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.

On Non-Linear operators for Geometric Deep Learning

This work studies operators mapping vector and scalar fields defined over a manifold $\mathcal{M}$, and which commute with its group of diffeomorphisms $\text{Diff}(\mathcal{M})$. We prove that in the case of scalar fields $L^p_\omega(\mathcal{M,\mathbb{R}})$, those operators correspond to point-wise non-linearities, recovering and extending known results on $\mathbb{R}^d$. In the context of Neural Networks defined over $\mathcal{M}$, it indicates that point-wise non-linear operators are the only universal family that commutes with any group of symmetries, and justifies their systematic use in combination with dedicated linear operators commuting with specific symmetries. In the case of vector fields $L^p_\omega(\mathcal{M},T\mathcal{M})$, we show that those operators are solely the scalar multiplication. It indicates that $\text{Diff}(\mathcal{M})$ is too rich and that there is no universal class of non-linear operators to motivate the design of Neural Networks over the symmetries of $\mathcal{M}$.

Regionalized optimization

Yedidia, Freeman, Weiss have shown in their reference article, "Constructing Free Energy Approximations and Generalized Belief Propagation Algorithms", that there is a variational principle underlying the General Belief Propagation, by introducing a region-based free energy approximation of the MaxEnt free energy, that we will call the Generalized Bethe free energy. They sketched a proof that fixed points of the General Belief Propagation are critical points of this free energy, this proof was completed in the thesis of Peltre. In this paper we identify a class of optimization problems defined as patching local optimization problems and associated message passing algorithms for which such correspondence between critical points and fix points of the algorithms holds. This framework holds many applications one of which being a PCA for filtered data and a region-based approximation of MaxEnT with stochastic compatibility constraints on the region probabilities. Such approach is particularly adapted for inference with multimodal integration, inference on scenes with multiple views.