This work analyzes the maximum-margin bias in quasi-homogeneous neural networks.
problem Analyzing the maximum-margin bias in quasi-homogeneous neural networks.
method Geometric analysis of gradient dynamics for quasi-homogeneous models.
result Gradient flow implicitly favors a subset of parameters, leading to asymmetric norm minimization.
The paper analyzes neural network dynamics after weights escape the origin.
problem Understanding gradient flow dynamics of neural networks after the origin.
method Analyzes gradient flow of homogeneous neural networks with locally Lipschitz gradients.
result Characterizes the first saddle point encountered after escaping the origin.
Neural network approach simplifies multiscale problem homogenization.
problem Homogenizing multiscale problems with varying microscale structures.
method Derivative-free neural network with Brownian walkers.
result Neural network method is computationally efficient and robust.
We embed KKT points in neural networks of different sizes.
problem Classifying data using homogeneous neural networks.
method Introducing KKT point embedding principle and proving it for different network types.
result KKT points of a smaller network can be mapped to those of a larger network via linear transformations.
The paper generalizes equivariant neural networks on homogeneous spaces to the non-linear setting.
problem Equivariant neural networks on homogeneous spaces.
method Deriving generalized steerability constraints for non-linear equivariant layers.
result The universality of the derived construction for non-linear equivariant layers.
Early training of deep neural networks leads to small, directionally converging weights.
problem Training dynamics of deep homogeneous neural networks with small initializations.
method Gradient flow analysis and study of KKT points for neural correlation function.
result Weights converge in direction to KKT points during early training stages.
The paper studies neural networks' convergence near origin and saddle points.
problem Directional convergence of neural networks near small initializations and saddle points.
method Gradient flow dynamics analysis of two-homogeneous neural networks.
result Neural networks' weights approximately converge in direction to KKT points for small initializations.
We develop a convex relaxation method for analyzing neural network generalization.
problem Analyzing the generalization of parallel positively homogeneous networks.
method Linking non-convex ERM to a convex optimization problem over prediction functions.
result Achieved generalization bounds with almost linear sample complexity in network width.
Neural networks can approximate positive homogeneous functions, especially with multiple hidden layers.
problem Approximating positive homogeneous functions with neural networks.
method Using scale-invariant ReLU networks with multiple hidden layers.
result Approximation of positive homogeneous functions is possible with neural networks, especially with two hidden layers.
SGD converges to critical points of normalized margin in late-stage training for homogeneous neural networks.
problem Analyzing the implicit bias of SGD on homogeneous neural networks.
method Interpreting SGD dynamics as an Euler-like discretization of a conservative field flow associated with the normalized classification margin.
result Normalized SGD iterates converge to the set of critical points of the normalized margin at late-stage training.
GCNNs on homogeneous spaces use vector bundles and Hilbert spaces.
problem Learning data on homogeneous spaces with global symmetry.
method Analysis of G-equivariant convolutional layers on homogeneous G/K spaces, using vector bundles and reproducing kernel Hilbert spaces. result A precise criterion for expressing G-equivariant layers as convolutional layers, leading to stronger results for some groups. In this paper, we study the implicit regularization of the gradient descent algorithm in homogeneous neural networks, including fully-connected and convolutional neural networks with ReLU or LeakyReLU activations. In particular, we study the gradient descent or gradient flow (i.e., gradient descent with infinitesimal s…
Study shows how steepest descent algorithms' geometric margin increases during training.
problem Understanding implicit bias in steepest descent algorithms for neural networks.
method Analysis of steepest descent algorithms with infinitesimal learning rates in homogeneous neural networks.
result Limit points of training trajectories correspond to KKT points of margin-maximization problems.
New regularizer improves neural network robustness and generalization.
problem Ineffective weight decay for networks with homogeneous activation functions.
method Proposes an invariant regularizer to penalize intrinsic weight norms.
result Improves generalization and adversarial robustness on various datasets.
Develops wavelet-based neural network approximation theory.
problem Analyzing neural network approximation capabilities over various activation functions.
method Wavelet frame theory on spaces of homogeneous type, sufficient conditions for approximation, error estimates.
result Derives sufficient conditions for neural networks to approximate any functions in a given space, including non-smooth activations.
Large GD stepsizes improve margins and speed up training for non-homogeneous networks.
problem Training efficiency and margin improvement in non-homogeneous two-layer networks.
method Investigation of two distinct phases in GD training, showing margin growth and empirical risk decrease.
result Large GD stepsizes lead to faster convergence and improved margins in non-homogeneous networks.
A new algorithm trains deep neural networks by adding neurons greedily.
problem Training deep neural networks efficiently and effectively.
method Neuron Pursuit (NP) algorithm, which alternates between neuron addition and loss minimization.
result The algorithm can train deep neural networks efficiently and effectively.
The paper connects flatness to generalization in learning multi-index models with neural networks.
problem Understanding the generalization of non-convex neural networks using flatness measures.
method Analyzes 2-layer non-convex homogeneous neural networks and their connection to multi-index models.
result Flattest interpolators achieve small population loss and generalize well, establishing a direct link between flatness and generalization.
Group equivariant neural networks simplify complex tasks with group representation theory.
problem Challenging tasks requiring input transformations like rotations.
method Group representation theory, non-commutative harmonic analysis, differential geometry.
result A neural network is group equivariant if and only if it has a convolutional structure.
The paper examines the neural tangent kernel for PINNs solving general PDEs and finds convergence conditions.
problem Analyzing the convergence of neural tangent kernel for PINNs solving general PDEs.
method Analysis of NTK initialization and convergence during training for general PDEs using PINNs.
result Homogeneity of differential operators is crucial for NTK convergence.
A new neural network model uses polynomial chaos theory to improve neural signal processing.
problem Redundant neural signal representation in DANNs.
method Employing arbitrary polynomial chaos theory to construct orthonormal representations in DANNs.
result Improves neural signal processing by reducing redundancy and enhancing orthogonality.
Braid theory optimizes neural network structures.
problem Optimizing the architecture of neural networks.
method Using braid theory to describe and construct neural network structures.
result Braid-based networks outperform other architectures in classification tasks.
Study on geodesic distances on SE(3)/SO(2) in machine learning.
problem Investigating the efficiency of computationally efficient sections in selecting geodesic distances.
method Analyzing geodesic distances on reductive homogeneous spaces, proving the efficiency of minimal distance sections.
result Minimal distance sections are not always geodesic minimizers, but minimal horizontal geodesics are.
Graph neural networks improve topology control of power grids.
problem Grid congestion due to renewable energy and electrification.
method Investigated the effect of graph representation on GNN effectiveness for topology control.
result Heterogeneous graph representation outperforms homogeneous in topology control tasks.
Heavy-tailed distributions emerge in SGD's parameter evolution.
problem Understanding heavy-tailed distributions in SGD parameter evolution.
method Continuous diffusion approximation of SGD (homogenized SGD) analysis.
result Explicit upper and lower bounds on tail-index of homogenized SGD.
We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic cla…
An overwhelming number of true and false news stories are posted and shared in social networks, and users diffuse the stories based on multiple factors. Diffusion of news stories from one user to another depends not only on the stories' content and the genuineness but also on the alignment of the topical interests betw…
Bayesian PINNs learn elliptic PDEs with near-minimax posterior contraction rate.
problem Learning elliptic PDEs with noisy data and non-homogeneous boundary conditions.
method Bayesian approach with a Hölder space prior on neural network weights.
result Posterior contracts at near-minimax rate without prior knowledge of solution smoothness.
Attention-based GNNs can't prevent oversmoothing, leading to homogeneous node representations.
problem The issue of oversmoothing in attention-based GNNs.
method Viewed attention-based GNNs as nonlinear time-varying dynamical systems and used tools from the theory of products of inhomogeneous matrices and the joint spectral radius.
result Graph attention mechanism cannot prevent oversmoothing and loses expressive power exponentially.
Unified theory of deep neural networks with diverse activations.
problem Understanding the relationship between depth and complexity in deep neural networks.
method Developed a unified function space theory for deep networks with various activations.
result Unified theory provides meaningful complexity for deep networks with diverse activations.
Gradient descent on normalized networks reveals sparsity preferences.
problem Understanding the inductive bias of gradient descent on normalized neural nets.
method Analysis of gradient descent on weight-normalized smooth homogeneous neural nets, focusing on SWN and EWN.
result EWN causes weights to be updated in a way that prefers asymptotic relative sparsity.
Survival models predict component failures using neural networks and resampled data.
problem Accurately predicting component failure times for maintenance planning.
method Neural network-based survival models trained on non-independent, homogeneously sampled data.
result Random resampling during training reduces dataset size and improves efficiency.
This work studies the entity-wise topical behavior from massive network logs. Both the temporal and the spatial relationships of the behavior are explored with the learning architectures combing the recurrent neural network (RNN) and the convolutional neural network (CNN). To make the behavioral data appropriate for th…
New method improves graph neural networks by considering different types of relations in sampling.
problem Current graph neural networks ignore relation types in biomedical graphs, leading to suboptimal performance.
method Proposes relation-dependent sampling for multi-relational graphs to balance relation frequency and importance.
result State-of-the-art graph neural networks achieve better accuracy and efficiency with relation-dependent sampling.
GD iterates for non-homogeneous deep nets increase margin and converge in direction.
problem Understanding implicit bias in non-homogeneous deep networks.
method Characterization of GD iterates' properties starting from small empirical risk.
result GD iterates converge in direction despite diverging norms, satisfying KKT conditions.
Study shows momentum-based optimizers like Muon and MomentumGD bias towards KKT points in smooth homogeneous models.
problem Understanding the implicit bias of momentum-based optimizers on smooth homogeneous models.
method Analysis of Muon, MomentumGD, Signum, and Adam optimizers under decaying learning rate schedules.
result Momentum-based optimizers approximate steepest descent trajectories and bias towards KKT points of margin maximization problems.
Steerable neural ODEs on homogeneous spaces for equivariant feature dynamics.
problem Learning continuous-time equivariant dynamics of vector-valued features on homogeneous spaces.
method Introduces steerable neural ordinary differential equations on homogeneous spaces, interpreting features as sections of associated vector bundles over M. result Steerable NODEs are G-equivariant when the flow and connection are G-invariant, and they incorporate existing models. Multivariate splines linked to infinitely-wide neural networks with improved numerical performance.
problem Understanding the relationship between multivariate splines and neural networks.
method Showed multivariate splines can be represented as random features in infinitely-wide neural networks with a homogeneous activation function.
result The function space of multivariate splines is a Sobolev space on a Euclidean ball with explicit norm bounds on derivatives.
Study reveals biases in gradient descent for GLNs, improving neural network performance.
problem Understanding and improving the inductive biases of deep neural networks.
method Derive infinite-time training limit of gated linear networks and generalize to other networks.
result Theoretical framework captures key inductive biases of ReLU networks.
New findings on the max margin problem in neural networks.
problem Understanding the max margin problem in neural networks.
method Analyzing gradient flow and max margin problem in linear and ReLU networks.
result The KKT point is not always an optimum of the max margin problem.
Graph neural networks (GNN) has been successfully applied to operate on the graph-structured data. Given a specific scenario, rich human expertise and tremendous laborious trials are usually required to identify a suitable GNN architecture. It is because the performance of a GNN architecture is significantly affected b…
Neural networks can learn kernel machines with a data-dependent kernel.
problem Can neural networks in the rich feature learning regime learn a kernel machine?
method Demonstrated silent alignment effect in neural networks, showing they can learn a kernel machine with a data-dependent kernel.
result Neural networks in the rich feature learning regime can learn a kernel machine with a data-dependent kernel due to silent alignment.
We propose a method to impose homogeneous linear inequality constraints of the form Ax≤0 on neural network activations. The proposed method allows a data-driven training approach to be combined with modeling prior knowledge about the task. One way to achieve this task is by means of a projection step at test time…
AEGCN uses autoencoder constraints to improve graph node classification.
problem Node classification on graph domains with reduced information loss.
method Autoencoder-constrained graph convolutional network (AEGCN).
result Adding autoencoder constraints significantly improves graph convolutional network performance.
The paper develops a neural network method for estimating drift functions of diffusion processes from discrete observations.
problem Nonparametric estimation of drift function for diffusion processes from high-frequency discrete observations.
method Neural network-based estimator for drift function estimation.
result Derives a non-asymptotic convergence rate for the neural network estimator.
Gradient descent biases towards stable rank networks for nearly-orthogonal data.
problem Understanding implicit bias in non-smooth neural networks trained by gradient descent.
method Analysis of two-layer ReLU and leaky ReLU networks trained by gradient descent on nearly-orthogonal data.
result Gradient descent biases towards networks with stable rank and uniform margin for nearly-orthogonal data.
This study investigates how gradient-based methods bias neural networks trained on high-dimensional data.
problem The implicit biases of gradient-based optimization algorithms in neural networks trained on high-dimensional data.
method Investigation of gradient flow and gradient descent in two-layer fully-connected neural networks with leaky ReLU activations.
result Gradient flow and gradient descent lead to neural networks with low-rank solutions and linear decision boundaries.
ETCNN uses neural networks to price American options accurately.
problem Accurately pricing American options with inequality constraints.
method ETCNN framework solving BSM equations with exact terminal condition.
result ETCNN achieves high accuracy and robustness across various scenarios.