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.
Method solves Gaussian graphical models on ladder graphs efficiently.
problem Solving Gaussian graphical models on ladder graphs efficiently.
method Proposes a method that depends on the position of zeros in local covariance matrices.
result Efficiently solves Gaussian graphical models on ladder graphs under certain conditions.
The paper proves local laws for non-separable sample covariance matrices.
problem Analyzing non-separable sample covariance matrices with dependent or nonlinearly transformed data.
method Tensor network framework for analyzing fluctuation averaging in the presence of higher-order cumulant structure.
result Optimal averaged local law and full anisotropic local law for non-separable sample covariance matrices.
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…
We simplify matrix computations for block matrices, especially useful for covariance and correlation matrices.
problem Complex computations for block matrices, especially for covariance and correlation matrices.
method Obtained a canonical representation for block matrices, facilitating computation of various matrix operations.
result Simplified computation of matrix operations for block matrices, particularly useful for covariance and correlation matrices.
GLSKF improves tensor completion by capturing both global and local variations.
problem Tensor completion with missing entries, especially in data with spatial or temporal side information.
method Integrates smoothness-constrained low-rank factorization with a locally correlated residual process.
result GLSKF achieves superior performance and scalability on real-world datasets.
Paper proposes a deep learning method for better covariance matrix forecasting.
problem Suboptimal predictive performance in traditional matrix volatility forecasting.
method Riemannian-geometry-aware deep learning framework for symmetric positive definite matrices.
result Our method outperforms traditional approaches in predictive accuracy.
Paper explores geometry of covariance matrices using associated bundles.
problem Geometry of fixed-rank covariance matrices.
method Associated bundle approach to Bures--Wasserstein geometry.
result Established a one-to-one correspondence between geodesics.
Paper solves a key problem in learning from high-dimensional covariance matrices.
problem Computing normalizing factors for Riemannian Gaussian distributions on high-dimensional covariance matrices.
method Equivalence with random matrix theory and log-normal matrix ensembles to approximate normalizing factors.
result Efficient approximation of normalizing factors with decreasing error as dimension increases.
Simple bounds for covariance and Gram matrices across various settings.
problem Capturing the behavior of smaller eigenvalues in covariance and Gram matrices.
method General-purpose theorem converting uniform bounds into relative bounds.
result Sharper control of eigenvalues across the spectrum.
New method reconstructs manifolds from data using Gaussian processes.
problem Reconstructing lower-dimensional structure from complex data.
method Local covariance matrices and Gaussian processes for probabilistic manifold reconstruction.
result Probabilistic manifold reconstruction with Gaussian processes.
Signatures of universality are detected by comparing individual eigenvalue distributions and level spacings from financial covariance matrices to random matrix predictions. A chopping procedure is devised in order to produce a statistical ensemble of asset-price covariances from a single instance of financial data sets…
We find a closed-form determinant for a specific sparse covariance matrix model.
problem Finding the determinant of a specific class of sparse positive definite matrices.
method Using Fourier transform of local factors, Normal Factor Graph Duality Theorem, and Matrix Determinant Lemma.
result We derive a closed-form expression for the determinant.
Paper optimizes federated PCA for covariance estimation under privacy constraints.
problem Privacy-preserving covariance estimation in federated learning.
method Federated PCA, matrix version of van Trees' inequality, three-layer spectral decomposition.
result Optimal rates of convergence for central server's estimation, robust to inconsistent local estimators.
Hybrid ResNet and RMT improve covariance matrix estimation for cryptocurrency portfolios.
problem Noisy, non-Gaussian financial data leads to unstable covariance matrices.
method Combines RMT regularization and ResNet learning for data-driven corrections.
result Hybrid estimator outperforms traditional methods in portfolio optimization.
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.
Diagonal transformations preserve independence structures in non-Gaussian distributions.
problem Preserving independence structures in non-Gaussian distributions.
method Diagonal nonlinear transformations of multivariate normal variables.
result Independence structures are preserved in non-Gaussian distributions under diagonal transformations.
Study extends bounds on sample covariance matrices with general dependence.
problem Quantitative bounds on sample covariance matrices with i.i.d. columns.
method Extends previous work on deterministic equivalent to rectangular random matrices with general dependence structure.
result Proves quantitative bounds involving dimensions and spectral parameter, including closer proximity to real positive semi-line.
This paper studies geodesics between covariance matrices of different ranks using the Bures-Wasserstein metric.
problem Geodesics between covariance matrices of varying ranks.
method Analyzes the Bures-Wasserstein distance on covariance matrices, completing previous work on geodesics and providing explicit formulas.
result The set of all minimizing geodesics between two covariance matrices is parametrized by a closed unit ball in R(k−r)imes(l−r). Paper analyzes ensemble Kalman updates for effective dimension and localization.
problem Why small ensemble sizes work well in inverse problems and data assimilation.
method Non-asymptotic analysis of ensemble Kalman updates, focusing on effective dimension and localization.
result Rigorously explains why a small ensemble size is sufficient when prior covariance has moderate effective dimension.
Optimizes clustering in Gaussian mixtures with varying covariance matrices.
problem Clustering with anisotropic Gaussian mixture models where covariance matrices vary.
method Proposes a computationally feasible hard EM type algorithm.
result Achieves optimal clustering rate with few iterations.
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.
Much recent work has concerned sparse approximations to speed up the Gaussian process regression from the unfavorable O(n3) scaling in computational time to O(nm2). Thus far, work has concentrated on models with one covariance function. However, in many practical situations additive models with multiple covariance func…
Better signal detection in undersampled data using joint and cross covariances.
problem Detecting shared signals in high-dimensional data with limited samples.
method Analysis of three covariance matrices: individual, cross, and joint.
result Joint and cross covariance matrices detect signals earlier than individual covariances.
The inverse covariance matrix provides considerable insight for understanding statistical models in the multivariate setting. In particular, when the distribution over variables is assumed to be multivariate normal, the sparsity pattern in the inverse covariance matrix, commonly referred to as the precision matrix, cor…
Regularized EM algorithm improves GMM clustering in low sample settings.
problem Numerical instability and convergence issues in EM-GMM for low sample support.
method Regularized EM algorithm that maximizes penalized GMM likelihood, ensuring positive definiteness and structured covariance matrices.
result The regularized EM algorithm leads to better performing EM for structured covariance matrix models or low sample settings.
Method estimates sparse inverse covariance and partial correlation matrices efficiently.
problem Sparse high-dimensional inverse covariance and partial correlation matrix estimation.
method Two-stage estimation method using partial regression with positive semi-definiteness.
result Efficient estimation of inverse covariance and partial correlation matrices with derived non-asymptotic rates.
The paper addresses portfolio allocation with uncertain covariance matrices, finding a logarithmic risk dependence.
problem Portfolio allocation with uncertain covariance matrices.
method Calculates the expected value of CARA utility function over a distribution of covariance matrices, considering uncertainty in future returns and covariances.
result Marginalization introduces a logarithmic dependence on risk, leading to lower allocation levels for higher uncertainties.
This paper presents a new method for estimating high dimensional covariance matrices. The method, permuted rank-penalized least-squares (PRLS), is based on a Kronecker product series expansion of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to…
Paper introduces a novel method for dynamic covariance estimation with random forests.
problem Estimating high-dimensional dynamic covariance matrices with multiple covariates.
method Nonparametric approach using random forests.
result Uniform consistency theory and error rates established for high-dimensional scenarios.
In this short note we provide an analytical formula for the conditional covariance matrices of the elliptically distributed random vectors, when the conditioning is based on the values of any linear combination of the marginal random variables. We show that one could introduce the univariate invariant depending solely …
Kalman filtering and smoothing algorithms are used in many areas, including tracking and navigation, medical applications, and financial trend filtering. One of the basic assumptions required to apply the Kalman smoothing framework is that error covariance matrices are known and given. In this paper, we study a general…
Riemannian geometry has been applied to Brain Computer Interface (BCI) for brain signals classification yielding promising results. Studying electroencephalographic (EEG) signals from their associated covariance matrices allows a mitigation of common sources of variability (electronic, electrical, biological) by constr…
Designing a covariance function that represents the underlying correlation is a crucial step in modeling complex natural systems, such as climate models. Geospatial datasets at a global scale usually suffer from non-stationarity and non-uniformly smooth spatial boundaries. A Gaussian process regression using a non-stat…
Magnetoencephalography and electroencephalography (M/EEG) can reveal neuronal dynamics non-invasively in real-time and are therefore appreciated methods in medicine and neuroscience. Recent advances in modeling brain-behavior relationships have highlighted the effectiveness of Riemannian geometry for summarizing the sp…
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.
Linear and Quadratic Discriminant analysis (LDA/QDA) are common tools for classification problems. For these methods we assume observations are normally distributed within group. We estimate a mean and covariance matrix for each group and classify using Bayes theorem. With LDA, we estimate a single, pooled covariance m…
Simplified optimization for structured matrices in deep learning.
problem Computational challenges in Riemannian submanifold optimization for structured symmetric positive-definite matrices.
method Proposed a generalized Riemannian normal coordinates that dynamically orthonormalizes the metric and converts the problem into an unconstrained Euclidean space problem.
result Simplified existing approaches for structured covariances and developed matrix-inverse-free 2nd-order optimizers for deep learning with low precision.
Denise learns a function to quickly decompose covariance matrices robustly.
problem Robustly decomposing covariance matrices for feature extraction.
method Deep learning for symmetric positive semidefinite matrices.
result Denise achieves state-of-the-art performance in decomposition quality and speed.
In this paper, we obtain a property of the expectation of the inverse of compound Wishart matrices which results from their orthogonal invariance. Using this property as well as results from random matrix theory (RMT), we derive the asymptotic effect of the noise induced by estimating the covariance matrix on computing…
Nonsingular estimation of high dimensional covariance matrices is an important step in many statistical procedures like classification, clustering, variable selection an future extraction. After a review of the essential background material, this paper introduces a technique we call slicing for obtaining a nonsingular …
The salient properties of large empirical covariance and correlation matrices are studied for three datasets of size 54, 55 and 330. The covariance is defined as a simple cross product of the returns, with weights that decay logarithmically slowly. The key general properties of the covariance matrices are the following…
Recently, there has been focus on penalized log-likelihood covariance estimation for sparse inverse covariance (precision) matrices. The penalty is responsible for inducing sparsity, and a very common choice is the convex l1 norm. However, the best estimator performance is not always achieved with this penalty. The …
Lower bounds on private estimation of Gaussian covariance matrices.
problem Private estimation of Gaussian covariance matrices under various parameter regimes.
method Stein-Haff identity and fingerprinting lemma extensions.
result Lower bounds match existing upper bounds in the widest known parameters.
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.
We consider the problem of joint estimation of structured inverse covariance matrices. We perform the estimation using groups of measurements with different covariances of the same unknown structure. Assuming the inverse covariances to span a low dimensional linear subspace in the space of symmetric matrices, our aim i…
This note improves correlation stress tests using geodesic distance.
problem Improving financial risk management through better covariance stress tests.
method Proposes a new geometrically invariant definition of correlation stress tests.
result Demonstrates a submanifold approach to stress testing covariance matrices.
The expectation-maximization (EM) algorithm is an iterative method for finding maximum likelihood estimates when data are incomplete or are treated as being incomplete. The EM algorithm and its variants are commonly used for parameter estimation in applications of mixture models for clustering and classification. This …