Related papers: Hamiltonian System Approach to Distributed Spectra…
In this paper, we introduce an algorithm for performing spectral clustering efficiently. Spectral clustering is a powerful clustering algorithm that suffers from high computational complexity, due to eigen decomposition. In this work, we…
This paper develops a closed-form spectral decomposition framework for the Gramian matrices of discrete-time linear dynamical systems. The main results provide explicit decompositions of the discrete-time controllability Gramian and its…
Many tools from the field of graph signal processing exploit knowledge of the underlying graph's structure (e.g., as encoded in the Laplacian matrix) to process signals on the graph. Therefore, in the case when no graph is available, graph…
In data science, hypergraphs are natural models for data exhibiting multi-way relations, whereas graphs only capture pairwise. Nonetheless, many proposed hypergraph neural networks effectively reduce hypergraphs to undirected graphs via…
We consider the minimum-cut partitioning of a graph into more than two parts using spectral methods. While there exist well-established spectral algorithms for this problem that give good results, they have traditionally not been well…
We draw attention to the fact that a Hermitian matrix is always diagonalizable and has real discrete spectrum whereas the Hermitian Schr{\"o}dinger Hamiltonian: $H=p^2/2\mu+V(x)$, may not be so. For instance when $V(x)=x, x^3, -x^2$, $H$…
Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these…
Metric networks are network-shaped, one-dimensional structures on which one can solve differential equations to simulate a wide range of physical systems including conjugated molecules, photonic crystals, quantum mechanics in waveguide…
We derive the spectral properties of adjacency matrix of complex networks and of their Laplacian by the replica method combined with a dynamical population algorithm. By assuming the order parameter to be a product of Gaussian…
Graphs are fundamental mathematical structures used in various fields to represent data, signals and processes. In this paper, we propose a novel framework for learning/estimating graphs from data. The proposed framework includes (i)…
We introduce a complex-plane generalization of the consecutive level-spacing distribution, used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of complex-valued ratios between nearest- and…
A generalized Fourier analysis on arbitrary graphs calls for a detailed knowledge of the eigenvectors of the graph Laplacian. Using the symmetries of the Cayley tree, we recursively construct the family of eigenvectors with exponentially…
When attempting to develop wavelet transforms for graphs and networks, some researchers have used graph Laplacian eigenvalues and eigenvectors in place of the frequencies and complex exponentials in the Fourier theory for regular lattices…
The spectral properties of disordered fully-connected graphs with a special type of the node-node interactions are investigated. The approximate analytical expression for the ensemble-averaged spectral density for the Hamiltonian defined on…
We present a scalable Bayesian framework for the analysis of confocal fluorescence spectroscopy data, addressing key limitations in traditional fluorescence correlation spectroscopy methods. Our framework captures molecular motion,…
Motivated by the recent advances in the field of quantum computing, quantum systems are modelled and analyzed as networks of decentralized quantum nodes which employ distributed quantum consensus algorithms for coordination. In the…
We develop the partitioning technique for quantum discrete systems. The graph consists of several subgraphs: a central graph and several branch graphs, with each branch graph being rooted by an individual node on the central one. We show…
Sampling-based algorithms are classical approaches to perform Bayesian inference in inverse problems. They provide estimators with the associated credibility intervals to quantify the uncertainty on the estimators. Although these methods…
Lattice Hamiltonian systems underpin models across condensed matter, nonlinear optics, and biophysics, yet learning their dynamics from data is obstructed by two unknowns: the interaction topology and whether node dynamics are homogeneous.…
Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view…