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. Different machine learning approaches are capable of classifying the harmonic-space realizations with accuracy greater than 99 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 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 provided enough training data. This work identifies the prospects and the main challenges for developing machine learning techniques that are capable of accurately classifying observationally viable topologies.