Product Manifold Machine Learning for Physics

Particle jets are collimated flows of partons which evolve into tree-like structures through stochastic parton showering and hadronization. The hierarchical nature of particle jets aligns naturally with hyperbolic space, a non-Euclidean geometry that captures hierarchy intrinsically. To leverage the benefits of non-Euclidean geometries, we develop jet analysis in product manifold ($\mathcal P \mathcal M$) spaces, Cartesian products of constant curvature Riemannian manifolds. We consider particle representations as configurable parameters and compare the performance of $\mathcal P \mathcal M$ neural network models across several possible representations. We find product manifold representations perform equal or better than fully Euclidean models of the same latent dimension and the same approximate number of parameters. These findings reinforce the view of optimizing geometric representations as a key parameter in maximizing both performance and efficiency.