The problem of low rank matrix completion is considered in this paper. To exploit the underlying low-rank structure of the data matrix, we propose a hierarchical Gaussian prior model, where columns of the low-rank matrix are assumed to follow a Gaussian distribution with zero mean and a common precision matrix, and a W…
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48 results for “Gaussian matrix”
This short note reviews so-called Natural Gradient Descent (NGD) for multivariate Gaussians. The Fisher Information Matrix (FIM) is derived for several different parameterizations of Gaussians. Careful attention is paid to the symmetric nature of the covariance matrix when calculating derivatives. We show that there ar…
This paper considers inference over distributed linear Gaussian models using factor graphs and Gaussian belief propagation (BP). The distributed inference algorithm involves only local computation of the information matrix and of the mean vector, and message passing between neighbors. Under broad conditions, it is show…
A new optimization algorithm for Gaussian Variational Inference on precision matrices.
problem Complex models with positive definite constraints on covariance matrices.
method Manifold Gaussian Variational Bayes (MGVBP) with natural gradient updates.
result Empirically validated as a feasible and efficient solution for VI in complex models.
Paper improves efficiency in matrix computations for Gaussian processes.
problem Efficiency in matrix computations for Gaussian processes.
method Variance reduction via matrix factorization.
result Factorized estimator can be up to 1,000 times more efficient.
We use matricial free energy to regularize autoencoders, producing Gaussian-like codes.
problem Generating Gaussian-like codes for autoencoders.
method Define a differentiable loss function based on singular values of the code matrix, minimizing matricial free energy.
result Minimizing matricial free energy results in Gaussian-like codes that generalize.
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 .
The paper analyzes how Gaussian kernel parameters affect posterior covariance in Gaussian processes.
problem Understanding the influence of Gaussian kernel parameters on posterior covariance in Gaussian processes.
method Geometric analysis and a posteriori error estimation techniques from adaptive finite element methods.
result The bandwidth parameter and spatial distribution of observations significantly influence posterior covariance and its matrix.
Generalizes randomized SVD for better matrix approximations using Gaussian vectors.
problem Computing accurate rank-k approximations of matrices with limited data.
method Extends randomized SVD to multivariate Gaussian vectors, incorporating prior knowledge and using Gaussian processes.
result Demonstrates improved accuracy in approximating matrices and Hilbert-Schmidt operators.
Efficiently computes matrix square roots and their inverses for large matrices.
problem Computing matrix square roots and inverses for large matrices efficiently.
method Combines Krylov subspace methods with rational approximation for quadratic-time computation.
result Achieves 4 decimal places of accuracy with fewer than 100 matrix-vector multiplications.
New bounds for private matrix approximation using Gaussian noise and Dyson Brownian Motion.
problem Private approximation of symmetric matrices with Gaussian noise.
method Viewing Gaussian noise as Dyson Brownian Motion to track eigenvalue and eigenvector evolution.
result Improved bounds on Frobenius-distance utility for private matrix approximation.
This work analyzes self-attention matrices using random matrix theory.
problem Understanding the theoretical behavior of self-attention layers in neural networks.
method Asymptotic spectral analysis of the attention matrix, Gaussian equivalence, and linearization.
result The singular value distribution of the attention matrix is asymptotically characterized by a linear model.
Develops a new MCMC-based Wishart prior for Gaussian Process covariance matrix.
problem Difficult inference for multivariate Gaussian Processes with multiple lengthscale parameters.
method Introduces a self-assembled Wishart prior and uses MCMC for Bayesian inference on kernel hyperparameters.
result Demonstrates the effectiveness of the new prior in GP-based learning with empirical results.
This paper presents a sequential randomized lowrank matrix factorization approach for incrementally predicting values of an unknown function at test points using the Gaussian Processes framework. It is well-known that in the Gaussian processes framework, the computational bottlenecks are the inversion of the (regulariz…
Develops a Gaussian-based message-passing algorithm for noisy matrix completion.
problem Noisy matrix completion problems.
method Gaussian parameterized belief propagation algorithm.
result Advantages over traditional methods in non-Gaussian noise scenarios.
In this paper, we introduce a new directed graphical model from Gaussian data: the Gaussian graphical interaction model (GGIM). The development of this model comes from considering stationary Gaussian processes on graphs, and leveraging the equations between the resulting steady-state covariance matrix and the Laplacia…
Training Gaussian process-based models typically involves an computational bottleneck due to inverting the covariance matrix. Popular methods for overcoming this matrix inversion problem cannot adequately model all types of latent functions, and are often not parallelizable. However, judicious choice of model…
Extends spectral number variance convergence to random matrix ensembles for twisted Laplacians.
problem Spectral number variance convergence for twisted Laplacians and Dirac operators.
method Extends Rudnick's approach to Gaussian ensembles for twisted Laplacians and Dirac operators.
result Convergence to Gaussian ensembles for twisted Laplacians and Dirac operators.
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 Gaussian ICAstat.ML
Independent component analysis (ICA) is a cornerstone of modern data analysis. Its goal is to recover a latent random vector S with independent components from samples of X=AS where A is an unknown mixing matrix. Critically, all existing methods for ICA rely on and exploit strongly the assumption that S is not Gaussian…
Develops a Gaussian model to compute the Alexander polynomial of knots.
problem Computing the Alexander polynomial of knots.
method Uses perturbed Gaussian functions, Heisenberg algebra, and tensor-contraction formalism.
result Associates a Gaussian function to a knot whose partition function recovers the Alexander polynomial.
Differential privacy mechanism design has traditionally been tailored for a scalar-valued query function. Although many mechanisms such as the Laplace and Gaussian mechanisms can be extended to a matrix-valued query function by adding i.i.d. noise to each element of the matrix, this method is often suboptimal as it for…
Paper proposes a generalized precision matrix for t-Student distributions to improve portfolio optimization.
problem Limitations of inverse covariance matrix in non-Gaussian settings.
method Exploits local dependence function to define generalized precision matrix (GPM) for multivariate t-Student distribution.
result GPM leads to statistically significant lower out-of-sample variances in minimum-variance portfolios.
PACE-GGM uses Gaussian mechanism for private covariance estimation.
problem Private estimation of covariance matrices in high dimensions.
method Data-adaptive selection of entries, Gaussian mechanism, maximum-entropy reconstruction.
result Consistent improvements in estimation error compared to Gaussian mechanism and baselines.
Transposable data represents interactions among two sets of entities, and are typically represented as a matrix containing the known interaction values. Additional side information may consist of feature vectors specific to entities corresponding to the rows and/or columns of such a matrix. Further information may also…
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.
Robust 1-bit Compressive Sensing with Partial Gaussian Circulant Matrices and Generative Priorscs.LG
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.
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.
We propose a novel hierarchical model for multitask bipartite ranking. The proposed approach combines a matrix-variate Gaussian process with a generative model for task-wise bipartite ranking. In addition, we employ a novel trace constrained variational inference approach to impose low rank structure on the posterior m…
Corrected whitening restores orthogonality in high-dimensional spherical Gaussian mixtures.
problem In high-dimensional data, standard whitening fails to preserve orthogonality of mixture means.
method Derived exact limits for whitened means dot products using random matrix theory, constructed a corrected whitening matrix.
result Corrected whitening allows for improved estimation of spherical Gaussian mixtures in the large-dimensional regime.
Proposes a new method for selecting regularization parameters in sparse precision matrix estimation.
problem Selecting an appropriate regularization parameter for sparse precision matrix estimation.
method Developed a closed-form matrix-valued regularization parameter based on the sampling distribution of optimality conditions.
result The proposed method achieves comparable estimation accuracy and superior support recovery to cross-validation, with significant runtime improvements.
A new method clusters rows of a matrix of point processes.
problem Challenges in analyzing structured point process data.
method Mixture model of multi-level marked point processes, combined with ES algorithm and FPCA.
result An efficient method for clustering rows of a matrix of point processes.
Paper connects neural networks to Gaussian processes for understanding double-descent.
problem Understanding the double-descent phenomenon in neural networks.
method Uses techniques from random matrix theory and Gaussian processes.
result Establishes a connection between NNGP and random matrix theory for neural networks.
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.
Improved perturbation reduces matrix condition number to O(n) with minimal storage.
problem Reducing the condition number of deterministic matrices for efficient algorithmic use.
method Introduced pattern matrices and sparse perturbations with dependent entries.
result Condition number reduced to O(n) with O(n) random numbers in O(log n) precision.
We derive an efficient method to perform clustering of nodes in Gaussian graphical models directly from sample data. Nodes are clustered based on the similarity of their network neighborhoods, with edge weights defined by partial correlations. In the limited-data scenario, where the covariance matrix would be rank-defi…
We present a new method for estimating multivariate, second-order stationary Gaussian Random Field (GRF) models based on the Sparse Precision matrix Selection (SPS) algorithm, proposed by Davanloo et al. (2015) for estimating scalar GRF models. Theoretical convergence rates for the estimated between-response covariance…
New method trains sparse Gaussian processes without matrix inversion.
problem Costly training of Gaussian processes at scale.
method Inverse-free approach using matmul-only natural-gradient updates.
result Significantly improved stability and convergence in training.
Quantum-assisted Gaussian process speeds up data regression.
problem High computational complexity of Gaussian process regression for large datasets.
method Quantum-assisted sparse Gaussian process regression using random Fourier features.
result Achieves polynomial-order computational speedup compared to classical methods.
Proposes a new Gaussian factor for probabilistic inference with degenerate settings.
problem Handling linear dependencies among random variables in Gaussian networks.
method Introduces a parametrised factor that relaxes the positive-definite constraint of the covariance matrix.
result Accurately accommodates degeneracies in probabilistic inference without significant computational overhead.
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.
We introduce Gaussian Process Topic Models (GPTMs), a new family of topic models which can leverage a kernel among documents while extracting correlated topics. GPTMs can be considered a systematic generalization of the Correlated Topic Models (CTMs) using ideas from Gaussian Process (GP) based embedding. Since GPTMs w…
Proposes a method to estimate sparse Gaussian graphical models with hidden clustering structure.
problem Modeling statistical relationships between variables with sparsity and clustering.
method Two-phase algorithm using sGS-ADMM for initial point and pALM for solution.
result Demonstrates good performance and efficiency of the proposed model and algorithm on synthetic and real data.
The paper shows how to infer conditional independence from non-Gaussian data.
problem Inferring conditional independence from non-Gaussian distributions.
method Developed a method to recover conditional independence structure from the precision matrix of generalized nonparanormal data.
result The conditional independence structure can be inferred from the precision matrix of generalized nonparanormal data.
Scalable Gaussian processes with latent Kronecker structure for large datasets.
problem Limited scalability of Gaussian processes for large datasets.
method Leveraging latent Kronecker structure, projecting kernel matrix onto latent Kronecker product, using iterative linear system solvers and pathwise conditioning.
result Outperforms state-of-the-art sparse and variational GPs on real-world datasets with up to five million examples.
Eigenvalue Distribution of Large Random Matrices Arising in Deep Neural Networks: Orthogonal Casestat.ML
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.
The paper examines how kernel approximations affect Gaussian process regression in large data applications.
problem Effect of kernel approximations on Gaussian process regression in large data applications.
method Unified framework to analyze Gaussian process regression under computational and epistemic misspecification.
result Theoretical analysis of Gaussian process regression under various misspecifications.
The random matrix theory method of planar Gaussian diagrammatic expansion is applied to find the mean spectral density of the Hermitian equal-time and non-Hermitian time-lagged cross-covariance estimators, firstly in the form of master equations for the most general multivariate Gaussian system, secondly for seven part…