RandNLA uses randomness for matrix problems in machine learning.
problem Matrix problems in machine learning.
method Randomized Numerical Linear Algebra.
result New challenges in RandNLA due to hardware trends and advances in ML.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,657 papers · 148 categories
Trend · papers per month
48 results for “random matrix model”
Random matrix theory predicts neural representations generalize well.
problem Understanding why neural representations generalize well in practice.
method Applied random matrix theory to kernel regression and neural networks.
result GCV estimator accurately predicts generalization risk in overparameterized settings.
Study on signal-plus-noise decomposition in nonlinear spiked random matrices.
problem Nonlinear spiked random matrix models with rank-one signal and noise.
method Signal-plus-noise decomposition and phase transition analysis.
result Identified precise phase transitions in signal components at critical thresholds.
The Random Parameters model was proposed to explain the structure of the covariance matrix in problems where most, but not all, of the eigenvalues of the covariance matrix can be explained by Random Matrix Theory. In this article, we explore other properties of the model, like the scaling of its PDF as one take larger …
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.
U-Net trained to recover acoustic interference striations from distorted data.
problem Recovering acoustic interference striations from distorted signals.
method Training a U-Net using a random mode-coupling matrix model to generate training data.
result U-Net successfully recovers AISs under various conditions.
Study evaluates thresholds for removing noise from DNN weights using random matrix theory.
problem Removing noise from deep neural network weights for better approximation.
method Model weights as signal + noise, use random matrix theory to estimate thresholds, evaluate using cosine similarity.
result Proposed threshold estimation method improves approximation quality.
Improved matrix approximation using randomized algorithms.
problem Finding better approximations of given matrices.
method Randomized algorithms to compute as an improved approximation.
result Computed provides a better approximation than given .
Paper uses Random Matrix Theory for optimal training-testing data split.
problem Finding ideal training-testing data split for linear regression.
method Random Matrix Theory applied to Gaussian multivariate data.
result Ideal training and test sizes derived for any model.
Paper proposes diagnostics for error and variance estimation in randomized matrix computations.
problem Safe use of randomized matrix algorithms in applications.
method Leave-one-out error estimator and jackknife resampling method.
result Provides rapid diagnostics to assess quality of randomized matrix computations.
Matrices satisfying the Restricted Isometry Property (RIP) play an important role in the areas of compressed sensing and statistical learning. RIP matrices with optimal parameters are mainly obtained via probabilistic arguments, as explicit constructions seem hard. It is therefore interesting to ask whether a fixed mat…
Generalization in Representation Models via Random Matrix Theory: Application to Recurrent Networksmath.ST
We analyze generalization in deep learning models using random matrix theory.
problem Understanding the generalization error in deep learning models with random feature representations.
method Applying Random Matrix Theory to derive asymptotic generalization error formulas for various architectures.
result Linear ESNs are equivalent to ridge regression with exponentially time-weighted input covariance, revealing an inductive bias towards recent inputs.
In this paper, we present a new framework to obtain tail inequalities for sums of random matrices. Compared with existing works, our tail inequalities have the following characteristics: 1) high feasibility--they can be used to study the tail behavior of various matrix functions, e.g., arbitrary matrix norms, the absol…
Matrix multiplication is a fundamental building block for large scale computations arising in various applications, including machine learning. There has been significant recent interest in using coding to speed up distributed matrix multiplication, that are robust to stragglers (i.e., machines that may perform slower …
This note gives a simple analysis of a randomized approximation scheme for matrix multiplication proposed by Sarlos (2006) based on a random rotation followed by uniform column sampling. The result follows from a matrix version of Bernstein's inequality and a tail inequality for quadratic forms in subgaussian random ve…
Paper studies tensor models using random matrix theory.
problem Analyzing asymmetric order-d spiked tensor models with Gaussian noise.
method Uses variational definition of singular vectors and values, constructs equivalent spiked symmetric block-wise random matrix from tensor contractions.
result Characterizes asymptotic singular values and alignments of singular vectors with true spike components.
New method for faster graph parameter inference from large random Kronecker graphs.
problem Efficiently infer graph parameters from large random Kronecker graphs.
method Decompose adjacency matrix into signal and noise components, then use denoising and solving approach.
result Proposed method achieves comparable or better performance than existing methods at lower computational cost.
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.
A new method uses SPDEs to efficiently model random fields on complex domains.
problem Efficient representation of random fields on complex domains for engineering and machine learning.
method Uses SPDEs to develop a scalable framework for statFEM and GP regression.
result Can model anisotropic, non-stationary random fields with arbitrary smoothness.
The paper analyzes data augmentation for precision matrix estimation in high dimensions.
problem Precision matrix estimation in high-dimensional settings.
method Linear shrinkage estimators and data augmentation methods.
result Concentration bounds for the quadratic error of estimators.
We consider a fundamental algorithmic question in spectral graph theory: Compute a spectral sparsifier of random-walk matrix-polynomial where is the adjacency matrix of a weighted, undirected graph, is the diagonal matrix of weighted degrees, and are nonn…
HD algorithm simulates dynamics on random matrix ensembles without generating full matrices.
problem Simulating dynamics on dense random matrix ensembles with high space and time complexity.
method Householder reflectors for adaptive and recursive construction, deferring decisions.
result Significant reductions in runtime and memory footprint for practical .
Model, sample, and epoch-wise descents: exact solution of gradient flow in the random feature modelcs.LG
Analyzes the generalization and training errors of the random feature model over time.
problem Understanding the temporal behavior of generalization and training errors in deep learning.
method Uses Cauchy complex integral representations and random matrix methods based on linear pencils.
result Analytical solution of the full time-evolution path of generalization and training errors.
Random feature maps are ubiquitous in modern statistical machine learning, where they generalize random projections by means of powerful, yet often difficult to analyze nonlinear operators. In this paper, we leverage the "concentration" phenomenon induced by random matrix theory to perform a spectral analysis on the Gr…
The high-order relations between the content in social media sharing platforms are frequently modeled by a hypergraph. Either hypergraph Laplacian matrix or the adjacency matrix is a big matrix. Randomized algorithms are used for low-rank factorizations in order to approximately decompose and eventually invert such big…
Study on signed graphs with random signs, focusing on community detection.
problem Community detection in signed stochastic block models.
method Strong concentration inequalities for adjacency and Laplacian matrices, applied to signed Laplacian matrix.
result The sign of the first eigenvector of the Laplacian matrix defines a weakly consistent estimator for balanced community detection.
Random matrix theory explains how neural networks adapt to data.
problem Understanding how neural networks learn and generalize from data.
method Random matrix analysis of two-layer neural networks.
result Sharp characterization of feature spectrum and generalization error.
Study reveals an equivalence principle for the spectrum of random inner-product kernel matrices in polynomial scaling.
problem Understanding the spectrum of random kernel matrices in polynomial scaling regimes.
method Investigates random matrices with nonlinear kernel functions applied to inner products of uniformly distributed vectors.
result The spectrum of the random kernel matrix is asymptotically equivalent to a simpler matrix model through free additive convolution.
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 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…
Gradient descent proves global convergence for 4-layer matrix factorization.
problem Global convergence of gradient descent on four-layer matrix factorization under random initialization.
method New techniques to show saddle-avoidance properties and extend eigenvalue theories.
result Polynomial-time global convergence guarantee for randomly initialized gradient descent on four-layer matrix factorization.
New matrix ensembles better match deep neural network spectral densities.
problem Theoretical spectral density models for deep networks do not match empirical observations.
method Introduced new matrix ensemble classes to better fit observed spectral densities.
result Theoretical models for deep networks are significantly flawed.
Analysing Multi-Task Regression via Random Matrix Theory with Application to Time Series Forecastingstat.ML
The paper uses random matrix theory for multi-task regression, improving time series forecasting.
problem Improving time series forecasting using multi-task regression.
method Applying random matrix theory to multi-task regression problems, deriving closed-form solutions for optimization.
result Provides a robust foundation for hyperparameter optimization in multi-task regression scenarios.
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.
Exponential Weighting and Random-Matrix-Theory-Based Filtering of Financial Covariance Matrices for Portfolio Optimizationcond-mat.stat-mech
We introduce a covariance matrix estimator that both takes into account the heteroskedasticity of financial returns (by using an exponentially weighted moving average) and reduces the effective dimensionality of the estimation (and hence measurement noise) via techniques borrowed from random matrix theory. We calculate…
For random matrix models, the parameter estimation based on the traditional likelihood functions is not straightforward in particular when we have only one sample matrix. We introduce a new parameter optimization method for random matrix models which works even in such a case. The method is based on the spectral distri…
Random Matrix Theory explains loss surface Hessians in neural networks.
problem Understanding the loss surfaces of neural networks.
method Investigation of local spectral statistics of neural network Hessians.
result Excellent agreement with Gaussian Orthogonal Ensemble statistics.
We discuss the applications of Random Matrix Theory in the context of financial markets and econometric models, a topic about which a considerable number of papers have been devoted to in the last decade. This mini-review is intended to guide the reader through various theoretical results (the Marcenko-Pastur spectrum …
New method for privacy amplification without sampling for matrix factorization.
problem Privacy amplification for differentially private model training with matrix factorization.
method Sampling-free bounds based on Rényi divergence and conditional composition.
result Stronger privacy guarantees for small ε, applicable to various matrices.
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.
Proposes a framework to balance supervised and unsupervised learning using random matrix theory.
problem Balancing supervised and unsupervised learning in high-dimensional data.
method QLDS model with quadratic margin maximization under low density separation assumption.
result Establishes a smooth bridge between supervised and unsupervised learning methods.
We study covariance matrix estimation for the case of partially observed random vectors, where different samples contain different subsets of vector coordinates. Each observation is the product of the variable of interest with a Bernoulli random variable. We analyze an unbiased covariance estimator under this mod…
Study proposes memory-efficient backpropagation for linear layers in neural networks.
problem Significant memory usage in backpropagation through linear layers in neural networks.
method Randomized matrix multiplications to reduce memory usage with a moderate decrease in test accuracy.
result Demonstrated benefits of the proposed method on fine-tuning pre-trained models.
Diffusion models' consistency across splits explained by random matrix theory.
problem Consistency of diffusion models trained on non-overlapping subsets.
method Random matrix theory framework to quantify dataset effects on denoiser and sampling map.
result The theory explains and predicts cross-split disagreement in diffusion models.
Study analyzes perturbations in singular subspaces under random noise.
problem Understanding singular vector and subspace changes in signal-plus-noise models.
method Generalized Davis-Kahan-Wedin theorem for any unitarily invariant norm, considering and bounds.
result Fine-grained insights into singular vector and subspace perturbations, including and bounds.
New approach uses random matrix theory to understand tensor estimation performance.
problem Understanding the performance of estimators for low-rank signals in noisy tensors.
method Developed a new approach using random matrix theory to study random tensors.
result Discovered a fixed-point equation that matches the performance of the maximum likelihood estimator.
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.
Tensor factorization arises in many machine learning applications, such knowledge base modeling and parameter estimation in latent variable models. However, numerical methods for tensor factorization have not reached the level of maturity of matrix factorization methods. In this paper, we propose a new method for CP te…