A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Study eigenvector overlaps in large Gaussian matrices, simplifying for GOE.
Machine learning models perform better with location coordinates alone, not Moran Eigenvectors.
Paper addresses eigenvector perturbation in small eigen-gap scenarios.
We calculate eigenvector overlaps between intersecting time periods of covariance matrices.
In many applications, one has side information, e.g., labels that are provided in a semi-supervised manner, about a specific target region of a large data set, and one wants to perform machine learning and data analysis tasks "nearby" that prespecified target region. For example, one might be interested in the clusteri…
In spectral clustering, one defines a similarity matrix for a collection of data points, transforms the matrix to get the Laplacian matrix, finds the eigenvectors of the Laplacian matrix, and obtains a partition of the data using the leading eigenvectors. The last step is sometimes referred to as rounding, where one ne…
We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in massive data analysis. Singer-Wu proposed the vector diffusion map which embeds mani…
New metric tensor field on symmetric matrices simplifies eigenvector computation.
Spectral clustering performance depends on eigenvector fluctuations, shown to be Gaussian.
New neural architectures invariant to sign flips and basis symmetries for graph representation learning.
New algorithm updates eigenvectors of evolving graphs efficiently.
How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite second moment and distributions supported in a centered Euclidean ball, we prove …
New theory for eigenvectors of generalized Laplacian matrices, addressing dependency issues.
Fast algorithm recovers principal eigenvector from noisy matrices.
This paper develops the exact linear relationship between the leading eigenvector of the unnormalized modularity matrix and the eigenvectors of the adjacency matrix. We propose a method for approximating the leading eigenvector of the modularity matrix, and we derive the error of the approximation. There is also a comp…
The paper explores how kernel eigenalignments affect generalization in KRR.
The problem of estimating sparse eigenvectors of a symmetric matrix attracts a lot of attention in many applications, especially those with high dimensional data set. While classical eigenvectors can be obtained as the solution of a maximization problem, existing approaches formulate this problem by adding a penalty te…
Improved spectral clustering with fewer eigenvectors performs better.
New insights into spectral clustering reveal strong connections within eigenvectors.
The original contributions of this paper are twofold: a new understanding of the influence of noise on the eigenvectors of the graph Laplacian of a set of image patches, and an algorithm to estimate a denoised set of patches from a noisy image. The algorithm relies on the following two observations: (1) the low-index e…
We characterize the contractions that are similar to the backward shift in the Hardy space . This characterization is given in terms of the geometry of the eigenvector bundles of the operators.
Graph convolutional networks fail to use eigenvectors beyond the first, unlike spectral embedding.
We provide new examples of diffusion operators in dimension 2 and 3 which have orthogonal polynomials as eigenvectors. Their construction rely on the finite subgroups of O(3) and their invariant polynomials.
Study eigenvalues and eigenvectors in neural networks, focusing on signal propagation.
The paper tackles learning symmetries in data without expert knowledge.
This paper aims to address two fundamental challenges arising in eigenvector estimation and inference for a low-rank matrix from noisy observations: (1) how to estimate an unknown eigenvector when the eigen-gap (i.e. the spacing between the associated eigenvalue and the rest of the spectrum) is particularly small; (2) …
New method improves covariance estimation for weighted samples.
We apply random matrix theory to compare correlation matrix estimators C obtained from emerging market data. The correlation matrices are constructed from 10 years of daily data for stocks listed on the Johannesburg Stock Exchange (JSE) from January 1993 to December 2002. We test the spectral properties of C against ra…
We prove a central limit theorem for the components of the largest eigenvectors of the adjacency matrix of a finite-dimensional random dot product graph whose true latent positions are unknown. In particular, we follow the methodology outlined in \citet{sussman2012universally} to construct consistent estimates for the …
Sign equivariant networks improve model expressiveness for spectral geometric learning.
Biological neurons learn tensor decompositions of higher-order correlations using nonlinear Hebbian plasticity.
LEGO estimates tangent spaces more robustly than LPCA in noisy data.
In this paper, we apply tools from the random matrix theory (RMT) to estimates of correlations across volatility of various assets in the S&P 500. The volatility inputs are estimated by modeling price fluctuations as GARCH(1,1) process. The corresponding correlation matrix is constructed. It is found that the distribut…
Algorithm estimates principal eigenvector with adaptive sensing, improving over non-adaptive methods.
The paper proves Lipschitz regularity of graph Laplacian eigenvectors on random data clouds.
Sep-SpectralNet improves SE for broader applicability and scalability.
We quantify uncertainty in Oja's algorithm's leading eigenvector estimation.
Repeated application of machine-learning, eigen-centric methods to an evolving dataset reveals that eigenvectors calculated by well-established computer implementations are not stable along an evolving sequence. This is because the sign of any one eigenvector may point along either the positive or negative direction of…
Kernel method is a very powerful tool in machine learning. The trick of kernel has been effectively and extensively applied in many areas of machine learning, such as support vector machine (SVM) and kernel principal component analysis (kernel PCA). Kernel trick is to define a kernel function which relies on the inner-…
New method detects global factors near BBP phase transition in high-dimensional data.
Study on signal recovery from low-rank matrix with sparse noise.
We develop and analyze efficient "coordinate-wise" methods for finding the leading eigenvector, where each step involves only a vector-vector product. We establish global convergence with overall runtime guarantees that are at least as good as Lanczos's method and dominate it for slowly decaying spectrum. Our methods a…
Spectral methods are popular in detecting global structures in the given data that can be represented as a matrix. However when the data matrix is sparse or noisy, classic spectral methods usually fail to work, due to localization of eigenvectors (or singular vectors) induced by the sparsity or noise. In this work, we …
We prove a central limit theorem for the components of the eigenvectors corresponding to the largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized La…
Recently, Mahoney and Orecchia demonstrated that popular diffusion-based procedures to compute a quick \emph{approximation} to the first nontrivial eigenvector of a data graph Laplacian \emph{exactly} solve certain regularized Semi-Definite Programs (SDPs). In this paper, we extend that result by providing a statistica…
The online problem of computing the top eigenvector is fundamental to machine learning. In both adversarial and stochastic settings, previous results (such as matrix multiplicative weight update, follow the regularized leader, follow the compressed leader, block power method) either achieve optimal regret but run slow,…
This paper focuses on obtaining clustering information about a distribution from its i.i.d. samples. We develop theoretical results to understand and use clustering information contained in the eigenvectors of data adjacency matrices based on a radial kernel function with a sufficiently fast tail decay. In particular, …
The smallest eigenvectors of the graph Laplacian are well-known to provide a succinct representation of the geometry of a weighted graph. In reinforcement learning (RL), where the weighted graph may be interpreted as the state transition process induced by a behavior policy acting on the environment, approximating the …