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.
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.
New bound for neural networks with full-rank weights, independent of network width.
problem Understanding generalization of neural networks with full-rank weight matrices.
method Using Koopman operators to derive a tighter generalization bound for full-rank weight matrices.
result The bound is tighter than existing norm-based bounds when condition numbers are small.
SGD and weight decay encourage neural networks to learn low-rank weight matrices.
problem The bias of SGD towards low-rank weight matrices in neural networks.
method The study investigates the effect of SGD and weight decay on the rank of weight matrices in neural networks, both theoretically and empirically.
result Training with SGD and weight decay induces a bias towards rank minimization in weight matrices, which becomes more pronounced with smaller batch sizes and stronger weight decay.
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.
This work compresses heavy-tailed weight matrices for tighter generalization bounds.
problem Empirical evidence linking heavy-tailed weight matrices to test set accuracy but lack of formal relationship with generalization bounds.
method Utilized the compression framework to show that heavy-tailed matrices can be compressed, resulting in sparse weight matrices.
result Demonstrated a non-vacuous generalization bound for compressed networks with heavy-tailed weight matrices.
Heavy-tailed regularization improves deep neural network performance.
problem Improving generalization of deep neural networks.
method Introducing Heavy-Tailed Regularization, using differentiable penalty terms and Bayesian statistics.
result Heavy-tailed regularization outperforms conventional regularization techniques.
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.
The study examines Fisher information matrices and neural tangent kernels for simple ReLU networks with random weights.
problem Understanding the relationship between Fisher information matrices and neural tangent kernels for 2-layer ReLU networks.
method Analyzes Fisher information matrices and neural tangent kernels for 2-layer ReLU networks with random hidden weights, focusing on spectral decomposition and eigenfunctions.
result Obtained an approximation formula for functions represented by 2-layer neural networks.
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…
Recently low displacement rank (LDR) matrices, or so-called structured matrices, have been proposed to compress large-scale neural networks. Empirical results have shown that neural networks with weight matrices of LDR matrices, referred as LDR neural networks, can achieve significant reduction in space and computation…
Weight Decay induces low-rank weight matrices in neural networks, improving generalization.
problem Improving generalization in neural networks.
method Training ReLU NN with Weight Decay and Stochastic Gradient Descent.
result The weight matrix of a trained NN is approximately rank-two.
Study uses graph Laplacians to analyze surface links.
problem Analyzing virtual genus of surface links.
method Laplacian matrices of weighted graphs in surfaces are used to define invariants.
result Obtained information about virtual genus.
Generative Adversarial Networks (GANs), though powerful, is hard to train. Several recent works (brock2016neural,miyato2018spectral) suggest that controlling the spectra of weight matrices in the discriminator can significantly improve the training of GANs. Motivated by their discovery, we propose a new framework for t…
One reflection suffices for orthogonal weights, reducing GPU usage.
problem Efficiently computing orthogonal weight matrices without high GPU utilization.
method Use an auxiliary neural network to compute one reflection instead of many.
result One reflection is sufficient for orthogonal weights, improving GPU utilization.
Connections between nodes of fully connected neural networks are usually represented by weight matrices. In this article, functional transfer matrices are introduced as alternatives to the weight matrices: Instead of using real weights, a functional transfer matrix uses real functions with trainable parameters to repre…
We discuss a weighted estimation of correlation and covariance matrices from historical financial data. To this end, we introduce a weighting scheme that accounts for similarity of previous market conditions to the present one. The resulting estimators are less biased and show lower variance than either unweighted or e…
We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems. Among these weight systems are those associated with the Conway and HOMFLYPT polynomials. We extend these ideas to looking at a space o…
The paper studies matrix normalization and graph balancing using a new functional and gradient descent.
problem Matrix normalization and graph balancing.
method A new functional called the non-normal energy, and gradient descent.
result Gradient descent of the non-normal energy converges to balanced graphs and preserves spectra and realness of weights.
BLAST optimizes deep model inference by learning efficient matrix structures.
problem Efficient inference of large-scale deep neural networks.
method Introduces BLAST matrix for adaptive structured weights.
result Significant compression and performance improvements for various model sizes.
Study of metrics on positive-definite matrices from power potential, linking to power means.
problem Understanding metrics on positive-definite matrices derived from power potential.
method Explicit expressions for geodesics and distance function derived from Hessian of power potential.
result Geodesics and distance function converge to weighted matrix geometric mean as β tends to zero.
Study heavy-tailed weights' impact on neural network's spectral distribution.
problem Analyzing spectral distribution of conjugate kernel matrices with heavy-tailed weights.
method Computed limiting eigenvalue distribution through moments, considering heavy-tailed distributions and nonlinear activation functions.
result Heavy-tailed weights induce strong correlations, leading to fundamentally different spectral behavior.
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.
New method corrects quantization errors in LLMs using low-rank matrices.
problem Correcting quantization errors in large language models.
method Introducing low-rank weight matrices to correct quantized activations in LLMs.
result Reduces accuracy gap with original model by more than 50% using low-rank matrices.
Spectral portfolio theory links neural networks to wealth dynamics via SGD weight matrices.
problem Understanding wealth dynamics from neural network training.
method Direct identification of weight matrices as portfolio allocation matrices, linking SGD forces to portfolio dynamics.
result Spectral properties of SGD weight matrices transition between additive and multiplicative regimes, influencing wealth dynamics.
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.
Extends graph encoder embedding to weighted graphs and matrices.
problem Classifying vertices in various graph types efficiently.
method Graph encoder embedding applied to weighted graphs, distance matrices, and kernel matrices.
result The method achieves asymptotic normality, enabling optimal classification.
Given two data matrices X and Y, sparse canonical correlation analysis (SCCA) is to seek two sparse canonical vectors u and v to maximize the correlation between Xu and Yv. However, classical and sparse CCA models consider the contribution of all the samples of data matrices and thus cannot identify an unde…
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.
Study of discrete period matrices on embedded graphs, relating to Riemann surfaces.
problem Understanding discrete conformal structures on surfaces via period matrices.
method Combinatorial interpretation of period matrices, using homological quasi-trees and Laplacian determinants.
result Derived a combinatorial analogue of the Weil-Petersson potential and related it to homological quasi-trees.
New method clusters hypergraphs using weighted random walks and Laplacians.
problem Clustering hypergraph data with edge-dependent weights.
method Random walks with edge-dependent vertex weights, constructing hypergraph Laplacians for clustering.
result Proposed methods outperform existing hypergraph clustering algorithms.
New method connects neural networks to diagrammatic algebra.
problem Constructing permutation equivariant neural networks.
method Schur-Weyl duality between symmetric group and partition algebra.
result Simple diagrammatic method for calculating weight matrices.
The low displacement rank (LDR) framework for structured matrices represents a matrix through two displacement operators and a low-rank residual. Existing use of LDR matrices in deep learning has applied fixed displacement operators encoding forms of shift invariance akin to convolutions. We introduce a class of LDR ma…
Low-rank matrix completion is an important problem with extensive real-world applications. When observations are uniformly sampled from the underlying matrix entries, existing methods all require the matrix to be incoherent. This paper provides the first working method for coherent matrix completion under the standard …
Autoencoder performance is predicted by eigenvalues of weight matrices.
problem Predicting an autoencoder's generalization ability without dataset knowledge.
method Analyze Jacobian matrices' eigenvalues to bound mean squared errors.
result Eigenvalues are good predictors of MSE on test points.
WE constructs GP kernels for mixed inputs using weighted EDMs.
problem Limitation of standard GP models in handling categorical variables.
method WEGP constructs kernel function using weighted EDMs for categorical inputs.
result WEGP improves GP model accuracy in both synthetic and real-world optimization problems.
Based on the property that solving the system of linear matrix equations via the column space and the row space projections boils down to an approximation in the least squares error sense, a formulation for learning the weight matrices of the multilayer network can be derived. By exploiting into the vast number of feas…
We present a continuous-time maximum likelihood estimation methodology for credit rating transition probabilities, taking into account the presence of censored data. We perform rolling estimates of the transition matrices with exponential time weighting with varying horizons and discuss the underlying dynamics of trans…
Enhances power of covariance matrix tests for high-dimensional data.
problem Testing large covariance matrices in high-dimensional data.
method Proposes a new Fisher's combined probability test for quadratic form and maximum form statistics.
result Boosts power against more general alternatives.
Neural networks are commonly trained to make predictions through learning algorithms. Contrastive Hebbian learning, which is a powerful rule inspired by gradient backpropagation, is based on Hebb's rule and the contrastive divergence algorithm. It operates in two phases, the forward (or free) phase, where the data are …
Recurrent neural networks (RNNs) are notoriously difficult to train. When the eigenvalues of the hidden to hidden weight matrix deviate from absolute value 1, optimization becomes difficult due to the well studied issue of vanishing and exploding gradients, especially when trying to learn long-term dependencies. To cir…
Study shows how feature weighting affects neural network regularization.
problem Understanding how feature weighting influences neural network regularization.
method Derived equivalence paths connecting different weighting matrices and ridge regularization levels.
result Ridge estimators trained on weighted features are asymptotically equivalent when evaluated against test vectors.
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. 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.
At the core of any inference procedure in deep neural networks are dot product operations, which are the component that require the highest computational resources. A common approach to reduce the cost of inference is to reduce its memory complexity by lowering the entropy of the weight matrices of the neural network, …
This paper shows universality in spectrum behavior for random inner-product kernel matrices in polynomial regime.
problem Understanding spectrum behavior of random inner-product kernel matrices in polynomial regime.
method Analyzing matrices formed by a nonlinear function applied entrywise to a sample-covariance matrix, considering i.i.d. entries with all finite moments.
result The spectrum of random inner-product kernel matrices is universally described by the free convolution of the semicircular and Marčenko-Pastur distributions, with relative weights given by expanding the nonlinear function in the Hermite basis.
Recurrent Neural Networks (RNNs) are designed to handle sequential data but suffer from vanishing or exploding gradients. Recent work on Unitary Recurrent Neural Networks (uRNNs) have been used to address this issue and in some cases, exceed the capabilities of Long Short-Term Memory networks (LSTMs). We propose a simp…
The backpropagation algorithm for calculating gradients has been widely used in computation of weights for deep neural networks (DNNs). This method requires derivatives of objective functions and has some difficulties finding appropriate parameters such as learning rate. In this paper, we propose a novel approach for c…