We consider a UV-complete field-theoretic model in general dimensions, including d=2+1, which consists of two copies of thelong-range vector models, with O(m) and O(N-m) global symmetry groups,perturbed by double-trace operators.Using conformal perturbation theorywe find weakly-coupled IR fixed points for N\geq 6 N ≥ 6 that reveal a spontaneousbreaking of global symmetry. Namely, at finite temperature the lower rank group is broken,with the pattern persisting at all temperatures due to scale-invariance. We provide evidence that the models in question are unitary and invariant under full conformal symmetry. Furthermore, we show that this model exhibits a continuous family of weakly interacting field theories at finite N.
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A General Framework for Robust G-Invariance in G-Equivariant Networks
We introduce a general method for achieving robust group-invariance in group-equivariant convolutional neural networks (G-CNNs), which we call the G-triple-correlation (G-TC) layer. The approach leverages the theory of the triple-correlation on groups, which is the unique, lowest-degree polynomial invariant map that is also complete. Many commonly used invariant maps--such as the max--are incomplete: they remove both group and signal structure. A complete invariant, by contrast, removes only the variation due to the actions of the group, while preserving all information about the structure of the signal. The completeness of the triple correlation endows the G-TC layer with strong robustness, which can be observed in its resistance to invariance-based adversarial attacks. In addition, we observe that it yields measurable improvements in classification accuracy over standard Max G-Pooling in G-CNN architectures. We provide a general and efficient implementation of the method for any discretized group, which requires only a table defining the group's product structure. We demonstrate the benefits of this method for G-CNNs defined on both commutative and non-commutative groups--SO(2), O(2), SO(3), and O(3) (discretized as the cyclic C8, dihedral D16, chiral octahedral O and full octahedral Oh groups)--acting on ℝ2 and ℝ3 on both G-MNIST and G-ModelNet10 datasets.
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- Award ID(s):
- 2313150
- PAR ID:
- 10529128
- Publisher / Repository:
- NeurIPS 2023
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract A twisted commutative algebra is (for us) a commutative ‐algebra equipped with an action of the infinite general linear group. In such algebras, the “‐prime” ideals assume the duties fulfilled by prime ideals in ordinary commutative algebra, and so it is crucial to understand them. Unfortunately, distinct ‐primes can have the same radical, which obstructs one from studying them geometrically. We show that this problem can be eliminated by working with super vector spaces: doing so provides enough geometry to distinguish ‐primes. This yields an effective method for analyzing ‐primes.
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A<sc>bstract</sc> We have investigated the modular binary octahedral group 2
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The scattering transform is a multilayered, wavelet-based transform initially introduced as a mathematical model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks’ stability and invariance properties. In subsequent years, there has been widespread interest in extending the success of CNNs to data sets with non- Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. Analogous to the original scattering transform, these works prove that these variants of the scattering transform have desirable stability and invariance properties and aim to improve our understanding of the neural networks used in geometric deep learning. In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on compact Riemannian manifolds without boundary and undirected graphs as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, a directed graph stochastic block model, and on high-dimensional single-cell data.more » « less
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Accepted Manuscript1.0
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