Computing homotopy groups of spheres has long been a fundamental objective in algebraic topology. Various theoretical and algorithmic approaches have been developed to tackle this problem. In this paper we take a step towards the goal of comprehending the group-theoretic structure of the generators of these homotopy groups by leveraging the power of machine learning. Specifically, in the simplicial group setting of Wu's formula, we reformulate the problem of generating simplicial cycles as a problem of sampling from the intersection of algorithmic datasets related to Dyck languages. We present and evaluate language modelling approaches that employ multi-label information for input sequences, along with the necessary group-theoretic toolkit and non-neural baselines.

The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier or to use the geometry to come up with a better distance metric. Manifold learning problems are often posed in a very high dimension, e.g. for spaces of images or spaces of words. Today, with deep representation learning on the rise in areas such as computer vision and natural language processing, many problems of this kind may be transformed into problems of moderately high dimension, typically of the order of hundreds. Motivated by this, we propose a manifold learning technique suitable for moderately high dimension and large datasets. The manifold is learned from the training data in the form of an intersection of quadric hypersurfaces—simple but expressive objects. At test time, this manifold can be used to introduce a computationally efficient outlier score for arbitrary new data points and to improve a given similarity metric by incorporating the learned geometric structure into it.