Generalization bounds derived for neural ODEs and deep residual networks.
problem Understanding the generalization capability of neural ODEs and deep residual networks.
method Lipschitz-based argument and analogy with deep residual networks.
result A generalization bound involving the magnitude of weight matrix differences.
This research explores principles of Lipschitz continuity in neural networks for robustness and generalization.
problem Ensuring robustness and generalization in neural networks, especially to small input perturbations and out-of-distribution data.
method Two complementary perspectives: internal (training dynamics) and external (frequency signal propagation).
result Advances in understanding the principles of Lipschitz continuity in neural networks.
BFNs use Bayesian inference and neural networks for generative modeling.
problem Learning from non-stationary data in continual learning.
method Bayesian Flow Networks (BFNs) combining neural network expressiveness and Bayesian inference.
result BFNs effectively model non-stationary data.
Generalizes PCA and ICA for continuous-time signals using neural networks.
problem Low-rank decomposition of continuous-time vector-valued signals.
method Implicit neural network framework to learn numerical approximations of PCA and ICA.
result Unified approach to PCA and ICA in continuous domain, enforcing decorrelation and independence.
Continuous-time SGD converges under certain conditions, useful for deep learning.
problem Minimizing population expected loss in learning problems.
method Continuous-time approximation of stochastic gradient descent.
result Establishes sufficient conditions for convergence, applicable to overparametrized neural networks.
We consider efficiency in the implementation of deep neural networks. Hardware accelerators are gaining interest as machine learning becomes one of the drivers of high-performance computing. In these accelerators, the directed graph describing a neural network can be implemented as a directed graph describing a Boolean…
VSDN models sporadic time series with neural SDEs.
problem Modeling irregular and sparse time series data.
method Variational Bayesian method and neural SDEs.
result VSDNs outperform state-of-the-art models in prediction and interpolation.
New framework generalizes neural network parameters to C∗-algebra for more efficient feature learning.
problem Efficient feature learning and adaptability of neural network models.
method Generalizes neural network parameters to C∗-algebra-valued parameters and combines models continuously. result Shows improved feature learning with limited data using the new framework.
A graph VAE framework optimizes neural architectures in a continuous space.
problem Discovering efficient neural architectures in a discrete space.
method Graph VAE framework with VAE and GNN components, joint learning of predictors and decoders.
result The framework discovers powerful neural architectures with both excellent performance and high computational efficiency.
This research studies affine invariance in continuous-domain convolutional neural networks.
problem Recognizing patterns and features under affine transformations in continuous domains.
method Introduces a new criterion for assessing affine invariance, embeds images into the affine Lie group, and analyzes convolution over this group.
result Extends the scope of geometrical transformations that deep-learning pipelines can handle.
Neural networks with integer weights approximate continuous functions efficiently.
problem Approximating continuous functions using neural networks with integer weights.
method Integrates superexpressive activation functions and integer weights.
result Convergence rate of order n2β+d−2βlog2n for neural network regression. 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.
CGNNs use wavelets for continuous function generation in infinite-dimensional spaces.
problem Generating continuous functions in infinite-dimensional spaces for applications like inverse problems.
method Inspired by DCGAN, CGNNs use wavelet multiresolution analysis with convolutional and nonlinear layers.
result CGNNs can be injective under certain conditions on filters and nonlinearity, leading to Lipschitz stability estimates.
Continuous formulation of machine learning models and algorithms.
problem Generalization error and implicit regularization in machine learning.
method Continuous formulation in calculus of variations and differential-integral equations, with new models and algorithms.
result Conventional models and algorithms can be recovered as particular discretizations.
Proposes VCNet for estimating ADRFs of continuous treatments.
problem Estimating ADRFs of continuous treatments from observational data.
method VCNet for improved model expressiveness and continuity; targeted regularization for finite sample performance.
result Improves model expressiveness and continuity of ADRFs.
Study on neural networks' performance in sequential task learning.
problem Understanding the performance of neural networks in sequential task learning.
method Theoretical analysis of generalization performance in continual learning using statistical mechanical analysis of kernel ridge-less regression.
result Characteristic transitions from positive to negative transfer observed in neural networks.
GroupSort neural networks can approximate Lipschitz continuous functions.
problem Understanding and improving the expressive power of neural networks with Lipschitz constraints.
method Introduced and studied GroupSort neural networks with constraints on weights, proving their ability to approximate Lipschitz continuous functions.
result GroupSort networks can represent any Lipschitz continuous piecewise linear functions and are well-suited for approximating general Lipschitz continuous functions.
New framework discovers non-affine continuous symmetries in neural networks.
problem Lack of efficient methods for detecting non-affine continuous symmetries in neural networks.
method Computational framework for discovering infinitesimal generators of multi-parameter group actions.
result Framework can discover non-affine continuous symmetries in neural networks.
TMNs model brain memory systems for continual learning.
problem Catastrophic forgetting in neural networks.
method Triple Network architecture of GANs, incorporating brain-inspired algorithms.
result New state-of-the-art performance on class-incremental learning benchmarks.
Researchers develop neural networks for approximating functions in Banach spaces.
problem Approximating Banach space valued continuous functions.
method Quasi-interpolation Banach space valued neural network operators using algebraic sigmoid functions.
result Jackson type inequalities for function approximation.
Proposes a new convolutional neural network for non-grid data.
problem Limited applicability of standard CNNs to non-grid structured data.
method Introduces Parametric Continuous Convolution (PCC) with learnable kernel functions.
result Significant improvement in point cloud segmentation and lidar motion estimation.
Neural networks approximate high-dimensional functions better than theory predicts.
problem Current theory struggles to explain why small neural networks work well in high-dimensional inverse problems.
method Bounding complexity required for neural networks to approximate Hölder or uniformly continuous functions on high-dimensional sets.
result A general theoretical framework explaining empirical successes of smaller networks in inverse problems.
CN-DPM model tackles task-free continual learning for neural networks.
problem Current continual learning methods are limited to task-boundary known settings.
method CN-DPM uses a neural Dirichlet process mixture model for expansion-based task-free learning.
result CN-DPM successfully performs task-free continual learning for image classification and generation.
Study on neural networks forgetting in continual learning.
problem Forgetting in neural networks during continual learning.
method Gradient descent analysis on XOR-cluster datasets.
result Explicit bounds on forgetting rate and generalization gap.
Lie groupoid equivariant neural networks are a new type of neural network.
problem Designing neural networks that respect the structure of Lie groupoids.
method Introducing Lie groupoid equivariant convolutions and layers, and showing their equivalence to Lie algebroid-equivariant networks.
result Lie groupoid equivariant neural networks are equivalent to certain Lie algebroid-equivariant networks.
Proposes a new method for continual learning in neural networks.
problem Challenges in applying sequential Bayesian inference to neural networks.
method Sequential function-space variational inference.
result Neural networks trained with the proposed method achieve better predictive accuracy.
Neural Ordinary Differential Equation (Neural ODE) has been proposed as a continuous approximation to the ResNet architecture. Some commonly used regularization mechanisms in discrete neural networks (e.g. dropout, Gaussian noise) are missing in current Neural ODE networks. In this paper, we propose a new continuous ne…
Paper proposes ensemble methods to prevent forgetting in neural networks.
problem Catastrophic forgetting in retraining neural networks.
method Gradient boosting and meta-learning approaches.
result Prevents forgetting in pre-trained neural network models.
New bound on neural nets complexity for approximating functions.
problem Approximating continuous functions with shallow neural networks.
method Inspired by Stone-Weierstrass theorem, constructive proof.
result General upper bound on neuron count for accuracy.
Continual learning is hard for AI, leading to forgetting old knowledge.
problem Catastrophic forgetting in AI when learning new data.
method Review of continual learning in deep learning.
result Challenges and insights in continual learning.
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.
Neural SDEs model continuous sequences using neural networks.
problem Modeling continuous-time dynamics in sequence data.
method Interprets time-series as samples from a continuous dynamical system, parameterized by Neural SDE.
result Demonstrates superior performance in diverse sequence modeling tasks.
Simple neural networks approximate any continuous function with fixed neurons.
problem Approximating arbitrary continuous functions with limited neurons.
method Developed simple feed-forward neural networks with a specific activation function.
result Proven that networks with 36d(2d+1) neurons and depth 11 can approximate any continuous function.
New method creates coresets for deep neural networks efficiently.
problem Efficiently handling large data streams with limited resources.
method Cardinality-constrained bilevel optimization for deep neural networks.
result Demonstrated efficient generation of coresets for deep neural networks.
Neural circuits integrate continuous dynamics efficiently.
problem Efficiently integrating continuous neural dynamics for simulation and learning.
method Compact neural circuits for Runge-Kutta and Adams-Bashforth-Moulton methods.
result Equivalence of neural and numerical integration for polynomial systems.
Proposes continuous graph neural networks to capture long-range dependencies.
problem Capturing long-range dependencies in graph data.
method Defines continuous dynamics for graph neural networks using diffusion-based methods.
result Proposed continuous graph neural networks are effective and deeper networks can capture long-range dependencies.
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constan…
Hybrid quantum neural networks predict continuous variables.
problem Predicting continuous variables using quantum computing.
method Quantum classical hybrid neural networks for continuous variable prediction.
result Quantum neural networks outperform classical methods in continuous variable prediction.
Three-hidden-layer neural networks can approximate Hölder continuous functions uniformly with exponential rate.
problem Approximating Hölder continuous functions with neural networks.
method Introduced Floor-Exponential-Step (FLES) networks with three hidden layers.
result Uniform approximation of Hölder continuous functions with an exponential rate.
New proof shows neural networks can represent all multivariate functions.
problem Representing all multivariate functions with neural networks.
method Proved that three-layer neural networks can represent both continuous and discontinuous functions.
result Three-layer neural networks can represent all multivariate functions, including discontinuous ones.
Investigates Lipschitz continuity in neural networks across various settings.
problem Understanding the Lipschitz behavior of neural networks.
method Empirical investigation of Lipschitz bounds in different neural network architectures and datasets.
result Remarkable fidelity of the lower Lipschitz bound and a Double Descent trend in both upper and lower bounds.
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.
This paper tackles discontinuous neural networks for better approximation of piecewise continuous functions.
problem Limitation of neural networks in approximating piecewise continuous functions due to discontinuities.
method Proposes a decoupled two-step procedure to train a discontinuous deep neural network model.
result Provides approximation guarantees for the proposed model in piecewise continuous function spaces.
CIPNN model tackles continuous latent variables, solving intractable posterior problems.
problem Solving intractable posterior calculation for continuous latent variables.
method Derives analytical solution for posterior of continuous latent variables, proposes CIPNN and CIPAE.
result CIPNN model demonstrates great classification capability, solving problems for continuous latent variables.
We prove a negative result for the approximation of functions defined on compact subsets of Rd (where d≥2) using feedforward neural networks with one hidden layer and arbitrary continuous activation function. In a nutshell, this result claims the existence of target functions that are as difficult to…
Neural models learn continuous-time Markov chain transition rates from data.
problem Learning transition rates for complex stochastic systems.
method Neural networks to model nonlinear transition rates from observed data.
result Neural models outperform traditional methods in accuracy.
New Banach spaces for ReLU networks enable better function approximation and gradient dynamics analysis.
problem Function approximation and gradient dynamics in multi-layer ReLU networks.
method Developed Banach spaces for ReLU networks, defined new function representations, and analyzed gradient flow dynamics.
result Gradient flow dynamics of the new representation is the continuous analog of gradient descent for ReLU networks.
Dense neural networks can't approximate all functions.
problem Approximation capabilities of dense neural networks.
method Model compression approach combining weak regularity lemma and graph neural networks.
result Existence of Lipschitz continuous functions not approximable by dense neural networks.