Smallest Eigenvalue

The smallest eigenvalue of a matrix is a crucial parameter with broad applications across diverse fields, driving research focused on efficient computation and understanding its implications for various models and algorithms. Current research emphasizes developing distributed and parallel algorithms, particularly for large-scale matrices arising in machine learning (e.g., neural tangent kernels, graph Laplacians) and signal processing, as well as exploring the role of smallest eigenvalues in characterizing model performance and convergence rates. Understanding the distribution and behavior of smallest eigenvalues is vital for improving the accuracy, efficiency, and interpretability of numerous methods in areas ranging from spectral clustering to neural network optimization and physics-informed machine learning.

Papers

November 19, 2024

Unlocking State-Tracking in Linear RNNs Through Negative Eigenvalues
Riccardo Grazzi, Julien Siems, Jörg K.H. Franke, Arber Zela, Frank Hutter, Massimiliano Pontil
Smallest Eigenvalue Linear Recurrent Linear RNN

October 8, 2024

Simple Relative Deviation Bounds for Covariance and Gram Matrices
Daniel Barzilai, Ohad Shamir
Covariance Conditioning Smallest Eigenvalue Gram Matrix

September 10, 2024

Distributed Cooperative AI for Large-Scale Eigenvalue Computations Using Neural Networks
Ronald Katende
Neural Network Decentralized Algorithm Smallest Eigenvalue Cooperative AI

May 23, 2024

Bounds for the smallest eigenvalue of the NTK for arbitrary spherical data of arbitrary dimension
Kedar Karhadkar, Michael Murray, Guido Montúfar
Neural Network Neural Tangent Kernel General Bound Smallest Eigenvalue Spherical Data Arbitrary Dimension Independent Variable Collinearity Hemisphere Neural Network

May 16, 2024

Distribution of Test Statistic for Euclidean Distance Matrices
Dawson Beatty
Product Distribution Smallest Eigenvalue Empirical Distribution Distance Matrix Test Statistic

February 15, 2024

Nonlinear spiked covariance matrices and signal propagation in deep neural networks
Zhichao Wang, Denny Wu, Zhou Fan
Deep Neural Network Nonlinear Model Propagation Environment Convolutional Kernel Dimensional Signal Smallest Eigenvalue Spiked Matrix

January 28, 2024

Improving Expressive Power of Spectral Graph Neural Networks with Eigenvalue Correction
Kangkang Lu, Yanhua Yu, Hao Fei, Xuan Li, Zixuan Yang, Zirui Guo, Meiyu Liang, Mengran Yin, Tat-Seng Chua
Expressive Power Spectral Graph Neural Network Spectral Graph Smallest Eigenvalue Polynomial Filter

November 24, 2023

Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal
Vincent P. Grande, Michael T. Schaub
Spectral Feature Graph Laplacian Smallest Eigenvalue Hodge Laplacian

September 29, 2023

Optimization on the smallest eigenvalue of grounded Laplacian matrix via edge addition
Xiaotian Zhou, Haoxin Sun, Wei Li, Zhongzhi Zhang
Optimization Purpose Faster Convergence Greedy Algorithm Leader Follower Smallest Eigenvalue Laplacian Matrix Edge Function

August 18, 2023

Eigenvalue-based Incremental Spectral Clustering
Mieczysław A. Kłopotek, Bartłmiej Starosta, Sławomir T. Wierzchoń
Data Clustering Spectral Clustering Smallest Eigenvalue Incremental Clustering

July 21, 2023

Universality of Linear Recurrences Followed by Non-linear Projections: Finite-Width Guarantees and Benefits of Complex Eigenvalues
Antonio Orvieto, Soham De, Caglar Gulcehre, Razvan Pascanu, Samuel L. Smith
\Sigma}{\Delta}$ Low Pas RNN K$ Universality Smallest Eigenvalue Causal Order Linear RNN Recurrence Relation

July 2, 2023

Moments, Random Walks, and Limits for Spectrum Approximation
Yujia Jin, Christopher Musco, Aaron Sidford, Apoorv Vikram Singh
Continuum Limit Wasserstein Distance Random Walk Temporal Moment Adjacency Matrix Smallest Eigenvalue Spectral Approximation

December 23, 2022

Eigenvalue initialisation and regularisation for Koopman autoencoders
Jack W. Miller, Charles O'Neill, Navid C. Constantinou, Omri Azencot
Regularization Model Koopman Operator Physic Informed Deep Learning Eigen Portfolio Smallest Eigenvalue Koopman Autoencoders

December 19, 2022

A note on the smallest eigenvalue of the empirical covariance of causal Gaussian processes
Ingvar Ziemann
Short Note Vector Autoregression Smallest Eigenvalue Tail Bound

December 8, 2022

A Distributed Block Chebyshev-Davidson Algorithm for Parallel Spectral Clustering
Qiyuan Pang, Haizhao Yang
Spectral Clustering Smallest Eigenvalue Chebyshev Davidson Algorithm Parallel Clustering

November 8, 2022

First principles physics-informed neural network for quantum wavefunctions and eigenvalue surfaces
Marios Mattheakis, Gabriel R. Schleder, Daniel T. Larson, Efthimios Kaxiras
Physic Informed Neural Network First Principle Smallest Eigenvalue Deep Learning Solution Wave Function Eigenfunction Decomposition

July 23, 2022

Exploration in Linear Bandits with Rich Action Sets and its Implications for Inference
Debangshu Banerjee, Avishek Ghosh, Sayak Ray Chowdhury, Aditya Gopalan
Scientific Inference Environment Exploration Linear Bandit Future Implication Sub Linear Regret Smallest Eigenvalue

July 12, 2022

Accelerating Certifiable Estimation with Preconditioned Eigensolvers
David M. Rosen
Robust Perception Semidefinite Relaxation Smallest Eigenvalue Outlier Robust Burer Monteiro

May 26, 2022

On the Eigenvalues of Global Covariance Pooling for Fine-grained Visual Recognition
Yue Song, Nicu Sebe, Wei Wang
Fine Grained Fine Grained Visual Smallest Eigenvalue Fine Grained Visual Recognition Global Covariance Pooling

January 27, 2022

Eigenvalues of Autoencoders in Training and at Initialization
Benjamin Dees, Susama Agarwala, Corey Lowman
Training Data Supervised Autoencoder New Initialization MNIST Dataset Random Matrix Smallest Eigenvalue Eigenvalue Distribution

Smallest Eigenvalue

Papers

Unlocking State-Tracking in Linear RNNs Through Negative Eigenvalues

Simple Relative Deviation Bounds for Covariance and Gram Matrices

Distributed Cooperative AI for Large-Scale Eigenvalue Computations Using Neural Networks

Bounds for the smallest eigenvalue of the NTK for arbitrary spherical data of arbitrary dimension

Distribution of Test Statistic for Euclidean Distance Matrices

Nonlinear spiked covariance matrices and signal propagation in deep neural networks

Improving Expressive Power of Spectral Graph Neural Networks with Eigenvalue Correction

Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal

Optimization on the smallest eigenvalue of grounded Laplacian matrix via edge addition

Eigenvalue-based Incremental Spectral Clustering

Universality of Linear Recurrences Followed by Non-linear Projections: Finite-Width Guarantees and Benefits of Complex Eigenvalues

Moments, Random Walks, and Limits for Spectrum Approximation

Eigenvalue initialisation and regularisation for Koopman autoencoders

A note on the smallest eigenvalue of the empirical covariance of causal Gaussian processes

A Distributed Block Chebyshev-Davidson Algorithm for Parallel Spectral Clustering

First principles physics-informed neural network for quantum wavefunctions and eigenvalue surfaces

Exploration in Linear Bandits with Rich Action Sets and its Implications for Inference

Accelerating Certifiable Estimation with Preconditioned Eigensolvers

On the Eigenvalues of Global Covariance Pooling for Fine-grained Visual Recognition

Eigenvalues of Autoencoders in Training and at Initialization