Maximizing a Tree Series in the Representation Space

This paper investigates the use of linear representations of trees (i.e. mappings from the set of trees into a finite dimensional vector space which are induced by rational series on trees) in the context of structured data learning. We argue that this representation space can be more appealing than the space of trees to handle machine learning problems involving trees. Focusing on a tree series maximization problem, we first analyze its complexity to motivate the use of approximation techniques. We then show how a tree series can be extended to the continuous representation space, we propose an adaptive Metropolis-Hastings algorithm to solve the maximization problem in this space, and we establish convergence guarantees. Finally, we provide some experiments comparing our algorithm with an implementation of the Metropolis-Hastings algorithm in the space of trees.