kth-order invariant graph networks are as powerful as kth-order WL in distinguishing graphs.
problem Measuring the expressive power of graph neural network formalisms.
method Considered kth-order invariant graph networks (k-IGNs) and compared their expressive power to kth-order WL.
result k-IGNs and k-WL are equally powerful in distinguishing graphs.
Geometrically interpolates rigid body motions with initial and terminal twists.
problem Finding spatial trajectories between prescribed initial and terminal poses.
method Derives solutions for k-IV-TIP and k-BV-TIP for k=1,...,4.
result Automatic cubic interpolation identical to minimum acceleration curve when twists are zero.
Bayesian network learns data invariances without augmentation.
problem Learning invariances in neural networks without manual design.
method Bayesian approach infers weight-sharing schemes from data.
result Model outperforms non-invariant networks on specific tasks.
Efficient neural network invariant to symmetry subgroups.
problem Designing neural networks invariant to symmetry subgroups for computational efficiency.
method A new G-invariant transformation module and multi-layer perceptron. result The proposed architecture is computationally and memory efficient, and universal.
Learn invariances in neural networks by optimizing over augmentation parameters.
problem Lack of knowledge about present invariances and their extent in data.
method Parameterize a distribution over augmentations and optimize network parameters and augmentation parameters simultaneously.
result Recover correct set and extent of invariances on various tasks from training data alone.
The study analyzes neural network predictions of knot invariants and finds that braid representations work best.
problem Understanding and predicting knot invariants using neural networks.
method Investigated different knot representations and invariants, proposed a cosine similarity score.
result Braid representations are best for predicting knot invariants, and some invariants are easier to learn than others.
Constraining linear layers in neural networks to respect symmetry transformations from a group G is a common design principle for invariant networks that has found many applications in machine learning. In this paper, we consider a fundamental question that has received little attention to date: Can these networks ap…
Proposes learning invariances in neural networks using a weight-space approach.
problem Learning invariances from data in neural networks remains an open problem.
method Minimizes a lower bound on the marginal likelihood in weight space.
result Results in higher performing models with naturally learned invariances.
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.
Frame Averaging makes neural networks invariant or equivariant to new symmetries.
problem Designing neural networks that respect symmetries while being expressive and efficient.
method Introduces Frame Averaging (FA) as a systematic framework to adapt architectures to become invariant or equivariant to new symmetries.
result Frame Averaging guarantees exact invariance or equivariance while being simpler to compute than full group averaging.
Numerous invariant (or equivariant) neural networks have succeeded in handling invariant data such as point clouds and graphs. However, a generalization theory for the neural networks has not been well developed, because several essential factors for the theory, such as network size and margin distribution, are not dee…
Treating neural network inputs and outputs as random variables, we characterize the structure of neural networks that can be used to model data that are invariant or equivariant under the action of a compact group. Much recent research has been devoted to encoding invariance under symmetry transformations into neural n…
This paper tackles non-vacuous generalization bounds in ReLU networks by resolving rescaling invariances.
problem Non-vacuous generalization guarantees for ReLU networks with rescaling invariances.
method Proposes a lifted representation to resolve rescaling invariances and studies KL-based rescaling-invariant PAC-Bayes bounds.
result KL-based rescaling-invariant PAC-Bayes bounds provide tighter guarantees and resolve discrepancies in network complexity.
A universal collection of 4 invariants improves neural network accuracy for molecular dynamics.
problem Improving accuracy of neural networks in molecular dynamics.
method Developed a universal collection of 4 smooth scalar invariants on M(3) x M(3) and evaluated their effectiveness in a PONITA neural network architecture.
result Using a universal collection of invariants significantly improves neural network accuracy.
In this paper, we develop a theory about the relationship between G-invariant/equivariant functions and deep neural networks for finite group G. Especially, for a given G-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip G-actions and each affine t…
Invariant polynomials improve machine learning performance.
problem Improving machine learning algorithms using invariant polynomials.
method Developed and implemented Lorentz- and permutation-invariant polynomial generators in neural networks.
result Reduction in loss on training and validation data with Hironaka decompositions.
Group-invariant neural networks improve approximation accuracy for symmetric functions.
problem Improving approximation accuracy for symmetric functions using neural networks.
method Investigates the generalization error of group-invariant neural networks within the Barron framework.
result Group invariance introduces a factor δ that can significantly improve approximation accuracy when it is small.
Deep neural networks approximate functions in shift-invariant spaces with controlled error.
problem Approximating functions in shift-invariant spaces with neural networks.
method Using deep ReLU neural networks, estimating approximation error bounds based on network width and depth.
result Deep neural networks achieve optimal approximation rates for Sobolev spaces up to a logarithmic factor.
Permutation invariant network learns Wasserstein metrics.
problem Understanding the space of probability measures and comparing distributions.
method Permutation invariant network mapping samples to a low-dimensional space.
result Network can generalize to compute distances between unseen densities and learn moments.
Framework adds invariance to pretrained networks without fine-tuning.
problem Adding invariance to pretrained networks without altering original behavior.
method Post-training augmentation invariance framework with Markov-Wasserstein minimization and Wasserstein correlation maximization losses.
result Adapter networks improve classification accuracy on rotated and noisy images.
Theoretical comparison of three invariance approaches in deep linear networks.
problem Understanding invariance in deep linear networks.
method Data augmentation, regularization, and hard-wiring approaches.
result Regularization introduces additional critical points, but they remain saddles except for the global optimum.
New method relaxes spatial invariance in locally connected layers, improving accuracy.
problem Improving classification accuracy with locally connected layers.
method Designing a low-rank locally connected layer with varying spatially varying combining weights.
result Relaxing spatial invariance improves classification accuracy over convolution and locally connected layers.
We add prior knowledge to deep networks to make them invariant to transformations.
problem Creating deep networks invariant to transformations like rotation.
method A novel layer based on invariant integration to enforce feature space invariances.
result State-of-the-art performance on the Rotated-MNIST dataset.
Three training regimes found for scale-invariant neural networks on the sphere.
problem Training scale-invariant neural networks on the sphere with varying effective learning rate.
method Investigated three regimes of training: convergence, chaotic equilibrium, and divergence.
result Discovered three distinct training regimes with unique characteristics.
One of the fundamental problems in supervised classification and in machine learning in general, is the modelling of non-parametric invariances that exist in data. Most prior art has focused on enforcing priors in the form of invariances to parametric nuisance transformations that are expected to be present in data. Le…
This research quantifies neural networks using magnitude, a topological invariant.
problem Understanding the generalization capabilities of neural networks.
method Using a novel topological invariant called magnitude to study neural network representations.
result Magnitude dimension is theoretically connected to generalisation error and can predict it.
The paper extends hypothesis testing to non-diagonalizable matrices, improving network statistics inference.
problem Testing on non-diagonalizable matrices for network statistics.
method Generalizes Wald and t-tests to non-symmetric matrices, controlling convergence rates.
result Improved inference on network statistics from directed networks.
Recently, researchers have started applying convolutional neural networks (CNNs) with one-dimensional convolutions to clinical tasks involving time-series data. This is due, in part, to their computational efficiency, relative to recurrent neural networks and their ability to efficiently exploit certain temporal invari…
New algorithm learns invariant representations for robust neural networks.
problem Learning robust neural network representations that are invariant to certain factors.
method Causal perspective and distribution matching approach.
result Empirically, the algorithm achieves state-of-the-art performance on domain generalization.
Training deep neural networks is known to require a large number of training samples. However, in many applications only few training samples are available. In this work, we tackle the issue of training neural networks for classification task when few training samples are available. We attempt to solve this issue by pr…
New approach connects 3D Chern-Simons theory to spectral networks.
problem Understanding Chern-Simons invariants in 3D manifolds.
method Constructing equivalences between bundles and spectral networks.
result New formulas for Chern-Simons invariants of 3D manifolds.
Machine learning classifies braids and discovers new invariants.
problem Classifying and discovering invariants of braids and flat braids.
method Supervised learning with neural networks to classify braids as trivial or non-trivial.
result Found new convenient invariants of braids, including a complete invariant of flat braids.
Deep neural networks predict knot invariants across dimensions with high accuracy.
problem Predicting knot invariants in different dimensions using machine learning.
method Two-layer feed-forward neural networks trained on various knot invariants.
result Neural networks achieve high accuracy in predicting knot invariants like s and g. This paper classifies G-invariant shallow neural networks.
problem Designing optimal G-invariant neural architectures for G-invariant target functions. method Proving theorems about the classification and morphisms of G-invariant single-hidden-layer neural networks. result Classification of G-invariant shallow neural networks and characterization of morphisms. Invariant and equivariant networks have been successfully used for learning images, sets, point clouds, and graphs. A basic challenge in developing such networks is finding the maximal collection of invariant and equivariant linear layers. Although this question is answered for the first three examples (for popular tra…
Automates machine learning of correlations between knot invariants.
problem Discovering and validating new relationships between knot invariants.
method Trained a neural network on 200,000 sets of knot invariants to predict an output invariant.
result Found novel correlations not explained by known results in knot theory.
Investigates spectral properties of neural networks, showing invariance under certain conditions.
problem Understanding the spectral evolution and invariance in linear-width neural networks.
method Empirical and theoretical analysis of spectra of weight matrices in high-dimensional settings.
result Spectra of weight matrices are invariant under certain training conditions, with implications for feature learning.
New neural architectures invariant to sign flips and basis symmetries for graph representation learning.
problem Learning invariant graph representations from eigenvectors.
method SignNet and BasisNet neural architectures that are invariant to sign flips and basis symmetries.
result Proven to be universal, approximating any continuous function of eigenvectors with desired invariances.
L-CNNs preserve gauge symmetry in lattice simulations.
problem Breaking gauge symmetry in neural network models.
method Lattice gauge equivariant convolutional neural networks (L-CNNs).
result L-CNNs represent gauge invariant functions on the lattice.
Performance of neural networks can be significantly improved by encoding known invariance for particular tasks. Many image classification tasks, such as those related to cellular imaging, exhibit invariance to rotation. We present a novel scheme using the magnitude response of the 2D-discrete-Fourier transform (2D-DFT)…
Human reasoning involves recognising common underlying principles across many examples. The by-products of such reasoning are invariants that capture patterns such as "if someone went somewhere then they are there", expressed using variables "someone" and "somewhere" instead of mentioning specific people or places. Hum…
Discover conservation laws from trajectories using a neural network.
problem Finding invariants and conservation laws from large-scale data without prior knowledge.
method ConservNet, a neural network trained with noise-variance loss to discover hidden invariants in grouped multi-dimensional observables.
result Successfully discovers underlying invariants from simulated and real-world systems.
Most neural networks are trained using first-order optimization methods, which are sensitive to the parameterization of the model. Natural gradient descent is invariant to smooth reparameterizations because it is defined in a coordinate-free way, but tractable approximations are typically defined in terms of coordinate…
Neural networks struggle with extrapolation, but a new framework allows them to learn counterfactual invariances.
problem Neural networks' inability to extrapolate beyond training data distribution.
method Introduces a learning framework that allows neural networks to extrapolate over group transformations based on counterfactual invariances.
result Neural networks can learn counterfactual invariances from a single environment, overcoming their limitations in extrapolation.
It is well known that neural networks with rectified linear units (ReLU) activation functions are positively scale-invariant. Conventional algorithms like stochastic gradient descent optimize the neural networks in the vector space of weights, which is, however, not positively scale-invariant. This mismatch may lead to…
Geometric deep learning predicts knot invariants.
problem Predicting knot invariants from knot data.
method Constructing a functor from knots to graphs and using graph neural networks.
result High generalization capabilities demonstrated.
An important goal in visual recognition is to devise image representations that are invariant to particular transformations. In this paper, we address this goal with a new type of convolutional neural network (CNN) whose invariance is encoded by a reproducing kernel. Unlike traditional approaches where neural networks …
Method learns invariances in deep nets without human validation.
problem Manual selection of data augmentation parameters is cumbersome.
method Differentiable Laplace approximation for Bayesian model selection.
result Method successfully recovers invariances and improves generalization.