The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important difference is that the population covariance matrices, which are assumed to be non-random…
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48 results for “data matrices”
Study on random matrices in deep neural networks with IID entries.
problem Distribution of singular values in product of random matrices for deep neural networks.
method Random matrix theory with a streamlined approach for non-Gaussian data.
result Generalization of macroscopic universality property to non-Gaussian data.
The article explains how to estimate confusion matrices for classifiers using unlabeled data.
problem Estimating sensitivity and specificity of binary medical diagnostic tests without gold standard tests.
method Modifying diagnostic test solutions to estimate confusion matrices for classifiers on unlabeled data.
result The approach can be used to estimate accuracy statistics for supervised or unsupervised binary classifiers on unlabeled data.
Testing for latent structure via the Wilcoxon--Wigner random matrix of normalized rank statisticsstat.ME
Paper develops new method for detecting latent structure in large symmetric data matrices.
problem Testing for latent structure in large symmetric data matrices.
method Introduces Wilcoxon--Wigner random matrices based on normalized rank statistics.
result Establishes asymptotic Gaussian fluctuations for leading eigenvalue and eigenvector of Wilcoxon--Wigner matrices.
A general framework for solving the subspace clustering problem using the CUR decomposition is presented. The CUR decomposition provides a natural way to construct similarity matrices for data that come from a union of unknown subspaces . The similarity matrices thus c…
Scalable methods for nonnegative matrix factorizations of near-separable tall-and-skinny matricescs.LG
Numerous algorithms are used for nonnegative matrix factorization under the assumption that the matrix is nearly separable. In this paper, we show how to make these algorithms efficient for data matrices that have many more rows than columns, so-called "tall-and-skinny matrices". One key component to these improved met…
Analyzes tt*-structures from -type Stokes data.
problem Classifying tt*-structures over .
method Isomonodromic deformations with upper unitriangular real Stokes matrices.
result Establishes a direct analytic realization of the classification.
-means on Positive Definite Matrices, and an Application to Clustering in Radar Image Sequencesstat.ML
New -means method clusters radar image sequences using SPD matrices.
problem Clustering radar image sequences efficiently.
method Developed -means on SPD matrices for non-Euclidean data.
result Effective clustering of radar image sequences via SPD matrices.
A network community detection method with integration of data from multiple layers and node attributesphysics.soc-ph
A method for community detection in multilayer networks using data matrices.
problem Community detection in multilayer networks with various node attributes.
method Data matrix representation and regular decomposition method extension for compression.
result Method identifies community structures well-aligned with real-world network hierarchies.
Gradient flow on softmax attention minimizes nuclear norm of weight matrices.
problem Classification with separate key and query weight matrices.
method Gradient flow on exponential loss, separability assumption, reparameterization, approximate KKT conditions.
result Gradient flow implicitly minimizes nuclear norm of weight matrices, contrasting with Frobenius norm minimization.
A new model uses Toeplitz matrices to analyze time-series data transitions.
problem Analyzing transitions in time-series data from nonautonomous systems.
method Deep Koopman-layered models with learnable Toeplitz matrices, leveraging Toeplitz matrices' universal property.
result The model demonstrates universality and generalization, outperforming existing methods.
Improved eigenvalue distribution method for financial data.
problem Noise and complexity in financial markets.
method Matrix H theory, hierarchical structure, informational cascade.
result Captures a larger fraction of data variance in financial markets.
A widespread approach in machine learning to evaluate the quality of a classifier is to cross -- classify predicted and actual decision classes in a confusion matrix, also called error matrix. A classification tool which does not assume distributional parameters but only information contained in the data is based on th…
The difficulty of classification affects the weight matrices' heavy tail appearance in deep learning networks.
problem Understanding the spectral properties of weight matrices in deep learning networks.
method Spectral analysis of weight matrices in different modules of DNNs, classification difficulty as a driving factor for heavy tail appearance.
result Higher classification difficulty leads to more frequent appearance of heavy tails in weight matrices spectra.
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.
Random feature matrices' singular values concentrate near their full expectation in high dimensions.
problem Characterizing the spectra of random feature matrices for regression problems.
method Analyzing two settings of input variables (random or well-separated) with conditions on dimension, complexity ratio, and sampling variance.
result The singular values of random feature matrices concentrate near their full expectation and near one with high probability.
Deep learning method clusters multi-view data matrices.
problem Clustering heterogeneous relational data matrices.
method Deep collective matrix tri-factorization (DCMTF).
result Discover latent clusters across input matrices and their associations.
Method estimates M-matrices in graphical models with improved accuracy.
problem Estimating M-matrices as precision matrices in Gaussian graphical models.
method Adaptive multiple-stage estimation method solving weighted ℓ1-regularized problems.
result Method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification.
New method classifies manifold-valued data using Riemannian geometry.
problem Classifying data on curved Riemannian manifolds.
method Probabilistic Learning Vector Quantization on Symmetric Positive Definite Matrices.
result The method outperforms traditional Euclidean methods on manifold-valued data.
We present a theory for Euclidean dimensionality reduction with subgaussian matrices which unifies several restricted isometry property and Johnson-Lindenstrauss type results obtained earlier for specific data sets. In particular, we recover and, in several cases, improve results for sets of sparse and structured spars…
Proposes a new Sliced-Wasserstein distance for covariance matrices in M/EEG signals.
problem Efficiently dealing with distributions of covariance matrices in M/EEG multivariate time series.
method Defines a Sliced-Wasserstein distance for symmetric positive definite matrices and applies it to brain-age prediction and Brain Computer Interface applications.
result Demonstrates computational efficiency and strong theoretical guarantees for the proposed distance.
rags2ridges simplifies graphical modeling of high-dimensional data.
problem Graphical modeling of high-dimensional precision matrices.
method Modular framework for extraction, visualization, and analysis of Gaussian graphical models.
result Provides a one-stop-shop for graphical modeling of high-dimensional precision matrices.
We analyze the spectral properties of correlation matrices between distinct statistical systems. Such matrices are intrinsically non symmetric, and lend themselves to extend the spectral analyses usually performed on standard Pearson correlation matrices to the realm of complex eigenvalues. We employ some recent random…
Bayesian Inference of Transition Matrices from Incomplete Graph Data with a Topological Priorstat.ME
Bayesian method infers transition matrices from incomplete graph data with topological constraints.
problem Inference of transition matrices from incomplete graph data with topological constraints.
method Bayesian approach using repeated interactions and a topological prior.
result Higher accuracy in inferring transition probabilities, improving downstream tasks.
A framework estimates multiple precision matrices with shared structures.
problem Estimating multiple precision matrices with shared structures.
method Penalized likelihood framework with iterative algorithm alternating between convex and clustering problems.
result The method outperforms competitors and performs similarly to methods using prior information.
New technique stabilizes singular values in concatenated matrices.
problem How singular values of concatenated matrices relate to individual components.
method Developed perturbation technique extending classical results to concatenated matrices.
result Dominant singular values remain stable under small perturbations in submatrices.
The paper uses spectral flow on SPD matrices to analyze multimodal data.
problem Analyzing data from multiple sensors with shared and unique sources.
method Combines manifold learning with Riemannian geometry of SPD matrices.
result Spectral analysis of kernels on SPD manifold reveals common and unique components.
New framework estimates eigenvalues of kernel matrices without full matrix construction.
problem Estimating eigenvalues of large kernel matrices efficiently.
method Eigenvalue quantile estimation framework for kernel matrices with quick decay.
result Validates framework with empirical evidence and proves interlacing theorem.
With the huge influx of various data nowadays, extracting knowledge from them has become an interesting but tedious task among data scientists, particularly when the data come in heterogeneous form and have missing information. Many data completion techniques had been introduced, especially in the advent of kernel meth…
Geometric framework for SPD matrices preserving subspace structures.
problem Processing SPD-valued data with preserved subspace structures.
method Thompson geometry of the semidefinite cone, extreme generalized eigenvalues, geodesic space structure.
result Novel inductive mean of SPD matrices based on Thompson geometry.
Twisted Neumann--Zagier matrices for quantum invariants.
problem Constructing quantum invariants from ideal triangulations.
method Define and compute twisted Neumann--Zagier matrices from combinatorics.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial.
The paper tackles joint learning of linear systems, improving accuracy with pooled data.
problem Estimating transition matrices of multiple related linear systems more accurately.
method Developed novel techniques to bound estimation errors and establish high probability bounds for singular values.
result Significant gains in accuracy achieved by pooling data across systems.
Proposes BONMI for integrating noisy matrices from multi-source data.
problem Integrating noisy matrices from multi-source data with block-wise missingness.
method Exploits orthogonal Procrustes problem to align eigenspaces and completes missing blocks.
result Statistical rate for eigenspace of underlying matrix comparable to independently missing assumption.
WISDoM (Wishart Distributed Matrices) is a new framework for the quantification of deviation of symmetric positive-definite matrices associated to experimental samples, like covariance or correlation matrices, from expected ones governed by the Wishart distribution WISDoM can be applied to tasks of supervised learning,…
We provide bounds for kernel matrices and new approximations for high-dimensional data.
problem Approximating high-dimensional empirical kernel matrices.
method Decoupling results for U-statistics and non-commutative Khintchine inequality.
result New tighter approximations for inner-product kernel matrices.
The paper uses deep learning to detect financial market regimes from correlation matrices.
problem Detecting financial market regimes from correlation dynamics.
method Representation learning on block hierarchical SPD correlation matrices using SPDNet, SPD-NetBN, and U-SPDNet models.
result Deep learning models overfit in financial market data, misleading performance metrics.
Paper introduces MSA for weakly supervised covariance alignment in MEG signals.
problem Limited labeled signals in target datasets for MEG applications.
method Mixing model Stiefel Adaptation (MSA) leveraging unlabeled data.
result MSA outperforms recent methods in brain-age regression with MEG signals.
Study optimizes shared singular subspace estimation from noisy matrices.
problem Estimating shared singular subspaces across multiple noisy matrices.
method Low-rank matrix denoising framework with Stack-SVD and novel estimators.
result Stack-SVD achieves minimax rate-optimality for identical shared subspaces, and novel estimators for partial sharing.
Estimates covariance matrices with correlations between samples.
problem Estimating large-dimensional covariance matrices with correlated samples.
method Generalized Marcenko-Pastur equation and Ledoit-Peche shrinkage estimator using random matrix theory and free probability. Developed an efficient algorithm based on Ledoit-Wolf kernel estimation.
result Efficient algorithm for estimating large covariance matrices with correlations.
The paper models financial correlation matrices using permutation invariant Gaussian models and predicts market anomalies.
problem Modeling and predicting financial correlation matrices from high-frequency data.
method Constructing permutation invariant Gaussian matrix models with 4 parameters, using graph theory and polynomial functions.
result The permutation invariant Gaussian matrix model predicts the expectation values of cubic and quartic polynomials with strong evidence of fit.
DiffeoCFM efficiently generates realistic brain connectivity matrices using pullback metrics.
problem Generating realistic brain connectivity matrices for population heterogeneity analysis.
method Conditional flow matching on matrix manifolds via pullback metrics induced by global diffeomorphisms.
result DiffeoCFM achieves state-of-the-art performance on large-scale fMRI and EEG datasets.
It is natural to ask: what kinds of matrices satisfy the Restricted Eigenvalue (RE) condition? In this paper, we associate the RE condition (Bickel-Ritov-Tsybakov 09) with the complexity of a subset of the sphere in , where is the dimensionality of the data, and show that a class of random matrices with indep…
Paper proposes a method to classify EEG signals with missing data.
problem Handling missing data in electroencephalogram (EEG) signals for classification.
method Uses an expectation-maximization algorithm with observed-data likelihood to compute covariance matrices, compares to imputed data and Riemannian averages.
result The proposed method generally performs better than existing methods on real EEG data.
Two algorithms estimate Wasserstein distance matrices from few entries for manifold learning.
problem Estimating Wasserstein distance matrices from limited data for manifold learning.
method Proposes two algorithms: matrix completion and Nyström completion for square Wasserstein matrices.
result Nyström completion can outperform matrix completion with a fixed sample budget and improve classification stability.
Bootstrapping regularizes singular correlation matrices, reducing the need for complex regularization.
problem Singular correlation matrices in large datasets.
method Averaging bootstrapped correlation matrices to ensure positive-definiteness.
result The averaged correlation matrix is almost surely positive-definite with a sufficient number of bootstraps.
We introduce a wrapped Gaussian for SPD matrices, enhancing data analysis.
problem Handling circular and non-flat data distributions on SPD manifolds.
method Introduced a non-isotropic wrapped Gaussian using the exponential map, derived theoretical properties, and proposed a maximum likelihood framework.
result Demonstrated the robustness and flexibility of the wrapped Gaussian model on synthetic and real-world datasets.
Kaleidoscope matrices improve model quality and inference speed.
problem Choosing structured linear transformations for efficiency and accuracy.
method Introduce kaleidoscope matrices that can capture any structured matrix with near-optimal space and time complexity. Learn these matrices automatically within end-to-end pipelines.
result Kaleidoscope matrices can improve model quality and inference speed.
A New Method to Estimate the Noise in Financial Correlation Matricescond-mat.stat-mech
Financial correlation matrices measure the unsystematic correlations between stocks. Such information is important for risk management. The correlation matrices are known to be ``noise dressed''. We develop a new and alternative method to estimate this noise. To this end, we simulate certain time series and random matr…