Polynomial Residual Networks are Generically Identifiable

Isabel Dahlgren, Giovanni Luca Marchetti, Kathlén Kohn

We prove identifiability results for constant-width residual networks with polynomial activations and bias terms. For activations of degree at least five with generic coefficients, the fiber over a generic parameter is a singleton. For activations of degree at least three with no linear term after a suitable translation of the input, the generic fiber is given by translating biases and then scaling weights and translated biases by real roots of unity. These results specialize to networks without bias terms. With the same proof technique, we further derive necessary conditions for critical points of the ResNet parametrization, that are central in singular learning theory.