Humans take advantage of real world symmetries for various tasks, yet capturing their superb symmetry perception mechanism with a computational model remains elusive. Motivated by a new study demonstrating the extremely high inter-person accuracy of human perceived symmetries in the wild, we have constructed the first deep-learning neural network for reflection and rotation symmetry detection (Sym-NET), trained on photos from MS-COCO (Microsoft-Common Object in COntext) dataset with nearly 11K consistent symmetry-labels from more than 400 human observers. We employ novel methods to convert discrete human labels into symmetry heatmaps, capture symmetry densely in an image and quantitatively evaluate Sym-NET against multiple existing computer vision algorithms. On CVPR 2013 symmetry competition testsets and unseen MS-COCO photos, Sym-NET significantly outperforms all other competitors. Beyond mathematically well-defined symmetries on a plane, Sym-NET demonstrates abilities to identify viewpoint-varied 3D symmetries, partially occluded symmetrical objects, and symmetries at a semantic level.

Symmetry, a central concept in understanding the laws of nature, has been used for centuries in physics, mathematics, and chemistry, to help make mathematical models tractable. Yet, despite its power, symmetry has not been used extensively in machine learning, until rather recently. In this article we show a general way to incorporate symmetries into machine learning models. We demonstrate this with a detailed analysis on a rather simple real world machine learning system - a neural network for classifying handwritten digits, lacking bias terms for every neuron. We demonstrate that ignoring symmetries can have dire over-fitting consequences, and that incorporating symmetry into the model reduces over-fitting, while at the same time reducing complexity, ultimately requiring less training data, and taking less time and resources to train.

Gens, Robert, Domingos, Pedro M.

The chief difficulty in object recognition is that objects' classes are obscured by a large number of extraneous sources of variability, such as pose and part deformation. These sources of variation can be represented by symmetry groups, sets of composable transformations that preserve object identity. Convolutional neural networks (convnets) achieve a degree of translational invariance by computing feature maps over the translation group, but cannot handle other groups. As a result, these groups' effects have to be approximated by small translations, which often requires augmenting datasets and leads to high sample complexity. In this paper, we introduce deep symmetry networks (symnets), a generalization of convnets that forms feature maps over arbitrary symmetry groups. Symnets use kernel-based interpolation to tractably tie parameters and pool over symmetry spaces of any dimension. Like convnets, they are trained with backpropagation. The composition of feature transformations through the layers of a symnet provides a new approach to deep learning. Experiments on NORB and MNIST-rot show that symnets over the affine group greatly reduce sample complexity relative to convnets by better capturing the symmetries in the data.

Lee, Jimmy H. M. (The Chinese University of Hong Kong) | Zhu, Zichen (The Chinese University of Hong Kong)

The pruning power of partial symmetry breaking depends on the given subset of symmetries to break as well as the interactions among symmetry breaking constraints. In the context of Partial Symmetry Breaking During Search (ParSBDS), the search order determines the set of symmetry breaking constraints to add and thus also makes an impact on node and solution pruning. In this paper, we give the first formal characterization of the pruning behavior of ParSBDS and its improved variants. Introducing the notion of Dominance-Completeness (DC-ness), we show that ParSBDS and variants eliminate the symmetry group of the given subset of symmetries if the resultant search tree is DC, and give an example scenario. Unfortunately, building a DC tree is not always possible. We propose two search heuristics with the aim of having more nodes dominated and thus also pruned during search. Extensive experimentation demonstrates how the proposed heuristics and their combination can drastically reduce the solution set size, search space and runtime when compared against the state-of-the-art static and dynamic symmetry breaking methods.