Controllability on landmark manifolds for shapes and neural ODEs

Abstract Landmark manifolds consist of finite collections of distinct points in an underlying space, referred to as landmark configurations. The dynamics of these configurations can be used to represent flows, such as solutions to ODEs or shape deformations. In this work, we consider landmark configurations in Euclidean space and study how they can be connected via flows of vector fields. For dimensions greater than or equal to two, we explicitly construct two vector fields whose flows can connect any pair of landmark configurations with the same cardinality. This property is known as exact universal interpolation. In dimension one, we show that the same result holds for pairs of landmark configurations that share the same relative ordering. In all dimensions, controllability is achieved using one constant vector field and one polynomial vector field of degree three.