Estimate of the Neural Network Dimension Using Algebraic Topology and Lie Theory

In this paper we present an approach to determine the smallest possible number of neurons in a layer neural network such way that topology input space can be learned sufficiently well. We introduce general procedure based on persistent homology investigate topological invariants manifold which suspect data set. specify required dimensions precisely, assuming there is smooth or near are located. Furthermore, require connected and has commutative group structure mathematical sense. These assumptions allow us derive decomposition underlying whose well known. use representatives $k$-dimensional groups from persistence landscape integer dimension for decomposition. This embedding capable capturing manifold. theory validate it experimentally toy sets.