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Deep Learning methods have been adopted in mobile networks, especially for network management automation where they provide means for advanced machine cognition. Deep learning methods utilize cutting-edge hardware and software tools, allowing complex cognitive algorithms to be developed. In a recent paper, we introduced the Bounding Sphere Quantization (BSQ) algorithm, a modification of the k-Means algorithm, that was shown to create better quantizations for certain network management use-cases, such as anomaly detection. However, BSQ required a significantly longer time to train than k-Means, a challenge which can be overcome with a neural network-based implementation. In this paper, we present such an implementation of BSQ that utilizes state-of-the-art deep learning tools to achieve a competitive training speed.

In this paper we study the regular prism tilings and construct ball packings by geodesic balls related to the above tilings in the projective model of $\mathbf{Nil}$ geometry. Packings are generated by action of the discrete prism groups $\mathbf{pq2_{1}}$. We prove that these groups are realized by prism tilings in $\mathbf{Nil}$ space if $(p,q)=(3,6), (4,4), (6,3)$ and determine packing density formulae for geodesic ball packings generated by the above prism groups. Moreover, studying these formulae we determine the conjectured maximal dense packing arrangements and their densities and visualize them in the projective model of $\mathbf{Nil}$ geometry. We get a dense (conjectured locally densest) geodesic ball arrangement related to the parameters $(p,q)=(6,3)$ where the kissing number of the packing is $14$, similarly to the densest lattice-like $\mathbf{Nil}$ geodesic ball arrangement investigated by the second author ., Comment: arXiv admin note: text overlap with arXiv:1105.1986

In this paper we study the locally optimal geodesic ball packings with equal balls to the $\mathbf{S}^2\!\times\!\mathbf{R}$ space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive geodesic ball arrangements for the above space groups, moreover we compute their optimal densities and radii. The densest packing is derived from the $\mathbf{S}^2\!\times\!\mathbf{R}$ space group $\mathbf{3qe.~I.~3}$ with packing density $\approx 0.7278$. E. Moln\'ar has shown, that the Thurston geometries have an unified interpretation in the real projective 3-sphere $\mathcal{PS}^3$. In our work we shall use this projective model of $\mathbf{S}^2\!\times\!\mathbf{R}$ geometry., Comment: arXiv admin note: substantial text overlap with arXiv:1206.0566, arXiv:1210.2202