TVS-FNNs can approximate any continuous function on expanded input spaces.
problem Processing a broader range of inputs like sequences and matrices.
method Proving a universal approximation theorem for TVS-FNNs.
result TVS-FNNs can approximate any continuous function on expanded input spaces.
New neural network models learn symmetric functions of varying input sizes.
problem Learning symmetric functions with varying input sizes.
method Functional perspective on neural networks, treating symmetric functions as functions over probability measures.
result Established approximation and generalization bounds for shallow architectures that extend across input sizes.
HyCNNs improve convex function learning and optimal transport.
problem Learning and optimizing convex functions efficiently.
method Combining Maxout networks and ICNNs to create a new neural architecture.
result HyCNNs require fewer parameters and outperform existing methods in convex tasks.
This work uses sampling theory to analyze smoothness and error bounds of finite neural networks.
problem Analyzing the function space of finite neural networks and providing error bounds.
method Applying sampling theory to finite neural networks with non-expansive activation functions, considering both deterministic and random sampling.
result Novel error bounds for univariate neural networks under band-limited input assumption, highlighting the advantage of deterministic uniform sampling.
New neural network approximates convex option prices.
problem Approximating prices of options with convex payoffs.
method Input Convex Neural Network (ICNN) architecture, with a scrambling phase.
result Validated convergence and effectiveness in estimating option prices.
Neural networks learn to mimic brain neurons with two-input activation functions, improving performance and robustness.
problem Training neural networks to mimic the complex interactions of brain neurons.
method Developed a network-in-network architecture with two-input activation functions, optimized hyperparameters, and compared to conventional ReLU networks.
result Two-input activation functions can learn soft XOR functions, improving network performance and robustness.
Package provides sensitivity analysis for neural networks.
problem Understanding how input variables affect neural network predictions.
method Partial derivatives method to calculate sensitivities.
result Allows evaluation of variable importance and input-output relationships.
RI-DeepONet learns neural operators from arbitrary sensor data.
problem Discretization of input functions limits practical applications of DeepONet.
method Introduces RI-DeepONet and two dictionary learning algorithms for INRs.
result RINO handles arbitrary sensor data robustly and applies to various problems.
Presented are two neural network architectures for convex functions, demonstrating competitive performance.
problem Approximating convex functions efficiently and accurately.
method Developed two neural network architectures: one based on linear-by-part representation and the other on cubic splines.
result Cubic ICKAN networks produce results similar to classical ICNNs in solving convex approximation problems.
Paper develops neural network for distribution regression.
problem Regression with probability measures.
method Develops a novel fully connected neural network (FNN) for distribution inputs.
result Almost optimal learning rates for distribution regression derived.
Functional input neural networks approximate continuous functions on weighted spaces.
problem Approximating continuous functions on infinite-dimensional weighted spaces.
method Additive family mapping, non-linear activation, linear readouts, Stone-Weierstrass theorem.
result Global universal approximation of continuous functions on weighted spaces.
New methods for uncertainty in neural networks with leaky ReLU activations.
problem Uncertainty in feed-forward neural networks with random input perturbations.
method Analytical expressions for PDF and moments of neural network output, linearization of leaky ReLU, Gaussian copula surrogate models.
result Accurate statistical results for large input perturbations, excellent agreement with Monte Carlo simulations.
Neural networks adapt to any input dimensionality.
problem Learning functions with inputs of varying dimensions.
method Equivariant neural networks using representation stability.
result Training on fixed dimensions allows extension to any input.
Paper explores neural network approximations on sphere domains.
problem Approximating functionals on sphere domains using neural networks.
method Encoder-decoder framework with spherical harmonics for infinite-dimensional domain.
result Approximation rates of neural networks with different encoder structures.
FoRDE uses input gradients to improve neural network ensembles.
problem Improving neural network ensembles for robustness and accuracy.
method Proposes FoRDE, an ensemble learning method based on ParVI, which repels function space by input gradients.
result FoRDE significantly outperforms DEs and other ensemble methods in accuracy and calibration.
Sobolev training helps neural nets fit function values and derivatives.
problem Training neural nets to match function values and derivatives accurately.
method Using Sobolev loss with gradient flow for overparameterized networks.
result Gradient flow from random initialization can fit any function and its derivatives.
Neural network predicts functional responses from scalar inputs.
problem Regression of functional responses with large scalar predictors and nonlinear relationships.
method Transform functional response to finite dimensions, design feed-forward neural network, modify output via objective functions, apply roughness penalty.
result Proposed neural network outperforms conventional methods in multiple scenarios.
Neural networks are usually not the tool of choice for nonparametric high-dimensional problems where the number of input features is much larger than the number of observations. Though neural networks can approximate complex multivariate functions, they generally require a large number of training observations to obtai…
New framework interprets deep neural networks through input structure.
problem Limited understanding of how input structure, network parameters, and optimization algorithms work together.
method Introducing a novel theoretical framework based on the compositional structure of piecewise linear activation functions.
result Shows that input instances can be grouped based on their similarity in the internal representation of the neural network.
Neural sorting network improves sorting accuracy for complex inputs.
problem Sorting complex inputs like multi-digit images.
method Differentiable swap function and permutation-equivariant Transformer network.
result Our method outperforms baseline methods on various sorting benchmarks.
Interprets neural network classifiers for categorical inputs.
problem Neural networks' interpretability in human-sensitive applications.
method Mapping to physical energy model, expansion of neural network layers.
result Each layer's contribution to classification can be analyzed.
Exact bounds derived for neural network outputs with noisy inputs.
problem Bounding the output distribution of neural networks with random inputs.
method Applying ReLU NNs to derive bounds for general NNs, then using these to find exact error guarantees.
result Exact upper and lower bounds for the output distribution of neural networks with random inputs.
Goldilocks activation functions improve neural network performance.
problem Improving neural network performance and understanding signal transformation.
method Introducing Goldilocks activation functions that locally deform input signals.
result Goldilocks networks outperform or match SELU and RELU on CIFAR-10 and CIFAR-100 datasets.
This research reverses feature visualization in neural networks to optimize for specific feature objectives.
problem The invertibility of feature visualization in neural networks is not well understood.
method The approach involves optimizing for the feature objective that generates the input used in feature visualization, using the gradient of a specific objective function.
result A closed-form solution is found to minimize the gradient, providing an alternative view on network sensitivity.
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…
Neural networks can approximate functions uniformly across various measures.
problem Universal approximation of functions across different probability measures.
method Proving neural networks are dense in Orlicz spaces, extending classical theorems.
result Neural networks uniformly approximate functions for weakly compact families of measures.
Graph Metanetworks process diverse neural architectures efficiently.
problem Processing diverse neural architectures efficiently.
method Builds metanetworks using graph neural networks to process graphs representing input neural networks.
result Proves GMNs are expressive and equivariant to parameter permutation symmetries.
Wide and shallow networks approximate convex functions well.
problem Understanding why wide and shallow neural networks perform well.
method Analyzing the epigraph of the input-output map of shallow and wide neural networks.
result The epigraph of the input-output map approximates a convex function.
We provide a theoretical explanation of the role of the number of nodes at each layer in deep neural networks. We prove that the largest variation of a deep neural network with ReLU activation function arises when the layer with the fewest nodes changes its activation pattern. An important implication is that deep neur…
Study analyzes catastrophic forgetting in continual learning using teacher-student networks.
problem Catastrophic forgetting in continuously learning systems.
method Teacher-student learning framework, similarity of input distributions and target functions.
result Network can avoid catastrophic forgetting with small input distribution similarity and large target function similarity.
Study on VC dimension of GCNNs with input resolution effects.
problem Understanding the generalization capabilities of GCNNs.
method Derived upper and lower bounds for VC dimension, analyzed factors affecting it.
result Extended previous results on VC dimension of GCNNs, providing insights into input resolution dependence.
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.
IANN visualizes all input variables effects simultaneously.
problem Inability to visualize all input variables effects simultaneously in black-box functions.
method Interpretable Architecture Neural Network (IANN) approach.
result Visualization of all input variables effects directly and simultaneously.
JacNet learns Jacobians to enforce structure on derivatives for invertibility and Lipschitz functions.
problem Enforcing structure on derivatives of neural network mappings.
method Proposes using a neural network to directly learn the Jacobian of the input-output function, allowing control over derivative structure.
result Demonstrates learning invertible approximations to simple and 1-Lipschitz functions.
Recurrent Neural Networks (RNNs) are among the most popular models in sequential data analysis. Yet, in the foundational PAC learning language, what concept class can it learn? Moreover, how can the same recurrent unit simultaneously learn functions from different input tokens to different output tokens, without affect…
The paper studies deep neural networks with Gaussian weights and finds their asymptotic behavior.
problem Understanding the behavior of deep neural networks with large width.
method Function-space perspective, Gaussian process analysis, weak convergence in large-width limit.
result Deep neural networks with large width converge to a continuous Gaussian process.
Adaptive neural networks learn functional data bases for improved performance.
problem Applying deep learning to functional data is challenging due to high dimensionality.
method Proposes adaptive neural networks with Basis Layers that learn relevant basis functions.
result Empirically outperforms other neural network approaches across various tasks.
A neural network finds causal relationships among latent variables.
problem Learning causal structure among latent variables in high-dimensional data.
method Redundant Input Neural Network (RINN) with modified architecture and regularized objective function.
result The RINN method successfully recovers latent causal structure between input and output variables.
Generalizes neural network approximation to infinite-dimensional manifolds and derivatives.
problem Approximating differentiable maps on infinite-dimensional manifolds.
method Proves a weighted Nachbin theorem to establish universal approximation for differentiable maps, including derivatives.
result Linear functions of the signature can approximate path space functionals including their derivatives.
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.
Sparse neural networks can match dense models on Lipschitz functions.
problem Sparse networks are more efficient but lack theoretical guarantees.
method Formal model of sparse networks, LSH-based routing function, Lipschitz function approximation.
result Sparse networks can approximate dense networks on Lipschitz functions.
Paper presents a method to accurately quantify neural network uncertainty without sampling.
problem Uncertainty quantification in neural networks for reliability and robustness.
method Sample-free moment propagation technique for mean vectors and covariance matrices.
result Analytic solution for covariance of nonlinear activation functions.
Neural networks learn discrete tasks on continuous data via emergent geometry.
problem Understanding how neural networks perform discrete computations on continuous data.
method Analysis of Riemannian pullback metric across neural network layers.
result Neural networks learn to discretize continuous inputs and perform logical operations on these discretized variables.
This paper improves RNN generalization without normalization conditions.
problem Generalization of over-parameterized RNNs without normalization constraints.
method Detailed analysis of neural tangent kernel matrix for improved generalization bounds.
result RNNs can learn functions without normalization conditions and with almost-polynomial scaling in input length.
Neural networks benefit from intermediate representations, reducing sample complexity.
problem Understanding how neural networks leverage intermediate representations for hierarchical learning.
method Fixed, randomly initialized neural network as a representation function, compared with raw inputs and other trainable networks.
result Neural representations can achieve improved sample complexities compared to raw inputs, especially for low-rank polynomials.
This paper introduces hierarchical Gaussian process priors for neural networks to capture weight correlations and inductive biases.
problem Capturing weight correlations and inductive biases in neural networks.
method Hierarchical Gaussian process priors with unit embeddings and input-dependent kernels.
result Hierarchical Gaussian process priors provide competitive predictive performance and desirable uncertainty estimates.
Proves depth 2 neural networks can't approximate certain functions as well as depth 3 networks.
problem Approximating functions with depth 2 networks in high dimensions.
method Lower bound proof using worst-to-average-case random self-reducibility.
result Proves depth 2 networks can't approximate certain functions as well as depth 3 networks, resolving an open problem.
We show that for neural network functions that have width less or equal to the input dimension all connected components of decision regions are unbounded. The result holds for continuous and strictly monotonic activation functions as well as for the ReLU activation function. This complements recent results on approxima…