相关论文: Isospectral Reductions of Non-negative Matrices

Singular values of a data in a matrix form provide insights on the structure of the data, the effective dimensionality, and the choice of hyper-parameters on higher-level data analysis tools. However, in many practical applications such as…

Covariance matrix estimates are an essential part of many signal processing algorithms, and are often used to determine a low-dimensional principal subspace via their spectral decomposition. However, exact eigenanalysis is computationally…

Graph neural network (GNN) is a popular tool to learn the lower-dimensional representation of a graph. It facilitates the applicability of machine learning tasks on graphs by incorporating domain-specific features. There are various options…

In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major…

Training of neural networks can be reformulated in spectral space, by allowing eigenvalues and eigenvectors of the network to act as target of the optimization instead of the individual weights. Working in this setting, we show that the…

In many machine learning and data related applications, it is required to have the knowledge of approximate ranks of large data matrices at hand. In this paper, we present two computationally inexpensive techniques to estimate the…

Graphs are central to modeling complex systems in domains such as social networks, molecular chemistry, and neuroscience. While Graph Neural Networks, particularly Graph Convolutional Networks, have become standard tools for graph learning,…

In the study of condensed matter physics, spectral information plays an important role for understand the mechanism of materials. However, it is difficult to obtain the spectrum directly through experiments or simulation. For example, the…

Techniques like speckle holography and shearography are rarely applied due to the complexity of instrument setup and lack of automated result analysis, despite their potential. By simulating speckle interferometric outcomes, we seek to…

This survey highlights the recent advances in algorithms for numerical linear algebra that have come from the technique of linear sketching, whereby given a matrix, one first compresses it to a much smaller matrix by multiplying it by a…

Spectral graph sparsification aims to find ultra-sparse subgraphs whose Laplacian matrix can well approximate the original Laplacian eigenvalues and eigenvectors. In recent years, spectral sparsification techniques have been extensively…

Anomaly detection in massive networks has numerous theoretical and computational challenges, especially as the behavior to be detected becomes small in comparison to the larger network. This presentation focuses on recent results in three…

There is a large variety of machine learning methodologies that are based on the extraction of spectral geometric information from data. However, the implementations of many of these methods often depend on traditional eigensolvers, which…

We develop a model reduction technique for non-smooth dynamical systems using spectral submanifolds. Specifically, we construct low-dimensional, sparse, nonlinear and non-smooth models on unions of slow and attracting spectral submanifolds…

We address the subset selection problem for matrices, where the goal is to select a subset of $k$ columns from a "short-and-fat" matrix $X \in \mathbb{R}^{m \times n}$, such that the pseudoinverse of the sampled submatrix has as small…

Relying on recent advances in statistical estimation of covariance distances based on random matrix theory, this article proposes an improved covariance and precision matrix estimation for a wide family of metrics. The method is shown to…

Community detection is one of the fundamental problems of network analysis, for which a number of methods have been proposed. Most model-based or criteria-based methods have to solve an optimization problem over a discrete set of labels to…

This paper studies two potential modifications of XTrace (Epperly et al., SIMAX 45(1):1-23, 2024), a randomized algorithm for estimating the trace of a matrix. The first is a variance reduction step that averages the output of XTrace over…

Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Iterative random sparsification methods allow for the estimation of a single dominant eigenvalue at reduced cost by…

We consider open systems generated from one-dimensional maps that admit a finite Markov partition and use the recently developed theory of isospectral graph transformations to estimate a system's survival probabilities. We show that these…