Data Symmetries and Learning in Fully Connected Neural Networks

ArXiv, 2020

Invariances to translations have imbued convolutional neural networks with powerful generalization properties. However, we often do not know a priori what invariances are present in the data, or to what extent a model should be invariant to a given symmetry group. We show how to \\emph{learn} invariances and equivariances by parameterizing a distribution over augmentations and optimizing the training loss simultaneously with respect to the network parameters and augmentation parameters. With this simple procedure we can recover the correct set and extent of invariances on image classification, regression, segmentation, and molecular property prediction from a large space of augmentations, on training data alone.

2021 International Joint Conference on Neural Networks (IJCNN), 2021

We introduce a method to design a computationally efficient G-invariant neural network that approximates functions invariant to the action of a given permutation subgroup G ≤ S n of the symmetric group on input data. The key element of the proposed network architecture is a new G-invariant transformation module, which produces a G-invariant latent representation of the input data. This latent representation is then processed with a multi-layer perceptron in the network. We prove the universality of the proposed architecture, discuss its properties and highlight its computational and memory efficiency. Theoretical considerations are supported by numerical experiments involving different network configurations, which demonstrate the effectiveness and strong generalization properties of the proposed method in comparison to other G-invariant neural networks.

Proceedings of the National Academy of Sciences

Exploiting data invariances is crucial for efficient learning in both artificial and biological neural circuits. Understanding how neural networks can discover appropriate representations capable of harnessing the underlying symmetries of their inputs is thus crucial in machine learning and neuroscience. Convolutional neural networks, for example, were designed to exploit translation symmetry, and their capabilities triggered the first wave of deep learning successes. However, learning convolutions directly from translation-invariant data with a fully connected network has so far proven elusive. Here we show how initially fully connected neural networks solving a discrimination task can learn a convolutional structure directly from their inputs, resulting in localized, space-tiling receptive fields. These receptive fields match the filters of a convolutional network trained on the same task. By carefully designing data models for the visual scene, we show that the emergence of this ...

2022

We consider Convolutional Neural Networks (CNNs) with 2D structured features that are symmetric in the spatial dimensions. Such networks arise in modeling pairwise relationships for a sequential recommendation problem, as well as secondary structure inference problems of RNA and protein sequences. We develop a CNN architecture that generates and preserves the symmetry structure in the network's convolutional layers. We present parameterizations for the convolutional kernels that produce update rules to maintain symmetry throughout the training. We apply this architecture to the sequential recommendation problem, the RNA secondary structure inference problem, and the protein contact map prediction problem, showing that the symmetric structured networks produce improved results using fewer numbers of machine parameters.

Physical Review E, 2004

This paper addresses the training of network models from data produced by systems with symmetry properties. It is argued that although general networks are global approximators, in practice some properties such as symmetry are very hard to learn from data. In order to guarantee that the final network will be symmetrical, constraints are developed for two types of models, namely, the multilayer perceptron ͑MLP͒ network and the radial basis function ͑RBF͒ network. In global modeling problems it becomes crucial to impose conditions for symmetry in order to stand a chance of reproducing symmetry-related phenomena. Sufficient conditions are given for MLP and RBF networks to have a set of fixed points that are symmetrical with respect to the origin of the phase space. In the case of MLP networks such conditions reduce to the absence of bias parameters and the requirement of odd activation functions. This turns out to be important from a dynamical point of view since some phenomena are only observed in the context of symmetry, which is not a structurally stable property. The results are illustrated using bench systems that display symmetry, such as the Duffing-Ueda oscillator and the Lorenz system.

arXiv (Cornell University), 2020

Conclusions 41 4.1 A remark on the standard formulation: deepening the ansatz 41 4.2 Outlooks and future developments 43

Cornell University - arXiv, 2022

Assumptions about invariances or symmetries in data can significantly increase the predictive power of statistical models. Many commonly used machine learning models are constraint to respect certain symmetries, such as translation equivariance in convolutional neural networks, and incorporating other symmetry types is actively being studied. Yet, learning invariances from the data itself remains an open research problem. It has been shown that the marginal likelihood offers a principled way to learn invariances in Gaussian Processes. We propose a weight-space equivalent to this approach, by minimizing a lower bound on the marginal likelihood to learn invariances in neural networks, resulting in naturally higher performing models. Kevin P Murphy. Machine learning: a probabilistic perspective. MIT press, 2012. Sebastian W Ober and Laurence Aitchison. Global inducing point variational posteriors for bayesian neural networks and deep gaussian processes. arXiv preprint arXiv:2005.08140, 2020.

2021

Group equivariant convolutional neural networks (G-CNNs) are generalizations of convolutional neural networks (CNNs) which excel in a wide range of technical applications by explicitly encoding symmetries, such as rotations and permutations, in their architectures. Although the success of G-CNNs is driven by their explicit symmetry bias, a recent line of work has proposed that the implicit bias of training algorithms on particular architectures is key to understanding generalization for overparameterized neural nets. In this context, we show that L-layer full-width linear G-CNNs trained via gradient descent for binary classification converge to solutions with lowrank Fourier matrix coefficients, regularized by the 2/L-Schatten matrix norm. Our work strictly generalizes previous analysis on the implicit bias of linear CNNs to linear G-CNNs over all finite groups, including the challenging setting of noncommutative groups (such as permutations), as well as band-limited G-CNNs over infinite groups. We validate our theorems via experiments on a variety of groups, and empirically explore more realistic nonlinear networks, which locally capture similar regularization patterns. Finally, we provide intuitive interpretations of our Fourier space implicit regularization results in real space via uncertainty principles.

2017

The properties of a representation, such as smoothness, adaptability, generality, equivariance/invariance, depend on restrictions imposed during learning. In this paper, we propose using data symmetries, in the sense of equivalences under transformations, as a means for learning symmetryadapted representations, i.e., representations that are equivariant to transformations in the original space. We provide a sufficient condition to enforce the representation, for example the weights of a neural network layer or the atoms of a dictionary, to have a group structure and specifically the group structure in an unlabeled training set. By reducing the analysis of generic group symmetries to permutation symmetries, we devise an analytic expression for a regularization scheme and a permutation invariant metric on the representation space. Our work provides a proof of concept on why and how to learn equivariant representations, without explicit knowledge of the underlying symmetries in the dat...

Learning problems of higher order, like detecting an unknown symmetry within a pattern, are difficult tasks for most neural networks. Even for small problem sizes a very large number of training examples is needed. We propose a self-organizing network which learns unsupervised to categorize the symmetry of new input patterns according to symmetries already seen. Single examples of a given symmetry are sufficient for learning. The network achieves a classification reliability of 96% percent (3 classes). The underlying mechanism is based on the self-organization of dynamical links which is driven by correlated activity of cell assemblies.