Polynomial time algorithm matches correlated Gaussian matrices without vanishing correlation.
problem Matching vertices in two correlated Erdős-Rényi graphs.
method Iterative matching algorithm for correlated Gaussian Wigner matrices.
result First polynomial time algorithm for graph matching with arbitrarily small constant correlation.
Paper offers robust recovery for 1-bit sensing with partial Gaussian circulant matrices.
problem Accurately recovering vectors from 1-bit measurements using structured matrices.
method Correlation-based optimization with randomly signed partial Gaussian circulant matrices and generative models.
result Recovery guarantees match those for i.i.d. Gaussian matrices but with faster computation.
Study on Gaussian ensemble of matrix products with mixed moments computed.
problem Understanding the statistical properties of matrix products of Gaussian matrices.
method Analysis of a multi-Wishart ensemble and enumeration of non-crossing pairings.
result Mixed moments of the product matrix are computed and found to be weighted by Fuss-Catalan numbers at large N. 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.
The paper proves a distribution claim for neural network Jacobians.
problem Distribution of singular values in deep neural networks.
method Free probability and random matrix theory techniques.
result Singular value distribution matches for specific cases.
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.
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.
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.
Banded matrices can be used as precision matrices in several models including linear state-space models, some Gaussian processes, and Gaussian Markov random fields. The aim of the paper is to make modern inference methods (such as variational inference or gradient-based sampling) available for Gaussian models with band…
In this paper, we study the problem of compressed sensing using binary measurement matrices and ℓ1-norm minimization (basis pursuit) as the recovery algorithm. We derive new upper and lower bounds on the number of measurements to achieve robust sparse recovery with binary matrices. We establish sufficient conditi…
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.
Unified bounds for iterative algorithms with Gaussian data matrices.
problem Establishing non-asymptotic bounds for iterative algorithms with Gaussian data.
method Explicit coupling between iterates and Gaussian process with deterministic covariance.
result Tight, dimension-free bounds for generalized first-order methods.
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.
The paper presents a probabilistic framework for SPD matrices in machine learning.
problem Machine learning on SPD matrices is fragmented; this paper aims to unify it.
method Unified probabilistic framework using Gaussian distributions and Bayes classifiers.
result Different SPD machine learning tools can be reinterpreted and extended using Gaussian distributions.
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…
Paper introduces a new distance measure for Gaussian Mixture Models.
problem Developing a new distance measure for Gaussian Mixture Models.
method Embedding K-component Gaussian Mixture Models into the manifold of symmetric positive definite matrices and calculating a lower bound for the Fisher-Rao metric.
result Demonstrated effectiveness through experiments on standard datasets.
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.
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.
Improves signal detection in non-Gaussian noise using transformed data.
problem Signal detection in rank-one signal-plus-noise data matrices.
method Pre-transforming matrix entries and using linear spectral statistics for hypothesis testing.
result Sharp phase transition of largest eigenvalues in spiked rectangular matrices.
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.
Motivated by a sampling problem basic to computational statistical inference, we develop a nearly optimal algorithm for a fundamental problem in spectral graph theory and numerical analysis. Given an n×n SDDM matrix M, and a constant −1≤p≤1, our algorithm gives efficient access to a…
Graph alignment problem solved with convex relaxations for correlated matrices.
problem Recovering hidden vertex permutations from correlated Gaussian matrices.
method Convex relaxations of the quadratic assignment problem over doubly stochastic matrices.
result The solution of the convex relaxation concentrates around the ground-truth permutation matrix for certain correlation parameters.
New Gaussian min-max theorem extends classical results to non-i.i.d. Gaussian matrices.
problem Extending classical Gaussian min-max theorems to non-i.i.d. Gaussian matrices.
method Identifying a new pair of Gaussian processes that satisfy comparison inequalities.
result New Gaussian min-max and convex Gaussian min-max theorems with applications in multi-source Gaussian regression and binary classification.
New method for estimating high-dimensional binary time series coefficients.
problem Statistical inference for high-dimensional binary time series.
method Post-selection estimator and second-order wild bootstrap algorithm.
result Good finite-sample performance of the proposed method.
Despite their successes, what makes kernel methods difficult to use in many large scale problems is the fact that storing and computing the decision function is typically expensive, especially at prediction time. In this paper, we overcome this difficulty by proposing Fastfood, an approximation that accelerates such co…
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.
We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the ℓ2 norm of such a product applied to any fixed vector is asymptotically Gaussian. The fluctuations we find can be thought of as a…
Method estimates multiple related Gaussian distributions using Laplacian regularization.
problem Jointly estimate multiple related zero-mean Gaussian distributions.
method Laplacian regularized stratified model fitting with hyper-parameters to encourage covariance closeness.
result The method performs well, especially in low data regimes, as demonstrated in finance, radar, and weather.
Deep Jump Gaussian Processes model high-dimensional piecewise functions.
problem Modeling high-dimensional piecewise continuous functions with limited accuracy.
method Integrates region-specific locally linear projections with Jump Gaussian Processes (JGP) to capture local low-dimensional subspace structures.
result DJGP achieves superior predictive accuracy and more reliable uncertainty quantification compared to existing methods.
Study finds the minimum number of finite Gaussian mixtures for best approximation.
problem Finding the minimum number of finite Gaussian mixtures for best approximation.
method Local moment matching for upper bound and spectral analysis for lower bound.
result Corrects a previous lower bound in the case of Gaussian mixing distributions.
We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as random, real, asymmetric matrices with a special structure superimposed due to the time-shift. We demonstrate that the associated eigen…
Improves detection of low-rank signals from noisy data matrices.
problem Statistical detection of low-rank signals in noisy data matrices.
method Entrywise pre-transforming data matrix for non-Gaussian noise, sharp phase transition thresholds, central limit theorem for linear spectral statistics, hypothesis test.
result Improves detection of low-rank signals from noisy data matrices, generalizing known results.
Quantum GBS boosts asset clustering for robust statistical arbitrage portfolios.
problem Identifying co-moving assets from correlation matrices for statistical arbitrage.
method Mapping S&P 500 correlation data to GBS-compatible adjacency matrices, benchmarking classical and quantum clustering algorithms.
result Quantum GBS generates superior alpha during high volatility periods, persisting under low-loss conditions.
Study finds Calabi-Yau models' operator spectra match random matrix theory.
problem Understanding spectra of Calabi-Yau sigma models.
method Numerical methods for Ricci-flat metrics, averaging over complex structure moduli space.
result Spectrum matches Gaussian orthogonal ensemble of random matrix theory.
In this paper we show that for the purposes of dimensionality reduction certain class of structured random matrices behave similarly to random Gaussian matrices. This class includes several matrices for which matrix-vector multiply can be computed in log-linear time, providing efficient dimensionality reduction of gene…
The paper develops efficient algorithms for variational inference with mixtures of isotropic Gaussians.
problem Efficiently approximating multimodal Bayesian posteriors.
method Develops a variational framework and efficient algorithms for mixtures of isotropic Gaussians.
result The approach provides accurate approximations of multimodal Bayesian posteriors while being memory and computationally efficient.
Directly proves Brioschi formula for Gaussian curvature.
problem Express Gaussian curvature in terms of local coordinates.
method Elementary proof without Christoffel symbols.
result Directly derived Brioschi formula for Gaussian curvature.
Orthogonal random features approximate a Bessel kernel, offering sharper bounds than random Fourier features.
problem Approximating Gaussian kernel efficiently for large datasets.
method Use of Haar orthogonal matrices to construct orthogonal random features and analyze their bias and variance.
result Orthogonal random features approximate a Bessel kernel, not the Gaussian kernel, with sharper bounds.
Sparse Gaussian processes with compact kernels for faster inference.
problem Efficient Gaussian process inference with high computational complexity.
method Parametric families of compactly-supported kernels for sparse matrix representations.
result Sub-quadratic inference complexity and improved performance on real-world tasks.
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.
Efficiently differentiate functions of large matrices using new adjoint systems.
problem Differentiating functions of large matrices in scientific and probabilistic machine learning models.
method Deriving and implementing new adjoint systems for Lanczos and Arnoldi iterations in JAX.
result Efficient differentiation of PDEs, Gaussian process models, and Bayesian neural networks.
Researchers approximate partition functions on Riemannian spaces in the large N limit.
problem Computing normalization factors (partition functions) on Riemannian symmetric spaces is challenging.
method Approximation techniques in the large N limit, including saddle-point equations.
result Formulas for leading order terms in the large N limit of SPD matrices and related spaces.
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 problem of filtering information from large correlation matrices is of great importance in many applications. We have recently proposed the use of the Kullback-Leibler distance to measure the performance of filtering algorithms in recovering the underlying correlation matrix when the variables are described by a mu…
Algorithm recovers multiple low-rank matrices from unlabeled data.
problem Learning mixtures of low-rank models from unlabelled data.
method Three-stage meta-algorithm that copes with non-convexity and noise.
result Near-optimal sample and computational complexities under Gaussian designs.
We study a new ensemble of random correlation matrices related to multivariate Student (or more generally elliptic) random variables. We establish the exact density of states of empirical correlation matrices that generalizes the Marcenko-Pastur result. The comparison between the theoretical density of states in the St…
The paper introduces a new method for tail bounds of random vectors and matrices.
problem Estimating norms of random vectors and matrices under moment assumptions.
method Variational tail bounds for norms of random vectors and matrices.
result Dimension-free concentration inequalities for various norms of random vectors and matrices.
Consider a random vector with finite second moments. If its precision matrix is an M-matrix, then all partial correlations are non-negative. If that random vector is additionally Gaussian, the corresponding Markov random field (GMRF) is called attractive. We study estimation of M-matrices taking the role of inverse sec…