Symmetries (transformations by group actions) are present in many datasets, and leveraging them holds significant promise for improving predictions in machine learning. In this work, we aim to understand when and how deep networks can learn symmetries from data. We focus on a supervised classification paradigm where data symmetries are only partially observed during training: some classes include all transformations of a cyclic group, while others include only a subset. We ask: can deep networks generalize symmetry invariance to the partially sampled classes? In the infinite-width limit, where kernel analogies apply, we derive a neural kernel theory of symmetry learning to address this question. The group-cyclic nature of the dataset allows us to analyze the spectrum of neural kernels in the Fourier domain; here we find a simple characterization of the generalization error as a function of the interaction between class separation (signal) and class-orbit density (noise). We observe that generalization can only be successful when the local structure of the data prevails over its non-local, symmetric, structure, in the kernel space defined by the architecture. This occurs when (1) classes are sufficiently distinct and (2) class orbits are sufficiently dense. Our framework also applies to equivariant architectures (e.g., CNNs), and recovers their success in the special case where the architecture matches the inherent symmetry of the data. Empirically, our theory reproduces the generalization failure of finite-width networks (MLP, CNN, ViT) trained on partially observed versions of rotated-MNIST. We conclude that conventional networks trained with supervision lack a mechanism to learn symmetries that have not been explicitly embedded in their architecture a priori. Our framework could be extended to guide the design of architectures and training procedures able to learn symmetries from data.

This paper presents a method for achieving equilibrium in the ISING Hamiltonian when confronted with unevenly distributed charges on an irregular grid. Employing (Multi-Edge) QC-LDPC codes and the Boltzmann machine, our approach involves dimensionally expanding the system, substituting charges with circulants, and representing distances through circulant shifts. This results in a systematic mapping of the charge system onto a space, transforming the irregular grid into a uniform configuration, applicable to Torical and Circular Hyperboloid Topologies. The paper covers fundamental definitions and notations related to QC-LDPC Codes, Multi-Edge QC-LDPC codes, and the Boltzmann machine. It explores the marginalization problem in code on the graph probabilistic models for evaluating the partition function, encompassing exact and approximate estimation techniques. Rigorous proof is provided for the attainability of equilibrium states for the Boltzmann machine under Torical and Circular Hyperboloid, paving the way for the application of our methodology. Practical applications of our approach are investigated in Finite Geometry QC-LDPC Codes, specifically in Material Science. The paper further explores its effectiveness in the realm of Natural Language Processing Transformer Deep Neural Networks, examining Generalized Repeat Accumulate Codes, Spatially-Coupled and Cage-Graph QC-LDPC Codes. The versatile and impactful nature of our topology-aware hardware-efficient quasi-cycle codes equilibrium method is showcased across diverse scientific domains without the use of specific section delineations.