Influence of the geometry of the feature space on curiosity based exploration

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. This collection of transformations makes 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 acting on the space, 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 (mutual information), a quantity known in active inference for driving an agent's curiosity about its environment, and the impact on 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.