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…
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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.
169,341 papers · 148 categories
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48 results for “eigenvector localization”
We calculate eigenvector overlaps between intersecting time periods of covariance matrices.
problem Analyzing overlapping time periods in covariance matrices.
method Girko linearisation and extended local laws.
result Computed eigenvector overlaps for intersecting time intervals.
New method detects global structures in sparse or noisy data.
problem Localization of eigenvectors in sparse or noisy data.
method Learn a regularization matrix from localized eigenvectors.
result Suppresses eigenvalues associated with localized eigenvectors.
LEGO estimates tangent spaces more robustly than LPCA in noisy data.
problem Estimating tangent spaces in high-noise settings.
method Spectral method using graph Laplacian eigenvectors and gradient orthogonization.
result LEGO yields more robust tangent space estimates than LPCA.
The paper embeds manifolds into finite Euclidean spaces using eigenvector fields of the connection Laplacian.
problem Embedding manifolds into finite-dimensional Euclidean spaces using eigenvector fields of the connection Laplacian.
method Constructing local coordinate charts with low distortion using eigenvector fields and proving estimates for eigenvector fields and the heat kernel.
result The distortion constants depend only on geometric properties of manifolds in the little Hölder space , allowing for embedding into a finite-dimensional Euclidean space.
We analyzed cross-correlations between price fluctuations of global financial indices (20 daily stock indices over the world) and local indices (daily indices of 200 companies in the Korean stock market) by using random matrix theory (RMT). We compared eigenvalues and components of the largest and the second largest ei…
New theory for eigenvectors of generalized Laplacian matrices, addressing dependency issues.
problem Dependency in random matrix theory hinders eigenvector analysis for latent embeddings.
method Introduces generalized Laplacian matrices and a new asymptotic theory framework.
result Established asymptotic normalities for spiked eigenvectors and eigenvalues.
LDLE embeds manifolds in lower dimensions with low distortion.
problem Embedding manifolds in lower dimensions with low distortion.
method Constructs local views using global eigenvectors of the graph Laplacian, registers them using Procrustes analysis, and tears manifolds apart for intrinsic dimension embedding.
result LDLE preserves distances up to a constant scale with low distortion.
SCI-PI solves scale invariant problems efficiently.
problem Solving scale invariant problems in optimization.
method Introduces SCI-PI and proves its convergence.
result SCI-PI achieves local linear convergence.
New method uses random signals to quickly approximate graph eigenvectors.
problem Estimating the first k eigenvectors of graph Laplacians efficiently.
method Filtering Gaussian random signals to recover eigenvectors.
result Only k random signals needed to recover k smallest eigenvectors.
CDP reduces point cloud dimensions by preserving detour-induced local non-convexity.
problem Preserving local non-convexity in point cloud dimensionality reduction.
method CDP builds a k-NN graph, identifies admissible pairs, aggregates normalized directions, and uses top-k eigenvectors for projection.
result CDP provides verifiable guarantees on post-projection distortion and direction energy.
Universal and non-universal properties of cross-correlations in financial time seriescond-mat.stat-mech
We use methods of random matrix theory to analyze the cross-correlation matrix C of price changes of the largest 1000 US stocks for the 2-year period 1994-95. We find that the statistics of most of the eigenvalues in the spectrum of C agree with the predictions of random matrix theory, but there are deviations for a fe…
Study eigenvector overlaps in large Gaussian matrices, simplifying for GOE.
problem Investigate eigenvector overlaps in large Gaussian matrices.
method Analysis of eigenvector flow under Dyson Brownian motion.
result Explicit computation of limiting rescaled mean squared overlaps.
Algorithm improves online canonical correlation analysis.
problem Online canonical correlation analysis.
method Stochastic Scaled-Gradient Descent (SSGD) for minimizing expectation over Riemannian manifolds.
result Achieved optimal one-time-scale algorithm with explicit rate of local asymptotic convergence.
Study shows how many samples are needed for eigenvector/eigenvalue accuracy.
problem Guaranteeing eigenvector and eigenvalue accuracy of sample vs actual covariance matrices.
method Proves inner product decrease proportional to eigenvalue distance for various distributions.
result Non-asymptotic concentration bounds and conditions for distinguishing principal components.
Machine learning models perform better with location coordinates alone, not Moran Eigenvectors.
problem Improving machine learning models for spatial data.
method Examined Moran Eigenvectors as additional spatial features in machine learning models using synthetic datasets.
result Machine learning models using only location coordinates achieve better accuracies than eigenvector-based approaches.
Paper addresses eigenvector perturbation in small eigen-gap scenarios.
problem Fine-grained behavior of eigenvectors in the presence of small eigen-gaps.
method Develops de-biased estimators for linear functions of an unknown eigenvector.
result Achieves minimax lower bounds for a family of scenarios, even with small eigen-gaps.
We efficiently find top k eigenvectors in streaming PCA with global convergence.
problem Finding top k eigenvectors in streaming PCA with limited space.
method Developed global convergence for Oja's algorithm and a faster variant Oja++.
result Achieved global convergence rate matching information theoretic lower bound.
New methods find eigenvectors faster than Lanczos's method.
problem Finding the leading eigenvector efficiently.
method Coordinate-wise methods combining shift-and-invert with linear regression.
result Global convergence with runtime guarantees better than Lanczos's method.
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…
New method estimates sparse eigenvectors without sacrificing orthogonality.
problem Estimating sparse eigenvectors of a symmetric matrix.
method Developed a new method using MM framework and Procrustes reformulation.
result Improves support recovery and explained variance compared to existing methods.
New metric tensor field on symmetric matrices simplifies eigenvector computation.
problem Complex eigenvector computation for 2x2 symmetric matrices.
method Introducing a metric tensor field on the space of symmetric matrices, resulting in a curved manifold.
result Parallel transport simplifies eigenvector computation for one-parameter families of matrices.
Spectral clustering performance depends on eigenvector fluctuations, shown to be Gaussian.
problem Predicting the performance of spectral clustering.
method General spike random matrix model and rotational invariance of noise.
result Fluctuations of eigenvector entries are Gaussian in large-dimensional regime.
New neural architectures invariant to sign flips and basis symmetries for graph representation learning.
problem Learning invariant graph representations from eigenvectors.
method SignNet and BasisNet neural architectures that are invariant to sign flips and basis symmetries.
result Proven to be universal, approximating any continuous function of eigenvectors with desired invariances.
New algorithm updates eigenvectors of evolving graphs efficiently.
problem Updating eigenvectors of dynamic graphs.
method Subspace projection based on Rayleigh-Ritz projections.
result Strong performance in eigenvector approximation and downstream tasks.
New method trains neural networks in spectral domain for improved performance.
problem Training deep neural networks in the space of nodes.
method Trains neural networks in the spectral domain, modifying eigenvalues and eigenvectors of transfer operators.
result Superior performance compared to standard methods, especially when adjusting eigenvalues.
We define a random-matrix ensemble given by the infinite-time covariance matrices of Ornstein-Uhlenbeck processes at different temperatures coupled by a Gaussian symmetric matrix. The spectral properties of this ensemble are shown to be in qualitative agreement with some stylized facts of financial markets. Through the…
New algorithm consistently orients eigenvectors for machine learning.
problem Inconsistent eigenvector orientation in machine learning.
method Postprocesses well-established eigen calls to create consistently oriented eigenvectors.
result Interpretable time series of training weights in machine learning models.
A new algorithm improves Wasserstein discriminant analysis for better data classification.
problem Improving data classification in machine learning.
method Bi-level nonlinear eigenvector algorithm (WDA-nepv) for optimal transport and trace ratio optimizations.
result WDA-nepv enhances classification accuracy and scalability.
The study proves a central limit theorem for eigenvectors of the normalized Laplacian in random graphs.
problem Understanding the distribution of eigenvectors of the normalized Laplacian in random graphs.
method Proving a central limit theorem for eigenvectors of the normalized Laplacian for random graphs.
result The components of the eigenvectors corresponding to the largest eigenvalues of the normalized Laplacian matrix converge to multivariate normals.
A new algorithm reduces online eigenvector computation time while maintaining optimal performance.
problem Online learning of top eigenvectors in both adversarial and stochastic settings.
method Follow the Compressed Leader (FTCL) framework, compressing the matrix strategy to dimensions 3 (adversarial) and 1 (stochastic).
result Achieves optimal regret without sacrificing running time, resolving open questions.
Fast algorithm recovers principal eigenvector from noisy matrices.
problem Recovering the first principal eigenvector from noisy positive semidefinite matrices.
method Cone projected power iteration algorithm.
result Achieves polynomial time complexity and small error for certain convex cones.
The paper explores how kernel eigenalignments affect generalization in KRR.
problem Achieving robust generalization in kernel methods.
method Direct connection between generalization and matrix eigenvectors/eigenvalues, focusing on finite-sample settings.
result Strong generalization requires increasing eigenvector alignment, eigenvalue magnitude, or gaps between eigenvalues.
New method improves subspace iteration for eigenvectors in machine learning.
problem Computing eigenvectors for large-scale problems in machine learning.
method Subspace iteration with norm convergence analysis.
result Deterministic bounds and practical stopping criterion for improved performance.
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…
A new method approximates Laplacian eigenvectors for RL efficiently.
problem Efficiently learning state representations in RL.
method General and scalable approach to approximating Laplacian eigenvectors.
result Empirically shows improved performance in RL tasks.
Paper tackles small eigen-gap estimation and inference for noisy symmetric matrices.
problem Estimating eigenvectors with small eigen-gap and fine-grained statistical reasoning.
method Eigen-decomposition of asymmetric data matrix, distribution-free procedures, adaptive to heteroscedastic noise.
result Minimax optimal under Gaussian noise, confidence intervals for eigenvalues, small eigen-gap handling.
Improved spectral clustering with fewer eigenvectors performs better.
problem Improving spectral clustering performance under weaker conditions.
method Tighter analysis and using fewer eigenvectors for embedding.
result Spectral clustering can produce better results with fewer eigenvectors.
New insights into spectral clustering reveal strong connections within eigenvectors.
problem Clustering on graphs when there are two underlying clusters.
method Analyzes the eigenvector corresponding to the second largest eigenvalue of the adjacency matrix.
result Vertices with extreme values in the eigenvector are more reliably classified.
Perturbation of the Eigenvectors of the Graph Laplacian: Application to Image Denoisingphysics.data-an
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…
This paper proves the convergence rate of Krasulina's estimator for least eigenvalue and eigenvector.
problem Finding the least eigenvalue and eigenvector of an unknown covariance matrix.
method Developed a convergence proof for Krasulina's estimator.
result Established the convergence rate of Krasulina's estimator for the least eigenvalue and eigenvector.
Graph convolutional networks fail to use eigenvectors beyond the first, unlike spectral embedding.
problem Understanding when graph convolutional networks fail compared to spectral embedding.
method Presented a simple generative model to illustrate failure.
result Graph convolutional networks fail to use eigenvectors beyond the first in certain graphs.
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.
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.
A new spectral clustering algorithm that avoids eigenvector computation.
problem Computational complexity in spectral clustering for large datasets.
method A mixing process on a graph to find a linear combination of eigenvectors without computing eigenvectors.
result Partitioning datasets achieves better accuracy than standard spectral clustering methods.
Study eigenvalues and eigenvectors in neural networks, focusing on signal propagation.
problem Characterize signal eigenvalues and eigenvectors in neural networks.
method Characterizes signal eigenvalues and eigenvectors for a nonlinear spiked covariance model.
result Provides precise quantitative characterizations of signal eigenvalues and eigenvectors in neural networks.
The paper tackles learning symmetries in data without expert knowledge.
problem Learning symmetries in data from raw data without prior knowledge.
method Develops methods to select eigenvectors for orthogonal symmetries and compares their effectiveness.
result The problem of learning symmetries is as hard as the graph automorphism problem in the worst case, but can be simplified with certain restrictions.
A new method for distributed PCA using matrix β-mean.
problem Efficiently aggregating PCA results across multiple machines with reduced computational overhead.
method Proposes a novel DPCA method that incorporates eigenvalue information using the matrix β-mean.
result The matrix β-mean method improves robustness and stability of eigenvector ordering.