Analyzes neural networks using linear models to understand their behavior.
problem Understanding multi-layer neural networks through linear models.
method Recalls and reviews four models: linear regression with concentrated features, kernel ridge regression, random feature model, and neural tangent model.
result Highlights limitations of linear theory and discusses approaches to overcome them.
Equivariant neural networks use symmetry to interpret complex data.
problem Interpreting and understanding the behavior of equivariant neural networks.
method Decompose layers into simple representations and analyze nonlinear activation functions.
result Equivariant neural networks can be interpreted using a filtration generalizing Fourier series.
Neural networks with DAGs show linearity as width increases.
problem Understanding linearity in neural networks with arbitrary DAG structures.
method Analyzing the transition to linearity in networks with arbitrary DAGs, characterizing width by minimum in-degree.
result General neural networks with DAGs exhibit linearity as width approaches infinity.
Neural networks can be simplified to linear regression for easier understanding by statisticians.
problem Introducing neural networks to statisticians who are not familiar with them.
method Describing neural networks that approximate linear regression and discussing customizations.
result Statisticians can now understand neural networks by focusing on linear regression.
The paper examines how deep linear neural networks behave as they become infinitely wide.
problem Understanding the behavior of deep linear neural networks as they approach infinite width.
method Analyzes the infinite-width limit of deep linear neural networks, proving convergence to deterministic models and providing precise laws for random weights.
result The training dynamics of deep linear neural networks converge to those of a deterministic model, and the weights' behavior is precisely described.
Neural nets trained with linear discriminant initialization converge faster and more accurately.
problem Training feed-forward neural networks efficiently and accurately.
method Initialize first layer weights with linear discriminants.
result Asymptotic higher accuracy and faster convergence.
Two new criteria help understand the advantage of deep neural networks.
problem Understanding the advantage of deepening neural networks.
method Proposed two new criteria to evaluate the expressivity of functions computable by deep neural networks.
result Increasing layers is more effective than increasing units in improving the expressivity of deep neural networks.
A new complexity measure for neural networks improves upon classical methods.
problem Lack of a refined complexity measure for comparing different neural network architectures, especially permutation-invariant ones.
method Introduced an equivalence relation among linear functions and counted them relative to this relation.
result The new complexity measure clearly distinguishes between different models and increases exponentially with depth.
Piecewise linear activations create many spurious local minima in neural networks.
problem Understanding the loss surface of neural networks with piecewise linear activations.
method Proved the existence of infinite spurious local minima and partitioned the loss surface into smooth cells.
result Piecewise linear activations create many spurious local minima that are invariant under a continuous path.
Wide neural networks become linear, with constant tangent kernel, due to Hessian scaling.
problem Understanding the linearity of large non-linear models and the tangent kernel.
method Analyzing the scaling properties of the Hessian matrix of neural networks as their width increases.
result The constancy of the tangent kernel is due to the scaling properties of the Hessian matrix.
Estimates the upper bound of linear regions in spheres centered at specific data points in ReLU neural networks.
problem Bounding the number of linear regions in specific areas of neural networks using ReLU activations.
method Developed a method to estimate the upper bound of linear regions in any sphere within the input space of a ReLU neural network.
result The boundaries of linear regions move away from training data points during training, and spheres centered at these points contain more regions than arbitrary points.
Use simplified layerwise linear models to understand neural dynamics.
problem Complex neural network dynamics are hard to grasp.
method Apply simplified layerwise linear models to explain neural phenomena.
result Simplified models explain neural collapse, emergence, etc.
DeConvNet, Guided BackProp, LRP, were invented to better understand deep neural networks. We show that these methods do not produce the theoretically correct explanation for a linear model. Yet they are used on multi-layer networks with millions of parameters. This is a cause for concern since linear models are simple …
New sample complexity bounds for linear predictors and neural networks, focusing on initialization.
problem Understanding sample complexity for vector-valued linear predictors and neural networks, especially under initialization-dependent conditions.
method Size-independent bounds on Frobenius norm distance from a fixed reference matrix, applying to vector-valued predictors and neural networks.
result Established new sample complexity bounds for feed-forward neural networks, resolving open questions and introducing a new learnable problem.
New neural network models for functional data.
problem Handling non-linear functional data.
method Functional Direct Neural Network (FDNN) and Functional Basis Neural Network (FBNN) with gradient-based optimization.
result Demonstrated effectiveness in complex functional models.
Neural networks improve geospatial data analysis by relaxing linearity assumptions.
problem Traditional geospatial analysis assumes linear models, limiting flexibility.
method Embedding neural networks within traditional geostatistical models for non-linear mean functions.
result NN-GLS algorithm provides consistent and scalable predictions for irregular spatial data.
Wavelets help compress neural networks efficiently.
problem Efficiently compressing linear layers in neural networks.
method Learnable wavelet transforms to compress RNNs.
result Wavelet compressed RNNs have fewer parameters and perform competitively.
New tensor formulation reveals gradient flow's bias in linear neural networks.
problem Understanding implicit bias in linear neural network training.
method Tensor formulation of neural networks, including fully-connected, diagonal, and convolutional networks.
result Gradient flow on linear tensor networks converges to solutions of specific optimization problems.
K-FAC speeds up training of modern neural networks with linear weight-sharing.
problem Efficiently training modern neural networks with linear weight-sharing layers.
method Kronecker-Factored Approximate Curvature (K-FAC) applied to linear weight-sharing layers.
result K-FAC-reduce is generally faster than K-FAC-expand for deep linear networks.
Analyzes minima of deep linear networks with weight decay.
problem Understanding the loss landscape of deep neural networks.
method Analytical solutions for global minima with weight decay and stochastic neurons.
result The origin is a special point with qualitatively different minima in networks with more than 1 hidden layer.
Paper proposes LANN to measure model complexity of neural networks with curve activation functions.
problem Measuring model complexity of neural networks with curve activation functions.
method Proposes LANN, a piecewise linear framework to approximate curve activation functions, and derives complexity measure based on the number of linear regions.
result Demonstrates positive correlation between overfitting and model complexity during training.
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.
Monotonic Linear Interpolation property in neural networks persists despite non-convexity.
problem Understanding the geometric properties of neural network loss landscapes.
method Tools from differential geometry to analyze the monotonicity of neural network weights.
result Sufficient conditions for the Monotonic Linear Interpolation property under mean squared error.
In recent years we see a rapidly growing line of research which shows learnability of various models via common neural network algorithms. Yet, besides a very few outliers, these results show learnability of models that can be learned using linear methods. Namely, such results show that learning neural-networks with gr…
DebiNet uses over-parameterized neural networks to improve linear model performance and debiasing.
problem Improving linear model performance and debiasing in high-dimensional settings.
method Incorporates over-parameterized neural networks into semi-parametric models to estimate parameters consistently.
result DebiNet offers valid inference and accurate prediction by leveraging neural networks' universal approximation and linear model's interpretability.
Quantum neural network and tensor network models outperform classical models in Japanese stock market predictions.
problem Improving stock return predictions using quantum and quantum-inspired machine learning.
method Evaluation of quantum neural network and tensor network models against classical models like linear and neural networks.
result Tensor network model outperforms classical models in Japanese stock market, including linear and neural network models.
Study clarifies Bayesian generalization error in CBM for 3-layered linear neural networks.
problem Understanding the generalization error in concept bottleneck models.
method Mathematical analysis of Bayesian generalization error and free energy in CBM for 3-layered linear neural networks.
result CBM significantly alters the parameter region and Bayesian generalization error compared to standard models.
Real-world problems of operations research are typically high-dimensional and combinatorial. Linear programs are generally used to formulate and efficiently solve these large decision problems. However, in multi-period decision problems, we must often compute expected downstream values corresponding to current decision…
Quadratic models explain neural network behavior during training.
problem Understanding neural network dynamics during training with large learning rates.
method Developed and tested Neural Quadratic Models.
result Neural Quadratic Models exhibit the 'catapult phase' similar to neural networks.
Wide neural networks become linear, but adding bottlenecks makes them bilinear or multilinear.
problem Understanding the transition of neural networks from linearity to higher-order functions.
method Analyzing the behavior of randomly initialized wide neural networks with and without bottleneck layers.
result Bottleneck layers transform the network's function from linear to bilinear or multilinear.
ReLU neural networks define piecewise linear functions of their inputs. However, initializing and training a neural network is very different from fitting a linear spline. In this paper, we expand empirically upon previous theoretical work to demonstrate features of trained neural networks. Standard network initializat…
Linearized neural networks provide a fast and interpretable way to adapt models to new settings.
problem Difficulty in understanding and adapting inductive biases of trained neural networks.
method Linearization of neural networks and embedding these biases into Gaussian processes through a kernel designed from the Jacobian.
result Domain adaptation becomes interpretable posterior inference with analytic and scalable computational speed-ups.
Recurrent neural networks can learn complex transduction problems that require maintaining and actively exploiting a memory of their inputs. Such models traditionally consider memory and input-output functionalities indissolubly entangled. We introduce a novel recurrent architecture based on the conceptual separation b…
Different neural networks trained on the same dataset often learn similar input-output mappings with very different weights. Is there some correspondence between these neural network solutions? For linear networks, it has been shown that different instances of the same network architecture encode the same representatio…
Study shows directional convergence for neural networks under spherical symmetry.
problem Learning linear predictors with neural networks under spherically symmetric data.
method Analysis of gradient flow and gradient descent for two-layer and deep linear networks.
result Directional convergence guarantees with exact convergence rate for specific network architectures.
Unified neural network for linear and nonlinear dimension reduction.
problem Efficiently perform linear and nonlinear sufficient dimension reduction.
method Belted and Ensembled Neural Network (BENN) framework.
result Unified framework for both linear and nonlinear dimension reduction.
DREAM model improves computational efficiency for non-linear effects in relational event models.
problem Efficiently modeling non-linear effects in dynamic relational networks.
method Introduces Deep Relational Event Additive Model (DREAM) using Neural Additive Models.
result Demonstrates superior computational efficiency compared to traditional REM approaches.
Gradient descent with growing learning rate enables learning non-linear features in neural networks.
problem Learning non-linear features in two-layer neural networks.
method Using gradient descent with a learning rate that grows with the sample size.
result Multiple rank-one components emerge, each corresponding to a specific polynomial feature.
Proposes a new linearity-based neural network compression method.
problem Reduction of neural network model size while maintaining accuracy.
method Integrates linearity-based intuition with ReLU activation functions to merge layers.
result Achieves up to 75% reduction in model size without loss of accuracy.
This paper explores loss landscapes of sparse neural networks, finding unique characteristics compared to dense networks.
problem Understanding the loss landscape of sparse neural networks, especially one-hidden-layer networks.
method Analyzes sparse networks with dense and sparse final layers, focusing on linear and non-linear models.
result Sparse networks can have no spurious valleys under certain conditions, but spurious valleys and minima can exist for wide sparse networks.
Study of deep linear neural networks with proportional width and depth.
problem Lack of descriptive power in Gaussian limit of deep linear neural networks.
method Proportional infinite-width infinite-depth limit for deep linear neural networks.
result Characterization of limiting distribution as a nontrivial mixture of Gaussians.
Deep networks become equivalent to linear models in large data regimes.
problem Understanding the behavior of deep neural networks in large data regimes.
method Information-theoretic analysis of fully-trained neural networks in proportional scaling regime.
result Proves deep Gaussian equivalence principle, showing deep networks can be simplified to linear models.
Upper bounds on fixed points in PWL neural networks with hyperplane analysis.
problem Analyzing the number of fixed points in neural networks with PWL activation.
method Hyperplane arrangements to bound the number of fixed points.
result Upper bounds on the number of fixed points for PWL networks, showing exponential growth in layers.
Develops exact convex optimization formulations for neural networks.
problem Training two-layer neural networks with rectified linear units.
method Uses semi-infinite duality and minimum norm regularization to develop exact convex optimization formulations.
result Shows equivalence of ReLU networks trained with weight decay to block ℓ1 penalized convex models. New bounds for nearly-linear networks without training.
problem Generalization of neural networks close to linearity.
method Perturbation of linear networks to derive bounds.
result First non-vacuous bounds for neural nets.
We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and s…
Sparse linear models improve neural network debuggability.
problem Improving neural network interpretability and debugging.
method Using sparse linear models over learned deep feature representations.
result The approach leads to more debuggable and accurate neural networks.
We propose \emph{Taylorized training} as an initiative towards better understanding neural network training at finite width. Taylorized training involves training the k-th order Taylor expansion of the neural network at initialization, and is a principled extension of linearized training---a recently proposed theory …