Abstract We study the global convergence of a Fisher–Rao policy gradient flow for infinite-horizon entropy-regularised Markov decision processes with Polish state and action spaces. The flow is a continuous-time analogue of a policy mirror descent method. We establish the global well-posedness of the gradient flow and demonstrate its exponential convergence to the optimal policy. Moreover, we prove the flow is stable with respect to gradient evaluation, offering insights into the performance of a natural policy gradient flow with log-linear policy parameterisation. To overcome challenges stemming from the lack of the convexity of the objective function and the discontinuity arising from the entropy regulariser, we leverage the performance difference lemma and the duality relationship between the gradient and mirror descent flows. Our analysis provides a theoretical foundation for developing various discrete policy gradient algorithms.

Abstract We give a conjectural characterisation of the stable reduction of plane quartics over local fields in terms of their Cayley octads. This results in p-adic criteria that efficiently give the stable reduction type amongst the 42 possible types, and whether the reduction is hyperelliptic or not. These criteria are in the vein of the machinery of “cluster pictures” for hyperelliptic curves. We also construct explicit families of quartic curves that realise all possible stable types, against which we test these criteria. We give numerical examples that illustrate how to use these criteria in practice.

Abstract We develop Conley’s theory for multivalued maps on finite topological spaces. More precisely, for discrete-time dynamical systems generated by the iteration of a multivalued map which satisfies appropriate regularity conditions, we establish the notions of isolated invariant sets and index pairs, and use them to introduce a well-defined Conley index. In addition, we verify some of its fundamental properties such as the Ważewski property and continuation.

Given a real inner product space V and a group G of linear isometries, we construct a family of G-invariant real-valued functions on V that we call max filters. In the case where V=Rd and G is finite, a suitable max filter bank separates orbits, and is even bilipschitz in the quotient metric. In the case where V=L2(Rd) and G is the group of translation operators, a max filter exhibits stability to diffeomorphic distortion like that of the scattering transform introduced by Mallat. We establish that max filters are well suited for various classification tasks, both in theory and in practice.

We present a sheaf-theoretic construction of shape space—the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transforms (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to “glue” PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision.