Fredrik Ohlsson

Oscar Carlsson, Jan E. Gerken, Hampus Linander et al.

High-resolution wide-angle fisheye images are becoming more and more important for robotics applications such as autonomous driving. However, using ordinary convolutional neural networks or vision transformers on this data is problematic due to projection and distortion losses introduced when projecting to a rectangular grid on the plane. We introduce the HEAL-SWIN transformer, which combines the highly uniform Hierarchical Equal Area iso-Latitude Pixelation (HEALPix) grid used in astrophysics and cosmology with the Hierarchical Shifted-Window (SWIN) transformer to yield an efficient and flexible model capable of training on high-resolution, distortion-free spherical data. In HEAL-SWIN, the nested structure of the HEALPix grid is used to perform the patching and windowing operations of the SWIN transformer, enabling the network to process spherical representations with minimal computational overhead. We demonstrate the superior performance of our model on both synthetic and real automotive datasets, as well as a selection of other image datasets, for semantic segmentation, depth regression and classification tasks. Our code is publicly available at https://github.com/JanEGerken/HEAL-SWIN.

Oskar Nordenfors, Fredrik Ohlsson, Axel Flinth

We investigate the optimization of neural networks on symmetric data, and compare the strategy of constraining the architecture to be equivariant to that of using data augmentation. Our analysis reveals that that the relative geometry of the admissible and the equivariant layers, respectively, plays a key role. Under natural assumptions on the data, network, loss, and group of symmetries, we show that compatibility of the spaces of admissible layers and equivariant layers, in the sense that the corresponding orthogonal projections commute, implies that the sets of equivariant stationary points are identical for the two strategies. If the linear layers of the network also are given a unitary parametrization, the set of equivariant layers is even invariant under the gradient flow for augmented models. Our analysis however also reveals that even in the latter situation, stationary points may be unstable for augmented training although they are stable for the manifestly equivariant models.

Per Åhag, Alexander Friedrich, Fredrik Ohlsson et al.

Neural ordinary differential equations (NODEs) are geometric deep learning models based on dynamical systems and flows generated by vector fields on manifolds. Despite numerous successful applications, particularly within the flow matching paradigm, all existing NODE models are fundamentally constrained to fixed-dimensional dynamics by the intrinsic nature of the manifold's dimension. In this paper, we extend NODEs to M-polyfolds (spaces that can simultaneously accommodate varying dimensions and a notion of differentiability) and introduce PolyNODEs, the first variable-dimensional flow-based model in geometric deep learning. As an example application, we construct explicit M-polyfolds featuring dimensional bottlenecks and PolyNODE autoencoders based on parametrised vector fields that traverse these bottlenecks. We demonstrate experimentally that our PolyNODE models can be trained to solve reconstruction tasks in these spaces, and that latent representations of the input can be extracted and used to solve downstream classification tasks. The code used in our experiments is publicly available at https://github.com/turbotage/PolyNODE .

Emma Andersdotter, Daniel Persson, Fredrik Ohlsson

We introduce steerable neural ordinary differential equations on homogeneous spaces $M=G/H$. These models constitute a novel geometric extension of manifold neural ordinary differential equations (NODEs) that transport associated feature vectors transforming under the local symmetry group $H$. We interpret features as sections of associated vector bundles over $M$, and describe their evolution as parallel transport. This results in a coupled system of ODEs consisting of a flow equation on $M$ and a steering equation acting on features. We show that steerable NODEs are $G$-equivariant whenever the vector field generating the flow and the connection governing parallel transport are both $G$-invariant. Furthermore, we demonstrate how steerable NODEs incorporate existing NODE models and continuous normalizing flows on Lie groups. Our framework provides the geometric foundation for learning continuous-time equivariant dynamics of general vector-valued features on homogeneous spaces.

Jan E. Gerken, Oscar Carlsson, Hampus Linander et al.

We analyze the role of rotational equivariance in convolutional neural networks (CNNs) applied to spherical images. We compare the performance of the group equivariant networks known as S2CNNs and standard non-equivariant CNNs trained with an increasing amount of data augmentation. The chosen architectures can be considered baseline references for the respective design paradigms. Our models are trained and evaluated on single or multiple items from the MNIST or FashionMNIST dataset projected onto the sphere. For the task of image classification, which is inherently rotationally invariant, we find that by considerably increasing the amount of data augmentation and the size of the networks, it is possible for the standard CNNs to reach at least the same performance as the equivariant network. In contrast, for the inherently equivariant task of semantic segmentation, the non-equivariant networks are consistently outperformed by the equivariant networks with significantly fewer parameters. We also analyze and compare the inference latency and training times of the different networks, enabling detailed tradeoff considerations between equivariant architectures and data augmentation for practical problems. The equivariant spherical networks used in the experiments are available at https://github.com/JanEGerken/sem_seg_s2cnn .

Emma Andersdotter, Daniel Persson, Fredrik Ohlsson

In this paper, we develop a manifestly geometric framework for equivariant manifold neural ordinary differential equations (NODEs) and use it to analyse their modelling capabilities for symmetric data. First, we consider the action of a Lie group $G$ on a smooth manifold $M$ and establish the equivalence between equivariance of vector fields, symmetries of the corresponding Cauchy problems, and equivariance of the associated NODEs. We also propose a novel formulation, based on Lie theory for symmetries of differential equations, of the equivariant manifold NODEs in terms of the differential invariants of the action of $G$ on $M$, which provides an efficient parameterisation of the space of equivariant vector fields in a way that is agnostic to both the manifold $M$ and the symmetry group $G$. Second, we construct augmented manifold NODEs, through embeddings into flows on the tangent bundle $TM$, and show that they are universal approximators of diffeomorphisms on any connected $M$. Furthermore, we show that universality persists in the equivariant case and that the augmented equivariant manifold NODEs can be incorporated into the geometric framework using higher-order differential invariants. Finally, we consider the induced action of $G$ on different fields on $M$ and show how it can be used to generalise previous work, on, e.g., continuous normalizing flows, to equivariant models in any geometry.

Jan E. Gerken, Jimmy Aronsson, Oscar Carlsson et al.

We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds $\mathcal{M}$ using principal bundles with structure group $K$ and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces $\mathcal{M}=G/K$, which are instead equivariant with respect to the global symmetry $G$ on $\mathcal{M}$. Group equivariant layers can be interpreted as intertwiners between induced representations of $G$, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case $\mathcal{M}=S^2=\mathrm{SO}(3)/\mathrm{SO}(2)$. Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch-Gordan coefficients for $G=\mathrm{SO}(3)$, illustrating the power of representation theory for deep learning.