In this paper, we propose a hybrid framework that heals corrupted finite semigroups, combining deterministic repair strategies with Machine Learning using a Random Forest Classifier. Corruption in these tables breaks associativity and invalidates the algebraic structure. Deterministic methods work for small cardinality n and low corruption but degrade rapidly. Our experiments, carried out on Mace4-generated data sets, demonstrate that our hybrid framework achieves higher healing rates than deterministic-only and ML-only baselines. At a corruption percentage of p=15%, our framework healed 95% of semigroups up to cardinality n=6 and 60% at n=10.

In this paper, we propose a framework to extract the algebra of the transformations of worlds from the perspective of an agent. As a starting point, we use our framework to reproduce the symmetry-based representations from the symmetry-based disentangled representation learning (SBDRL) formalism proposed by [1]; only the algebra of transformations of worlds that form groups can be described using symmetry-based representations. We then study the algebras of the transformations of worlds with features that occur in simple reinforcement learning scenarios. Using computational methods, that we developed, we extract the algebras of the transformations of these worlds and classify them according to their properties. Finally, we generalise two important results of SBDRL - the equivariance condition and the disentangling definition - from only working with symmetry-based representations to working with representations capturing the transformation properties of worlds with transformations for any algebra. Finally, we combine our generalised equivariance condition and our generalised disentangling definition to show that disentangled sub-algebras can each have their own individual equivariance conditions, which can be treated independently.

We present a neural network architecture, Bispectral Neural Networks (BNNs) for learning representations that are invariant to the actions of compact commutative groups on the space over which a signal is defined. The model incorporates the ansatz of the bispectrum, an analytically defined group invariant that is complete--that is, it preserves all signal structure while removing only the variation due to group actions. Here, we demonstrate that BNNs are able to simultaneously learn groups, their irreducible representations, and corresponding equivariant and complete-invariant maps purely from the symmetries implicit in data. Further, we demonstrate that the completeness property endows these networks with strong invariance-based adversarial robustness. This work establishes Bispectral Neural Networks as a powerful computational primitive for robust invariant representation learning. A fundamental problem of intelligence is to model the transformation structure of the natural world. In the context of vision, translation, rotation, and scaling define symmetries of object categorization--the transformations that leave perceived object identity invariant. In audition, pitch and timbre define symmetries of speech recognition. Biological neural systems have learned these symmetries from the statistics of the natural world--either through evolution or accumulated experience. Here, we tackle the problem of learning symmetries in artificial neural networks. At the heart of the challenge lie two requirements that are frequently in tension: invariance to transformation structure and selectivity to pattern structure. In deep networks, operations such as max or average are commonly employed to achieve invariance to local transformations. Such operations are invariant to many natural transformations; however, they are also invariant to unnatural transformations that destroy image structure, such as pixel permutations.

We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we find that questions such as identification of simple groups by "looking" at the Cayley table or correctly matching addition and multiplication tables for finite rings can, at least for structures of small size, be performed by the AI, even after having been trained only on small number of cases. These results are in tandem with recent investigations on whether AI can solve certain classes of problems in algebraic geometry.

3D Convolutional Neural Networks are sensitive to transformations applied to their input. This is a problem because a voxelized version of a 3D object, and its rotated clone, will look unrelated to each other after passing through to the last layer of a network. Instead, an idealized model would preserve a meaningful representation of the voxelized object, while explaining the pose-difference between the two inputs. An equivariant representation vector has two components: the invariant identity part, and a discernable encoding of the transformation. Models that can't explain pose-differences risk "diluting" the representation, in pursuit of optimizing a classification or regression loss function. We introduce a Group Convolutional Neural Network with linear equivariance to translations and right angle rotations in three dimensions. We call this network CubeNet, reflecting its cube-like symmetry. By construction, this network helps preserve a 3D shape's global and local signature, as it is transformed through successive layers. We apply this network to a variety of 3D inference problems, achieving state-of-the-art on the ModelNet10 classification challenge, and comparable performance on the ISBI 2012 Connectome Segmentation Benchmark. To the best of our knowledge, this is the first 3D rotation equivariant CNN for voxel representations.