions to the classical random walk on the graph. Then, from an analysis of functions supported only on the interior of edges, we develop a method for explicitly calculating the edge-interior eigenfunctions of the edge-based Laplacian. This reveals a connection between the edge-based Laplacian and the adjacency matrix of backtrackless random walk on the graph. The edge-based eigenfunctions therefore correspond to some eigenfunctions of the normalised Hashimoto matrix.

cs.DM↗

arXiv2007

Some families of density matrices for which separability is easily tested

We reconsider density matrices of graphs as defined in [quant-ph/0406165]. The density matrix of a graph is the combinatorial laplacian of the graph normalized to have unit trace. We describe a simple combinatorial condition (the "degree condition") to test separability of density matrices of graphs. The condition is directly related to the PPT-criterion. We prove that the degree condition is necessary for separability and we conjecture that it is also sufficient. We prove special cases of the conjecture involving nearest point graphs and perfect matchings. We observe that the degree condition appears to have value beyond density matrices of graphs. In fact, we point out that circulant density matrices and other matrices constructed from groups always satisfy the condition and indeed are separable with respect to any split. The paper isolates a number of problems and delineates further generalizations.

quant-ph↗

arXiv2006

A matrix representation of graphs and its spectrum as a graph invariant

We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the first two cases, we compute the spectrum explicitly and show that it is determined by the spectrum of the adjacency matrix of the original graph. We then show by computation that the isomorphism classes of many known families of strongly regular graphs (up to 64 vertices) are characterized by the spectrum of this matrix. We conjecture that this is always the case for strongly regular graphs and we show that the conjecture is not valid for general graphs. We verify that the smallest regular graphs which are not distinguished with our method are on 14 vertices.

quant-ph↗