To infer a multilayer representation of high-dimensional count vectors, we propose the Poisson gamma belief network (PGBN) that factorizes each of its layers into the product of a connection weight matrix and the nonnegative real hidden units of the next layer. The PGBN's hidden layers are jointly trained with an upwar…
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48 results for “one hidden layer”
3-layer NTK models generalize better than 2-layer models, especially with large input dimensions.
problem Understanding the generalization of overparameterized neural networks.
method Analyzing the 3-layer NTK model's test error and comparing it to 2-layer NTK models.
result 3-layer NTK models have a faster descent in test error with respect to the number of neurons in the second hidden layer.
Gamma Belief Networksstat.ML
To infer multilayer deep representations of high-dimensional discrete and nonnegative real vectors, we propose an augmentable gamma belief network (GBN) that factorizes each of its hidden layers into the product of a sparse connection weight matrix and the nonnegative real hidden units of the next layer. The GBN's hidd…
Formula found for neural network error with fixed weights.
problem Understanding error in neural networks with fixed weights.
method Provided an explicit formula for approximation error.
result Explicit formula for neural network error with fixed weights.
Universal MLPs with a single hidden layer can learn any function.
problem Learning on various data structures like sequences, images, sets, and graphs.
method Using group theory, the paper proves the universality of a broad class of equivariant MLPs with a single hidden layer.
result Having a hidden layer on which the group acts regularly is sufficient for universal equivariance (invariance).
A neural network with a single hidden layer can't represent certain multivariable functions.
problem Representing certain multivariable functions with a neural network having only one hidden layer.
method Developed a continuum version of a one-hidden-layer neural network with ReLU activation, and proved constraints on its parameters and second derivative.
result Existence of a smooth binary function that cannot be precisely represented by any such neural network.
Chemical networks outperform spiking neural networks in classification tasks.
problem Learning tasks with spiking neural networks require hidden layers, which are computationally expensive.
method Used deterministic mass-action kinetics to prove chemical reaction networks without hidden layers can solve tasks previously solved by spiking neural networks.
result A chemical reaction network without hidden layers outperforms a spiking neural network with hidden layers in a handwritten digit classification task.
Deep Convolutional Neural Networks (CNN) enforces supervised information only at the output layer, and hidden layers are trained by back propagating the prediction error from the output layer without explicit supervision. We propose a supervised feature learning approach, Label Consistent Neural Network, which enforces…
Binary autoencoder with sparse hidden layer preserves information and zero reconstruction error.
problem Preserving information and zero reconstruction error in binary neural networks.
method Binary autoencoder with random binary weights, sparse hidden layer, and varying neuron thresholds.
result Zero reconstruction error for any input with a large hidden layer and varying neuron thresholds.
The muti-layer information bottleneck (IB) problem, where information is propagated (or successively refined) from layer to layer, is considered. Based on information forwarded by the preceding layer, each stage of the network is required to preserve a certain level of relevance with regards to a specific hidden variab…
Recent results in nonparametric regression show that deep learning, i.e., neural network estimates with many hidden layers, are able to circumvent the so-called curse of dimensionality in case that suitable restrictions on the structure of the regression function hold. One key feature of the neural networks used in the…
New method amplifies hidden structure in complex networks.
problem Difficulty in uncovering hidden structure in complex networks.
method Iterative weakening of dominant structure through randomization.
result Theoretical support for the effectiveness of structure amplification.
This paper extends depth separation results to piece-wise oscillatory functions.
problem Approximating functions with piece-wise oscillatory structure using neural networks.
method Extends existing results to piece-wise oscillatory functions using proof strategy from (Eldan and Shamir, 2016).
result Approximation by one-hidden-layer networks holds at a poly(d) rate for functions with constant domain radius and oscillation rate.
New insights into hidden minima in neural networks.
problem Identifying hidden minima in two-layer ReLU networks.
method Analyzing curves along which loss is minimized, focusing on eigenvalue contributions.
result Distinctive structural and symmetry properties of arcs emanating from hidden minima.
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.
In this paper, we introduce transformations of deep rectifier networks, enabling the conversion of deep rectifier networks into shallow rectifier networks. We subsequently prove that any rectifier net of any depth can be represented by a maximum of a number of functions that can be realized by a shallow network with a …
Study on neural networks' sample complexity with one hidden layer.
problem Understanding how sample complexity is affected by network architecture and norm constraints.
method Norm-based uniform convergence bounds for scalar-valued one-hidden-layer networks, focusing on spectral and Frobenius norms.
result Spectral norm control is insufficient for uniform convergence guarantees, but Frobenius norm control is sufficient, with conditions.
Algorithm predicts performance of learning in multi-layer networks with matrix-valued hidden variables.
problem Signal recovery and learning in multi-layer neural networks with matrix-valued hidden variables.
method Unified approximation algorithm for MAP and MMSE inference, extending ML-VAMP to handle matrix-valued unknowns.
result Performance of ML-Mat-VAMP algorithm can be predicted in a random large-system limit.
Shallow supervised 1-hidden layer neural networks have a number of favorable properties that make them easier to interpret, analyze, and optimize than their deep counterparts, but lack their representational power. Here we use 1-hidden layer learning problems to sequentially build deep networks layer by layer, which ca…
George Cybenko's landmark 1989 paper showed that there exists a feedforward neural network, with exactly one hidden layer (and a finite number of neurons), that can arbitrarily approximate a given continuous function on the unit hypercube. The paper did not address how to find the weight/parameters of such a networ…
Proves SQ lower bounds for learning two-hidden-layer neural networks.
problem Learning two-hidden-layer ReLU networks with Gaussian inputs.
method Refined lifting procedure to reduce Boolean PAC learning to Gaussian.
result Superpolynomial SQ lower bounds for Gaussian inputs.
Deep nonlinear models pose a challenge for fitting parameters due to lack of knowledge of the hidden layer and the potentially non-affine relation of the initial and observed layers. In the present work we investigate the use of information theoretic measures such as mutual information and Kullback-Leibler (KL) diverge…
Revisits the connection between neural networks and the Kolmogorov-Arnold theorem.
problem Explains the limitations of using the Kolmogorov-Arnold theorem to explain neural networks with multiple hidden layers.
method Derives modifications of the Kolmogorov-Arnold representation that transfer smoothness properties to the outer function and can be well approximated by ReLU networks.
result Shows that a deep neural network with most layers approximating the interior function is a more natural interpretation of the Kolmogorov-Arnold representation.
We analyze deep neural networks using convex duality to reveal hidden layer structures.
problem Understanding the structure of deep neural networks.
method Introducing a convex analytic framework to characterize hidden layer weights.
result Optimal hidden layer weights align with previous layers via duality.
Enhanced ELM reduces randomness in neural network training.
problem Challenges in ELM architecture design and sensitivity to random weight initialization.
method Introduces Effective Non-Random ELM (ENR-ELM) incorporating signal processing concepts.
result ENR-ELM simplifies architecture design and eliminates random weight selection.
We present an attention mechanism inspired from definition of screened Coulomb potential. This attention mechanism was used to interpret the Graph Attention (GAT) model layers and training dataset by using a flexible and scalable framework (CoulGAT) developed for this purpose. Using CoulGAT, a forest of plain and resne…
Initialization Matters: On the Benign Overfitting of Two-Layer ReLU CNN with Fully Trainable Layerscs.LG
Two-layer CNNs can overfit well if initialized correctly.
problem Understanding the conditions for benign overfitting in over-parameterized CNNs.
method Extending analysis to fully trainable two-layer CNNs, examining initialization scaling effects.
result Initialization scaling of the output layer is crucial; large scales lead to fixed output behavior, small scales to complex interactions.
Supervised training of neural networks for classification is typically performed with a global loss function. The loss function provides a gradient for the output layer, and this gradient is back-propagated to hidden layers to dictate an update direction for the weights. An alternative approach is to train the network …
RBM models reveal how hidden unit tail behavior affects pattern reconstruction.
problem Understanding how the tail behavior of hidden units in RBMs influences pattern reconstruction.
method Identified an effective energy function for RBMs and studied its local minima.
result The ability to reconstruct patterns depends on the tail behavior of the hidden unit prior distribution.
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.
The performance of Feedforward neural network (FNN) fully de-pends upon the selection of architecture and training algorithm. FNN architecture can be tweaked using several parameters, such as the number of hidden layers, number of hidden neurons at each hidden layer and number of connections between layers. There may b…
We examine the effect of the Group Lasso (gLasso) regularizer in selecting the salient nodes of Deep Neural Network (DNN) hidden layers by applying a DNN-HMM hybrid speech recognizer to TED Talks speech data. We test two types of gLasso regularization, one for outgoing weight vectors and another for incoming weight vec…
We study the sample complexity of learning one-hidden-layer convolutional neural networks (CNNs) with non-overlapping filters. We propose a novel algorithm called approximate gradient descent for training CNNs, and show that, with high probability, the proposed algorithm with random initialization grants a linear conve…
Training deep neural networks with the error backpropagation algorithm is considered implausible from a biological perspective. Numerous recent publications suggest elaborate models for biologically plausible variants of deep learning, typically defining success as reaching around 98% test accuracy on the MNIST data se…
A new optimizer for deep learning improves accuracy and reduces training time.
problem Training deep neural networks for classification tasks.
method Hybrid Newton/Gradient Descent (NGD) method exploiting convexity of cross-entropy loss.
result Improves validation error and provides qualitative differences in hidden layer basis functions.
In this paper, a simple, general method of adding auxiliary stochastic neurons to a multi-layer perceptron is proposed. It is shown that the proposed method is a generalization of recently successful methods of dropout (Hinton et al., 2012), explicit noise injection (Vincent et al., 2010; Bishop, 1995) and semantic has…
Kolmogorov neural networks can represent various types of functions.
problem Representing different types of functions with neural networks.
method Continuous, discontinuous bounded or unbounded activation functions in a two hidden layer model.
result Kolmogorov neural networks can represent continuous, discontinuous bounded and all unbounded multivariate functions.
We study the problem of learning one-hidden-layer neural networks with Rectified Linear Unit (ReLU) activation function, where the inputs are sampled from standard Gaussian distribution and the outputs are generated from a noisy teacher network. We analyze the performance of gradient descent for training such kind of n…
LIFE framework improves model accuracy and interpretability.
problem Achieving high prediction accuracy and interpretability in neural networks.
method Three-step process: subset definition, feature creation, and linear model combination.
result LIFE consistently outperforms other models in prediction accuracy and interpretability.
Study Transformer layers under cross-entropy training using mean field control.
problem Understanding the behavior of Transformer layers in cross-entropy training.
method Continuous-depth mean field control analysis, treating depth as time and layer parameters as controls.
result Derivation of a Pontryagin condition for the limiting population problem, involving the softmax residual.
Detects adversarial examples using autoencoders at hidden layers.
problem Identifying adversarial examples in deep neural networks.
method Trains autoencoders on intermediate layers to detect deviations from true data.
result Outperforms state-of-the-art methods in both supervised and unsupervised settings.
Neural networks can approximate any continuous function with a single hidden layer.
problem Approximating any continuous function using neural networks.
method Direct algebraic proof and explicit quantification of hidden units required.
result Explicit bounds on the number of hidden units needed for approximation.
Generative model initializes 2-layer network weights for small datasets.
problem Approximating functions with 2-layer networks using small datasets and gradient-based training.
method Initialize hidden weights with a learned proposal distribution parameterized as a deep generative model. Refine with gradient-based post-processing and regularization.
result Demonstrates effectiveness of the approach with numerical examples.
Deep neural networks converge to Gaussian mixtures as layer width increases.
problem Understanding the distribution of outputs from deep neural networks.
method Proof and experiments with a simple model showing the convergence of neural network outputs to Gaussian mixtures.
result Neural networks converge to Gaussian mixtures as the width of the last hidden layer increases.
Wide neural networks' last hidden layers split into groups of redundant neurons.
problem Understanding why wide neural networks generalize well despite overfitting.
method Analyzed the last hidden layer representations of various convolutional neural networks.
result Wide hidden layers split into groups of redundant neurons, which help generalize.
Stable unactivated neurons reduce expressiveness in ReLU networks.
problem Reducing expressiveness in ReLU neural networks due to stably unactivated neurons.
method Investigated the probability of neurons being stably unactivated in ReLU networks with symmetric weight and bias distributions.
result Proved the probability of a neuron being stably unactivated in the second hidden layer of a ReLU network.
Two-hidden-layer Feedforward Neural Networks are Universal Approximators: A Constructive Approachcs.LG
It is well known that Artificial Neural Networks are universal approximators. The classical result proves that, given a continuous function on a compact set on an n-dimensional space, then there exists a one-hidden-layer feedforward network which approximates the function. Such result proves the existence, but it does …
New findings on hidden symmetries in ReLU networks.
problem Understanding the redundancy and symmetries in ReLU network parameter space.
method Analyzing parameter settings and function classes for various network architectures.
result For certain network architectures, there are no hidden symmetries.