Implicit Hypersurface Approximation Capacity in Deep ReLU Networks

We develop a geometric approximation theory for deep feed-forward neural networks with ReLU activations. Given a d-dimensional hypersurface in R^d+1 represented as the graph of a C²-function, we show that a deep fully-connected ReLU network of width d+1 can implicitly construct an approximation as its zero contour with a precision bound depending on the number of layers. This result is directly applicable to the binary classification setting where the sign of the network is trained as a classifier, with the network's zero contour as a decision boundary. Our proof is constructive and relies on the geometrical structure of ReLU layers provided in doi: 10. 48550/arXiv. 2310. 03482. Inspired by this geometrical description, we define a new equivalent network architecture that is easier to interpret geometrically, where the action of each hidden layer is a projection onto a polyhedral cone derived from the layer's parameters. By repeatedly adding such layers, with parameters chosen such that we project small parts of the graph of from the outside in, we, in a controlled way, construct a network that implicitly approximates the graph over a ball of radius R. The accuracy of this construction is controlled by a discretization parameter and we show that the tolerance in the resulting error bound scales as (d-1) R^3/2^1/2 and the required number of layers is of order d (32R) ^d+1{2}.