Cosmic topology. Part IVa. Classification of manifolds using machine learning: a case study with small toroidal universes

Non-trivial spatial topology of the Universe may give rise to potentially measurable signatures in the cosmic microwave background. We explore different machine learning approaches to classify harmonic-space realizations of the microwave background in the test case of Euclidean $E_1$ topology (the 3-torus) with a cubic fundamental domain of a size scale significantly smaller than the diameter of the last scattering surface. This is the first step toward developing a machine learning approach to classification of cosmic topology and likelihood-free inference of topological parameters. Different machine learning approaches are capable of classifying the harmonic-space realizations with accuracy greater than 99% if the topology scale is half of the diameter of the last-scattering surface and orientation of the topology is known. For distinguishing random rotations of these sky realizations from realizations of the covering space, the extreme gradient boosting classifier algorithm performs best with an accuracy of 88%. Slightly lower accuracies of 83% to 87% are obtained with the random forest classifier along with one- and two-dimensional convolutional neural networks. The techniques presented here can also accurately classify non-rotated cubic $E_1$ topology realizations with a topology scale slightly larger than the diameter of the last-scattering surface, if enough training data are provided. While information compressing methods like most machine learning approaches cannot exceed the statistical power of a likelihood-based approach that captures all available information, they potentially offer a computationally cheaper alternative. A principle challenge appears to be accounting for arbitrary orientations of a given topology, although this is also a significant hurdle for likelihood-based approaches.