Using and Abusing Equivariance: Investigating Differences between Exact and Approximate Equivariance in Computer Vision

In this work we show how Group Equivariant Convolutional Neural Networks use subsampling to learn to break equivariance to their symmetries. We focus on the 2D roto-translation group and investigate the impact of broken equivariance on network performance. We show that changing the input dimension of a network by as little as a single pixel can...

Learned equivariance in Convolutional Neural Networks

Aside from developing methods to embed the equivariant priors into the architectures, one can also study how the networks learn equivariant properties. In this work, we conduct a study on the influence of different factors on learned equivariance. We propose a method to quantify equivariance and argue why using the correlation to compare...

Best integer equivariant position estimation for multi-GNSS RTK: A multivariate normal and t-distributed performance comparison

The best integer equivariant (BIE) estimator for the multivariate t-distribution was introduced in Teunissen (J Geod, 2020. https://doi.org/10.1007/s00190-020-01407-2), where it was shown that the BIE-weights will be different from that of the normal distribution. In this contribution, we analyze these BIE estimators while making use of multi...

Group Equivariant Video Action Recognition: Making action-recognition networks equivariant to temporal direction and discrete spatial rotations

This work applies the theory of group equivariance to the domain of video action recognition replacing standard 3Dconvolutions with group convolutions which are equivariant to temporal direction, and multiples of 90-degree spatial rotations. We propose a temporal direction symmetry group T2 and extend the standard planar rotations group to three...

Exploring Convolutional Neural Networks

In this thesis we have looked into the complexity of neural networks. Especially convolutional neural networks (CNNs), which are useful for image recognition, are looked into. In order to better understand the process in the neural networks, in the first half of this report a mathematical foundation for neural networks and CNNs is constructed....

Normal form of equivariant maps in infinite dimensions

Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds...

Graph-Adaptive Activation Functions for Graph Neural Networks

Network data are essential in applications such as recommender systems, social networks, and sensor networks. A unique characteristic that these data encompass is the coupling between the data values and the underlying network structure on which these data are defined. Graph Neural Networks (GNNs) have been designed as tools to extend the...

CNNs on Surfaces using Rotation-Equivariant Features

This paper is concerned with a fundamental problem in geometric deep learning that arises in the construction of convolutional neural networks on surfaces. Due to curvature, the transport of filter kernels on surfaces results in a rotational ambiguity, which prevents a uniform alignment of these kernels on the surface. We propose a network...

Plannable Approximations to MDP Homomorphisms: Equivariance under Actions

This work exploits action equivariance for representation learning in reinforcement learning. Equivariance under actions states that transitions in the input space are mirrored by equivalent transitions in latent space, while the map and transition functions should also commute. We introduce a contrastive loss function that enforces action...

Best integer equivariant estimation: performance analysis using real data collected by low-cost, single- and dual-frequency, multi-GNSS receivers for short- to long-baseline RTK positioning

The key to precise global navigation satellite system (GNSS) positioning is carrier phase integer ambiguity resolution with a high success rate. On the other hand when the success rate is too low, the user will normally prefer the float solution. The alternative can be to use the best integer equivariant (BIE) estimator, since it is optimal in...

Best integer equivariant estimation for elliptically contoured distributions

This contribution extends the theory of integer equivariant estimation (Teunissen in J Geodesy 77:402–410, 2003) by developing the principle of best integer equivariant (BIE) estimation for the class of elliptically contoured distributions. The presented theory provides new minimum mean squared error solutions to the problem of GNSS carrier...

Harmonic Surface Networks

We present a new approach for deep learning on surfaces, combining geometric convolutional networks with rotationally equivariant networks. Existing work either learns rotationally invariant filters, or learns filters in the tangent plane without correctly relating orientations between different tangent planes (orientation ambiguity). We propose...

Best prediction in linear models with mixed integer/real unknowns: Theory and application