Invariance properties of the natural gradient in overparametrised systems

The natural gradient field is a vector field that lives on a model equipped with a distinguished Riemannian metric, e.g. the Fisher–Rao metric, and represents the direction of steepest ascent of an objective function on the model with respect to this metric. In practice, one tries to obtain the corresponding direction on the parameter space by multiplying the ordinary gradient by the inverse of the Gram matrix associated with the metric. We refer to this vector on the parameter space as the natural parameter gradient. In this paper we study when the push forward of the natural parameter gradient is equal to the natural gradient. Furthermore we investigate the invariance properties of the natural parameter gradient. Both questions are addressed in an over parametrised setting.