Action of the Euclidean versus projective group on an agent’s internal space in curiosity driven exploration

Biological Cybernetics (2025) 119:4 https://doi.org/10.1007/s00422-024-01001-1 ORIGINAL ARTICLE Action of the Euclidean versus projective group on an agent’s internal space in curiosity driven exploration Grégoire Sergeant-Perthuis1 · Nils Ruet2 · Dimitri Ognibene3,4 · Yvain Tisserand5 · Kenneth Williford6 · David Rudrauf2 Received: 30 December 2023 / Accepted: 18 October 2024 © The Author(s) 2025 Abstract According to the Projective Consciousness Model (PCM), in human spatial awareness, 3-dimensional projective geometry structures information integration and action planning through perspective taking within an internal representation space. The way different perspectives are related to and transform a world model defines a specific perception and imagination scheme. In mathematics, such a collection of transformations corresponds to a ‘group’, whose ‘actions’ characterize the geometry of a space. Imbuing world models with a group structure may capture different agents’ spatial awareness and affordance schemes. We used group action as a special class of policies for perspective-dependent control. We explored how such a geometric structure impacts agents’ behaviors, comparing how the Euclidean versus projective groups act on epistemic value in active inference, drive curiosity, and exploration. We formally demonstrate and simulate how the groups induce distinct behaviors in a simple search task. The projective group’s nonlinear magnification of information transformed epistemic value according to the choice of frame, generating behaviors of approach toward objects with uncertain locations due to limited sampling. The Euclidean group had no effect on epistemic value: no action was better than the initial idle state. In structuring a priori an agent’s internal representation, we show how geometry can play a key role in information integration and action planning. Our results add further support to the PCM. Keywords Geometric world model · Exploration · Embodied cognitive science · Cognitive modeling · Perception-action coupling 1 Introduction Communicated by Karl Friston. B Grégoire Sergeant-Perthuis 1 LCQB Sorbonne Université & OURAGAN team, Inria Paris Paris, France 2 CIAMS, Université Paris-Saclay, Orsay & Université d’Orléans, Orléans, France 3 Department of Psychology, University of Milano-Bicocca, Piazza dellÁteneo Nuovo, 1-20126, Milan, Italy 4 Institute of Cognitive Sciences and Technologies, National Research Council, Rome, Italy 5 CISA, University of Geneva, Geneva, Switzerland 6 Department of Philosophy and Humanities, The University of Texas at Arlington, Arlington, USA In artificial agent learning and control, intrinsic and extrinsic rewards can be combined to optimize the balance between exploration and exploitation. Intrinsic rewards in Reinforcement Learning (RL) (Hester and Stone 2017; Merckling et al. 2022; Oudeyer et al. 2007) or terms of epistemic value in active inference (Friston et al. 2015) have been suggested as mechanisms that might account for curiosity and exploratory drives, e.g. by integrating prediction error or uncertainty in order to drive actions favoring their reduction. However, efficient exploration is a computationally hard task. Recent neural planning models have increased planning flexibility and generality (Sekar et al. 2020). Yet, it is well known that models’ structures heavily impact planning performance and tractability (Geffner and Bonet 2013) as well as learning complexity (Goyal and Bengio 2022). A good representation of information may improve learning and search efficiency. 0123456789().: V,-vol 123 4 Page 2 of 18 These issues are particularly salient for computationheavy, highly recursive machine learning algorithms and applications, e.g., reinforcement learning (RL) in artificial agents (Bonet and Geffner 2019) or recursive modeling methods (RMM) in multi agent systems (MAS) and partially observable stochastic games (POSG) (Geffner and Bonet 2013). Although generic neural world models can support explo ration-related processes, incorporating prior knowledge that shapes internal representations to more effectively support exploration across a broad range of environments, such as 3D environments, may enable autonomous agents to explore more complex and realistic settings on a larger scale (Goyal and Bengio 2022). The exploration planning problem can thus be approached by considering how the structure of representation impacts exploration behaviors. In this article, the structure of representations is encoded into the geometry of the state space of an agent; we quantify the impact of changing this geometry on the behavior of the agent. Here, we do not consider mechanisms of representation learning, e.g., in which world dynamics and action effects need to be learned and represented, as typically done in RL. We consider control and execution in agents with an a priori, encoded by a generative model (POMDP), on the temporal evolution of their environment conditioned on their actions. In particular, an agent has an a priori on how its observations are related to the state of its environment, and on the consequences of its actions on the environment. However, the agent is uncertain about the exact state of the environment and updates its belief about this state through successive observations. We focus on action selection for environment exploration and mapping. We adopt the active inference framework, i.e., an implementation of the Bayesian Brain Hypothesis aimed at generating adaptive behaviors in agents (Friston et al. 2006), which has found applications in neuroscience (Da Costa et al. 2020) and was proposed for modeling molecular machines (Timsit and Sergeant-Perthuis 2021; Timsit et al. 2021). It relies on an internal representation of the environment that an agent is driven to explore and exploit. The agent continually updates its beliefs about plausible competing internal hypotheses about the state of its environment. Under common sensory limitations, active inference relates to optimal control (Kaelbling et al. 1998; Ognibene et al. 2019). The epistemic value of states is a quantity that arises in active inference (Friston et al. 2015). Its maximization drives the agent’s curiosity and actions. For exploration or search in 3-D space, it is warranted to consider how geometrical principles could be embedded into efficient control mechanisms in order to regularize the internal representation of information and mediate exploration under a drive of uncertainty reduction or information maximization. Geometrical considerations have previously been 123 Biological Cybernetics (2025) 119:4 integrated into a variety of optimization and machine learning approaches, such as RL, active inference, and Bayesian inference (see Related Works below), but not in the specific perspective we introduce here. We build upon the hypothesis that 3-D internal representations of space in agents performing active infe (...truncated)