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.
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.
New method improves covariance estimation for weighted samples.
problem Improving covariance estimation for weighted sample data.
method Asymptotic non-linear shrinkage formulas for covariance and precision matrix estimators of weighted sample covariances.
result Asymptotic non-linear shrinkage formulas for covariance and precision matrix estimators of weighted sample covariances.
We obtain general, exact formulas for the overlaps between the eigenvectors of large correlated random matrices, with additive or multiplicative noise. These results have potential applications in many different contexts, from quantum thermalisation to high dimensional statistics. We find that the overlaps only depend …
Study on overlaps of singular vectors in Gaussian matrix submatrices.
problem Analyzing overlaps of singular vectors in submatrices of Gaussian matrices.
method Utilizes dynamics of singular vectors and specific resolvents for Brownian trajectories.
result Explicit forms for limiting rescaled mean squared overlaps in the bulk of spectra.
Sparse spectral decomposition identifies overlapping communities in networks.
problem Estimating overlapping community memberships in networks where nodes can belong to multiple communities.
method Sparse principal subspace estimation with iterative thresholding.
result The fixed point of the algorithm corresponds to correct node memberships under the stochastic block model.
Study on signal recovery from low-rank matrix with sparse noise.
problem Inference of a rank-one signal in the presence of sparse noise.
method Replica method from statistical physics, recursive distributional equations, population dynamics algorithm.
result Critical signal strength for recovery via top eigenvector identified.
We propose a general framework to study the stability of the subspace spanned by P consecutive eigenvectors of a generic symmetric matrix H0, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (H0 is then the Hamiltonian) and financial ris…
We investigate the problem of estimating a given real symmetric signal matrix C from a noisy observation matrix M in the limit of large dimension. We consider the case where the noisy measurement M comes either from an arbitrary additive or multiplicative rotational invariant perturbati…
Optimal classifiers derived from GMMs are approximated by deep neural networks.
problem Binary classification of high-dimensional overlapping Gaussian mixtures.
method Closed-form expressions for Bayes optimal decision boundaries derived from GMMs' eigenstructure. Empirical validation through synthetic and real-world data.
result Deep neural networks approximate optimal classifiers for GMMs, with decision thresholds related to covariance eigenvectors.
Clustering of data sets is a standard problem in many areas of science and engineering. The method of spectral clustering is based on embedding the data set using a kernel function, and using the top eigenvectors of the normalized Laplacian to recover the connected components. We study the performance of spectral clust…
We consider the problem of estimating community memberships of nodes in a network, where every node is associated with a vector determining its degree of membership in each community. Existing provably consistent algorithms often require strong assumptions about the population, are computationally expensive, and only p…
Our work connects parameter magnitudes and Hessian eigenspaces in deep neural nets.
problem Understanding the relationship between parameter magnitudes and Hessian curvature in deep learning models.
method Developed a matrix-free algorithm based on sketched SVDs to measure similarity between parameter masks and Hessian eigenspaces.
result Top Hessian eigenvectors tend to be concentrated around larger parameters, indicating a connection between parameter magnitudes and loss curvature.
Study spectral estimators for multi-index models to recover low-dimensional signal subspaces.
problem Recovering low-dimensional signal subspaces in multi-index models.
method Spectral estimators for multi-index models.
result Precise asymptotic characterization of spectral methods' performance, revealing a phase transition for weak recovery.
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.
This paper uncovers the low-rank structure of neural network Hessians.
problem Understanding the structure of Hessians in neural networks.
method Proposes a decoupling conjecture to decompose layer-wise Hessians into Kronecker products of smaller matrices.
result Proves the structure of top eigenspaces in 2-layer networks and shows high overlap in top eigenvectors across different models.
Optimal spectral method found for inhomogeneous spiked Wigner model.
problem Structured noise in learning scenarios.
method Random matrix theory and spectral analysis.
result Optimal threshold for phase transition in block-structured Wigner model.
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.
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.
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.
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.
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.
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.
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…
New method improves subspace iteration for eigenvectors in machine learning.
problem Computing eigenvectors for large-scale problems in machine learning.
method Subspace iteration with ℓ2o∞ norm convergence analysis. result Deterministic bounds and practical stopping criterion for improved performance.
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.
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.
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.
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.
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.
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 H2. 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.
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 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.
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.
Overlapping clustering problem is an important learning issue in which clusters are not mutually exclusive and each object may belongs simultaneously to several clusters. This paper presents a kernel based method that produces overlapping clusters on a high feature space using mercer kernel techniques to improve separa…
Deconfounding scores improve causal effect estimation with weak overlap.
problem Challenges in causal treatment effect estimation due to weak overlap in high-dimensional data.
method Propose deconfounding scores to preserve identification and target estimation while improving overlap.
result Prognostic scores are overlap-optimal under a broad family of generalized linear models with Gaussian features.
A new method speeds up overlapping group lasso computations.
problem Time-consuming optimization of overlapping group lasso on large-scale problems.
method Non-overlapping statistical approximation to overlapping group lasso.
result The proposed penalty is statistically equivalent to overlapping group lasso.
In medicine, visualizing chromosomes is important for medical diagnostics, drug development, and biomedical research. Unfortunately, chromosomes often overlap and it is necessary to identify and distinguish between the overlapping chromosomes. A segmentation solution that is fast and automated will enable scaling of co…
New method improves CATE estimation in low overlap regions.
problem Low overlap in CATE estimation leads to poor performance of meta-learners.
method Overlap-Adaptive Regularization (OAR) that regularizes models proportionally to overlap weights.
result OAR significantly improves CATE estimation in low-overlap settings.
Proposes a sensitivity framework to handle limited overlap in causal inference.
problem Limited overlap between treated and control groups in observational studies.
method Sensitivity framework based on worst-case confidence bounds on bias introduced by trimming.
result Protects against spurious findings by quantifying uncertainty in regions with limited overlap.
Producing overlapping schemes is a major issue in clustering. Recent proposed overlapping methods relies on the search of an optimal covering and are based on different metrics, such as Euclidean distance and I-Divergence, used to measure closeness between observations. In this paper, we propose the use of another meas…