Extends capacity analysis to neural networks, showing how capacity is distributed across layers.
problem How capacity is distributed in neural networks with non-linear layers.
method Introduces layer decoupling to quantify non-linear activation's impact, and uses a markovian rule for capacity propagation in deep networks.
result Shows that under certain conditions, capacity allocation in neural networks is equivalent to linear capacity allocation in an extended input space.
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.
This article demonstrates that convolutional operation can be converted to matrix multiplication, which has the same calculation way with fully connected layer. The article is helpful for the beginners of the neural network to understand how fully connected layer and the convolutional layer work in the backend. To be c…
We propose a new notion of `non-linearity' of a network layer with respect to an input batch that is based on its proximity to a linear system, which is reflected in the non-negative rank of the activation matrix. We measure this non-linearity by applying non-negative factorization to the activation matrix. Considering…
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.
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.
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.
New neural network models for complex functional data analysis.
problem Complex relations between functional predictors and responses.
method Function-on-Function regression models using neural networks with continuous hidden layers.
result Demonstrated power and flexibility in handling complex functional models.
Traditional convolutional layers extract features from patches of data by applying a non-linearity on an affine function of the input. We propose a model that enhances this feature extraction process for the case of sequential data, by feeding patches of the data into a recurrent neural network and using the outputs or…
Deep models improve factor analysis by capturing non-linearity and interaction effects.
problem Improving factor models to better capture complex asset relationships.
method Developed deep fundamental factor models with uncertainty quantification and hidden layers.
result Generated information ratios approximately 1.5x greater than traditional models.
This paper explores how MoE layers improve deep learning performance.
problem Understanding the Mixture-of-Experts (MoE) layer in deep learning.
method Formal study of MoE layer's effectiveness and mechanism.
result MoE layer improves performance by leveraging cluster structure and non-linearity.
New findings show DNC is not optimal for deep models, revealing a low-rank bias.
problem Theoretical limitations of DNC in non-linear models and multi-class classification.
method Analysis of non-linear models of arbitrary depth in multi-class classification.
result DNC stops being optimal for DUFM when going beyond two layers or two classes, due to a low-rank bias.
Deep learning improves sensor performance optimization.
problem Optimizing sensor performance based on key metrics.
method Re-approach non-linear regression using deep learning with Keras and Tensorflow.
result Deep learning models improve sensor performance optimization.
New proof shows deep neural nets can have sub-optimal local minima.
problem Can over-parameterization eliminate sub-optimal local minima in deep neural networks?
method Counter-example with generic input data and non-linear activation functions.
result Sub-optimal local minima exist in deep neural networks regardless of width.
Reveals the first layer of deep networks with high activation thresholds.
problem Learning guarantees for deep neural networks with multiple layers.
method Strengthening parameter recovery guarantees for deep networks with a high threshold assumption.
result Reveals the first layer of a deep neural network under specific activation conditions.
Gradient descent fails to train two-layer ReLU networks, leading to poor performance.
problem Gradient descent training of two-layer ReLU networks initialized by He et al. (2015) fails to find optimal solutions.
method Gradient descent on a least-squares loss for training two-layer (Leaky)ReLU networks.
result Gradient descent only finds bad local minima, leading to linear regression for non-linear target functions.
Langevin algorithms improve training of very deep neural networks, especially for image classification.
problem Training very deep neural networks is challenging due to increased non-linearity and the risk of getting stuck in local minima.
method Comparison of Langevin and non-Langevin algorithms for training deep neural networks, introduction of Layer Langevin algorithm.
result Langevin algorithms, especially Layer Langevin, lead to significant improvements in training deep neural networks, particularly for image classification tasks.
A new linear GCN model improves recommendation performance for large graphs.
problem Training difficulties and over-smoothing in GCN-based CF models.
method Proposes a linear residual graph convolutional network (LRGCCF) to address training difficulties and over-smoothing issues.
result The proposed model yields better efficiency and effectiveness on real datasets.
The paper extends mean field results to three-layer neural networks using SGD.
problem Understanding the dynamics of training three-layer neural networks with SGD.
method Extending mean field results from two-layer networks to three-layer networks with two hidden layers, using non-linear partial differential equations.
result The distributions of weights in the two hidden layers are independent.
Improved texture synthesis using wavelet-based statistics with rectifier non-linearity.
problem Improving texture synthesis quality using wavelet representations.
method Proposes a family of statistics based on non-linear wavelet representations with a generalized rectifier non-linearity.
result Significantly improves visual quality of texture synthesis compared to classical wavelet-based models.
We consider the problem of reconstructing a signal from multi-layered (possibly) non-linear measurements. Using non-rigorous but standard methods from statistical physics we present the Multi-Layer Approximate Message Passing (ML-AMP) algorithm for computing marginal probabilities of the corresponding estimation proble…
Deep networks without non-linearities are equivalent to shallow ones.
problem Training deep orthogonal linear networks with no non-linearity.
method Riemannian gradient descent and gradient descent on factorization.
result Training deep overparametrized networks is equivalent to shallow ones.
A new method bypasses regularization for disentangled latent variables without tuning.
problem Learning disentangled latent variables in unsupervised settings.
method Projection strategy to modify Gaussian encoder, ensuring zero cross-correlation among latent sub-coordinates.
result The method achieves maximal disentanglement theoretically and without loss in expressiveness.
Deep Gaussian processes reduce uncertainty in porous media flow modeling.
problem Uncertainty quantification in flow through heterogeneous porous media.
method Multi-layer hierarchical Gaussian process with variational approximation.
result Automatic selection of hidden layer dimensions and uncertainty propagation.
Analytical solution found for a three-layer network with a specific activation function.
problem Understanding the power of depth in neural networks.
method Found analytical solutions for a three-layer network with a matrix exponential activation function.
result Analytical solutions for equations involving a three-layer network with a matrix exponential activation function.
Over the past decade a wide spectrum of machine learning models have been developed to model the neurodegenerative diseases, associating biomarkers, especially non-intrusive neuroimaging markers, with key clinical scores measuring the cognitive status of patients. Multi-task learning (MTL) has been commonly utilized by…
Layer normalization with activations prevents Gram matrix rank collapse at initialization.
problem Rank collapse in Gram matrices at initialization slows training in deep networks.
method Proved that layer normalization, with activation layers, biases Gram matrix towards identity matrix at exponential rate.
result Layer normalization with activations biases Gram matrix towards identity matrix at exponential rate with depth at initialization.
Transformer attention layers solve single-location regression tasks.
problem Understanding token-wise sparsity and internal linear representations in attention-based models.
method Introduce single-location regression task and a simplified predictor based on self-attention layers.
result Transformer attention layers are asymptotically Bayes optimal and can learn underlying structures effectively.
Study shows depth improves trainability of neural networks by improving kernel conditioning.
problem Improving trainability of neural networks with random initialization and overparameterization.
method Analyzes the role of depth in training neural networks, proving that depth improves conditioning of kernel matrices.
result General result showing depth improves trainability of neural networks by improving the conditioning of kernel matrices.
Study spectral density of neural networks using resolvent method.
problem Investigate spectral density of neural networks with random feature matrices.
method Use resolvent method from random matrix theory, cumulant expansion.
result Impossible to preserve singular value distribution with additive bias.
Our paper explains deep neural collapse in multiple layers.
problem Understanding deep neural collapse in multi-layered neural networks.
method Generalized unconstrained features model for deep networks.
result Deep unconstrained features model exhibits deep neural collapse.
We propose here a multiplex network approach to investigate simultaneously different types of dependency in complex data sets. In particular, we consider multiplex networks made of four layers corresponding respectively to linear, non-linear, tail, and partial correlations among a set of financial time series. We const…
We propose a new way of thinking about deep neural networks, in which the linear and non-linear components of the network are naturally derived and justified in terms of principles in probability theory. In particular, the models constructed in our framework assign probabilities to uncertain realizations, leading to Ku…
Layer-wise relevance propagation (LRP) is a recently proposed technique for explaining predictions of complex non-linear classifiers in terms of input variables. In this paper, we apply LRP for the first time to natural language processing (NLP). More precisely, we use it to explain the predictions of a convolutional n…
Learning weights in a spiking neural network with hidden neurons, using local, stable and online rules, to control non-linear body dynamics is an open problem. Here, we employ a supervised scheme, Feedback-based Online Local Learning Of Weights (FOLLOW), to train a network of heterogeneous spiking neurons with hidden l…
This paper presents a phase diagram for two-layer neural networks under different initialization scales.
problem Understanding the behavior of neural networks under varying scales of initialization.
method Analysis of a phase diagram for two-layer neural networks.
result Condensation of weight vectors on isolated orientations during training.
Transformers use a unique Hessian structure that differs from classical networks, affecting optimization.
problem Understanding the unique optimization landscape of Transformers.
method Theoretical Hessian analysis of a single self-attention layer in Transformers.
result Transformers have a highly non-linear Hessian structure, distinguishing them from classical networks.
Simplifies deep learning scaling analysis without sacrificing accuracy.
problem Interpreting feature learning mechanisms and determining network implicit bias in high-dimensional settings.
method Developed a heuristic approach for predicting data and width scales of feature learning patterns.
result Predictions align with known results and extend to complex architectures.
Develops a generic two-layer framework for adaptive ABMs.
problem Bi-level adaptation problem in ABMs: agents adapt to environment, and environment adapts to agents.
method Formalizes bi-level problem as a Stackelberg game with conditional policies, solving coupled non-linear equations.
result Unified framework for adaptive ABMs, addressing traditional ABM limitations.
Spiking neural networks (SNNs) have garnered a great amount of interest for supervised and unsupervised learning applications. This paper deals with the problem of training multi-layer feedforward SNNs. The non-linear integrate-and-fire dynamics employed by spiking neurons make it difficult to train SNNs to generate de…
We provide an overview of several non-linear activation functions in a neural network architecture that have proven successful in many machine learning applications. We conduct an empirical analysis on the effectiveness of using these function on the MNIST classification task, with the aim of clarifying which functions…
L-CNNs preserve gauge symmetry in neural networks.
problem Applying machine learning to lattice gauge theory while preserving gauge symmetry.
method L-CNNs use gauge equivariance to construct a gauge equivariant convolutional layer and bilinear layer.
result L-CNNs achieve higher accuracy in non-linear regression tasks compared to non-equivariant CNNs.
Optimal neuron activation functions improve neural network performance.
problem Limited expressive power of standard neuron activation functions in neural networks.
method Additive Gaussian process regression to construct individual neuron activation functions.
result Optimal neuron activation functions lead to better performance and reduced overfitting.
This research explores the dynamics of linearised neural nets, revealing distinct learning phases and layer growth rates.
problem Understanding the fundamental mechanics of neural nets and their learning dynamics.
method Derivation of properties of learning dynamics in general multi-layer linear neural nets, including orthogonal networks.
result Linear multi-layer neural nets exhibit distinct phases of learning with different layer growth rates, and nonlinearity affects these dynamics.
ThriftyNet uses a single convolutional layer recursively to maximize parameter usage.
problem Maximizing the use of parameters in deep convolutional neural networks.
method A single convolutional layer is used recursively, with normalization, non-linearities, downsampling, and shortcuts to maintain model expressivity.
result ThriftyNet achieves competitive performance with significantly fewer parameters.
Deep random feature models are analyzed for their performance with exact asymptotic expressions.
problem Understanding the performance of deep random feature models.
method Established a novel universality result and used the convex Gaussian Min-Max theorem.
result Exact asymptotic expressions for the performance of deep random feature models are derived.
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.
Analyzes generalization error in generalized linear models, explaining double descent phenomenon.
problem Understanding generalization of machine learning models in high dimensions.
method Develops a framework to characterize asymptotic generalization error for generalized linear models.
result Rigorously explains the double descent phenomenon in generalized linear models.