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Spectral techniques are popular and robust approaches to data analysis. A prominent example is the use of eigenvectors of a Laplacian, constructed from data affinities, to identify natural data groupings or clusters, or to produce a…
Eigenvector continuation is a computational method that finds the extremal eigenvalues and eigenvectors of a Hamiltonian matrix with one or more control parameters. It does this by projection onto a subspace of eigenvectors corresponding to…
Topological data analysis (TDA) detects geometric structure in biological data. However, many TDA algorithms are memory intensive and impractical for massive datasets. Here, we introduce a statistical protocol that reduces TDA's memory…
Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…
Topological Data Analysis (TDA) is a rising field of computational topology in which the topological structure of a data set can be observed by persistent homology. By considering a sequence of sublevel sets, one obtains a filtration that…
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…
The rapid growth of graph data creates significant scalability challenges as most graph algorithms scale quadratically with size. To mitigate these issues, Graph Condensation (GC) methods have been proposed to learn a small graph from a…
Topological data analysis (TDA) is a tool from data science and mathematics that is beginning to make waves in environmental science. In this work, we seek to provide an intuitive and understandable introduction to a tool from TDA that is…
The standard notion of the Laplacian of a graph is generalized to the setting of a graph with the extra structure of a ``transmission`` system. A transmission system is a mathematical representation of a means of transmitting…
In this paper, we propose a speed-up approach for subclass discriminant analysis and formulate a novel efficient multi-view solution to it. The speed-up approach is developed based on graph embedding and spectral regression approaches that…
We propose and study a novel graph clustering method for data with an intrinsic network structure. Similar to spectral clustering, we exploit an intrinsic network structure of data to construct Euclidean feature vectors. These feature…
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…
Topology identification and inference of processes evolving over graphs arise in timely applications involving brain, transportation, financial, power, as well as social and information networks. This chapter provides an overview of graph…
In this paper we study two classes of graphs, the (m,k)-stars and l-dependent graphs, investigating the relation between spectrum characteristics and graph structure: conditions on the topology and edge weights are given in order to get…
Solving different types of optimization models (including parameters fitting) for support vector machines on large-scale training data is often an expensive computational task. This paper proposes a multilevel algorithmic framework that…
In this study, we present and analyze a framework for geometric and topological estimation for mapping of unknown environments. We consider agents mimicking motion behaviors of cyborg insects, known as biobots, and exploit coordinate-free…
Graph embeddings have become a key and widely used technique within the field of graph mining, proving to be successful across a broad range of domains including social, citation, transportation and biological. Graph embedding techniques…
We introduce an unsupervised graph embedding that trades off local node similarity and connectivity, and global structure. The embedding is based on a generalized graph Laplacian, whose eigenvectors compactly capture both network structure…
Choosing a meaningful subset of features from high-dimensional observations in unsupervised settings can greatly enhance the accuracy of downstream analysis, such as clustering or dimensionality reduction, and provide valuable insights into…
We propose a novel algorithm for supervised dimensionality reduction named Manifold Partition Discriminant Analysis (MPDA). It aims to find a linear embedding space where the within-class similarity is achieved along the direction that is…