CPL-functions expressibility by neural circuits on ReLU bases

The paper considers the question of the expressibility of any continuous particle-linear function of several real variables in the form of a neural circuit over a basis with nonlinearities of the max type. Then the result is transferred to neural circuits built over a basis with a single non-linear RELU function. Before proving the result, several auxiliary, technical lemmas are formulated and proved, expanding the existing knowledge about the properties of particle-linear functions and equivalence classes generated by a certain set of hyperplanes. The paper also gives estimates of nonlinear complexity and depth for the constructed neural circuits in two given bases. Finally, the paper proves the equality of the class of continuous particle-linear functions, the class of functions representable by neural circuits over a basis of the first type, and the class of functions representable by neural circuits over a basis of the second type.