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Manifold learning and dimensionality reduction techniques are ubiquitous in science and engineering, but can be computationally expensive procedures when applied to large data sets or when similarities are expensive to compute. To date,…

Many important problems are characterized by the eigenvalues of a large matrix. For example, the difficulty of many optimization problems, such as those arising from the fitting of large models in statistics and machine learning, can be…

Graph signal processing (GSP) is a prominent framework for analyzing signals on non-Euclidean domains. The graph Fourier transform (GFT) uses the combinatorial graph Laplacian matrix to reveal the spectral decomposition of signals in the…

UMAP is a non-parametric graph-based dimensionality reduction algorithm using applied Riemannian geometry and algebraic topology to find low-dimensional embeddings of structured data. The UMAP algorithm consists of two steps: (1) Compute a…

Learning a suitable graph is an important precursor to many graph signal processing (GSP) pipelines, such as graph spectral signal compression and denoising. Previous graph learning algorithms either i) make some assumptions on connectivity…

We design two variational algorithms to optimize specific 2-local Hamiltonians defined on graphs. Our algorithms are inspired by the Quantum Approximate Optimization Algorithm. We develop formulae to analyze the energy achieved by these…

In this paper we present a decentralized algorithm to estimate the eigenvalues of the Laplacian matrix that encodes the network topology of a multi-agent system. We consider network topologies modeled by undirected graphs. The basic idea is…

Embedding of a knowledge graph(KG) entities and relations in the form of vectors is an important aspect for the manipulation of the KG database for several downstream tasks, such as link prediction, knowledge graph completion, and…

The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and…

For a regular polyhedron (or polygon) centered at the origin, the coordinates of the vertices are eigenvectors of the graph Laplacian for the skeleton of that polyhedron (or polygon) associated with the first (non-trivial) eigenvalue. In…

We show that, with very high probability, the random graph Laplacian has simple spectrum. Our method provides a quantitatively effective estimate of the spectral gaps. Along the way, we establish results on affine no-gaps delocalization,…

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…

The limited connectivity of current and next-generation quantum annealers motivates the need for efficient graph-minor embedding methods. These methods allow non-native problems to be adapted to the target annealer's architecture. The…

This paper proposes a hybrid quantum-classical algorithm that learns a suitable quantum feature map that separates unlabelled data that is originally non linearly separable in the classical space using a Variational quantum feature map and…

We propose a variational method for constructing the eigenvalues and generalized eigenvalues for an arbitrary $N\times N$ complex matrix. The quantum part of our algorithm is based on encoding the matrix elements into the pure state of a…

We note that building a magnetic Laplacian from the Markov transition matrix, rather than the graph adjacency matrix, yields several benefits for the magnetic eigenmaps algorithm. The two largest benefits are that the embedding becomes more…

Graph analytics can lead to better quantitative understanding and control of complex networks, but traditional methods suffer from high computational cost and excessive memory requirements associated with the high-dimensionality and…

Motivated by discrete Laplacian differential operators with various accuracy orders in numerical analysis, we introduce new matrices attached to a simple graph that can be considered graph Laplacians with higher accuracy. In particular, we…

Node embeddings map graph vertices into low-dimensional Euclidean spaces while preserving structural information. They are central to tasks such as node classification, link prediction, and signal reconstruction. A key goal is to design…

Graph is an important data representation which appears in a wide diversity of real-world scenarios. Effective graph analytics provides users a deeper understanding of what is behind the data, and thus can benefit a lot of useful…