Action of the Euclidean versus Projective group on an agent's internal space in curiosity driven exploration: a formal analysis

In human spatial awareness, information appears to be represented according to 3-D projective geometry. It structures information integration and action planning within an internal representation space. The way different first person perspectives of an agent relate to each other, through transformations of a world model, defines a specific perception scheme for the agent. In mathematics, this collection of transformations is called a `group' and it characterizes a geometric space by acting on it. We propose that imbuing world models with a `geometric' structure, given by a group, is one way to capture different perception schemes of agents. We explore how changing the geometric structure of a world model impacts the behavior of an agent. In particular, we focus on how such geometrical operations transform the formal expression of epistemic value in active inference as driving an agent's curiosity about its environment, and impact exploration behaviors accordingly. We used group action as a special class of policies for perspective-dependent control. We compared the Euclidean versus projective groups. We formally demonstrate that the groups induce distinct behaviors. The projective group induces nonlinear contraction and dilatation that transform entropy and epistemic value as a function of the choice of frame, which fosters exploration behaviors. This contribution opens research avenues in which a geometry structures \textit{a priori} an agent's internal representation space for information integration and action planning.