Wireless Channels Topology Invariant as Mathematical Foundation of Neural Network Channel Estimation Transfer Learning Properties

We applied residual deep neural network for high noise power region uplink channel estimation. Proposed residual deep neural network channel estimation show more 1 dB gain under EPA/ETU channels compare to MMSE channel estimation and ability to work on wireless channels that have not previously been trained. Based on fundamental group topological data analysis, we obtained the reason for such transfer learning property in the problem of channel estimation, as well as in machine learning in general. Using achievements of deductive Langlands program, inductive Forneys program and dialectic Topological programs of Frobenius und Weierstrass(students Ajtai, Tyrtyshnikov-Zamarashkin, Oseledets, GelfandBernstein-Frenkel, Pontryagin-Anosov, Berezin, Müller, Bugaev-Kolmogorov-Arnold-Gusein-Zade, Lyusternik-SigalovGlebskiy-Shevchenko, Sinai-Margulis)-Hermite(Poincaré)Rossetti(Witten)-Chebyshev(Manin, Korkin-Zolotarev-Voronoy- Delaunay-Aleksandrov-Ryshkov, Baranovskii, Perelman, Fadeev-Wenkow, Stark-Lagarias)-Poisson(Schnorr, Gallager, Robinson, Boumal)-Möbius(Andronov-Aranson-GrinesPochinka, Gurevich)-Duffin(Bott-Smale)-Kronecker(Rokhlin- Gromov)-Neumann (Ghrist, Hilbert-Moser-Mehrmann, Sloane, Nebe, Hopf-Stiefel-Gander, Gunning-Hamilton, Zagier, Katz, Sturmfels-Siegal) schools which combine: Margulis-Gromov's dynamic theory of Hyperbolic group, Lattice/Ricci-Flow dynamic, Computation Theory: limitation of quantum and other models for solving hidden subgroup problem, Geometric integration theory, methods to singularities of differentiable maps, Information Theory, Non-linear optimization was shown that the main problem of Data-Driven and Model-Driven signal processing approaches is to preserve the topological properties of the input data: compensate for the space's curvature using the correct geometry, choose a suitable bundle with the trade-off between error and complexity, make prior on weight (regularization), solve the problem of nonlinear optimization.