(Oxford University)

Introducing cheap function proxies for quickly producing approximate random numbers, we show convergence of modified numerical schemes, and coupling between approximation and discretisation errors. We bound the cumulative roundoff error introduced by floating-point calculations, valid for 16-bit half-precision (FP16). We combine approximate distributions and reduced-precisions into a nested simulation framework (via multilevel Monte Carlo), demonstrating performance improvements achieved without losing accuracy. These simulations predominantly perform most of their calculations in very low precisions. We will highlight the motivations and design choices appropriate for SVE and FP16 capable hardware, and present numerical results on Arm, Intel, and NVIDIA based hardware.

We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean-Vlasov jump-diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution. Based on a joint work with V. Ignazio, M. Reppen and H. M. Soner

We can see the simplest setting of persistence from a functional point of view: given a fixed finite simplicial complex, we have the barcode function which, given a filter function over this complex, returns the corresponding persistent diagram. The bottleneck distance induces a topology on the space of persistence diagrams, and makes the barcode function a continuous map: this is a consequence of the stability Theorem. In this presentation, I will present ongoing work that seeks to deepen our understanding of the analytic properties of the barcode function, in particular whether it can be said to be smooth. Namely, if we smoothly vary the filter function, do we get smooth changes in the resulting persistent diagram? I will introduce a notion of differentiability/smoothness for barcode valued maps, and then explain why the barcode function is smooth (but not everywhere) with respect to the choice of filter function. I will finally explain why these notions are of interest in practical optimisation/learning situations. 

The amount and complexity of biological data has increased rapidly in recent years with the availability of improved biological tools. When applying persistent homology to large data sets, many of the currently available algorithms however fail due to computational complexity preventing many interesting biological applications. De Silva and Carlsson (2004) introduced the so called Witness Complex that reduces computational complexity by building simplicial complexes on a small subset of landmark points selected from the original data set. The landmark points are chosen from the data either at random or using the so called maxmin algorithm. These approaches are not ideal as the random selection tends to favour dense areas of the point cloud while the maxmin algorithm often selects outliers as landmarks. Both of these problems need to be addressed in order to make the method more applicable to biological data. We study new ways of selecting landmarks from a large data set that are robust to outliers. We further examine the effects of the different subselection methods on the persistent homology of the data.

In a robust decision, we are pessimistic toward our decision making when the probability measure is unknown. In particular, we optimise our decision under the worst case scenario (e.g. via value at risk or expected shortfall).  On the other hand, most theories in reinforcement learning (e.g. UCB or epsilon-greedy algorithm) tell us to be more optimistic in order to encourage learning. These two approaches produce an apparent contradict in decision making. This raises a natural question. How should we make decisions, given they will affect our short-term outcomes, and information available in the future?

In this talk, I will discuss this phenomenon through the classical multi-armed bandit problem which is known to be solved via Gittins' index theory under the setting of risk (i.e. when the probability measure is fixed). By extending this result to an uncertainty setting, we can show that it is possible to take into account both uncertainty and learning for a future benefit at the same time. This can be done by extending a consistent nonlinear expectation  (i.e. nonlinear expectation with tower property) through multiple filtrations.

At the end of the talk, I will present numerical results which illustrate how we can control our level of exploration and exploitation in our decision based on some parameters.
 

In this talk I will introduce the concept of the path signature and motivate its recent use in analysis of time-ordered data. I will then describe a new feature map for barcodes in persistent homology by first realizing each barcode as a path in a vector space, and then computing its signature which takes values in the tensor algebra over that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness.

I will give an introduction to the asymptotic behaviour of random geometric complexes. In the specific case of a simplicial complex realised as the Cech complex of a point process sampled from a closed Riemannian manifold, we will explore conditions which guarantee the homology of the Cech complex coincides with the homology of the underlying manifold. We will see techniques which were originally developed to study random geometric graphs, which together with ideas from Morse Theory establish homological connectivity thresholds.

I will give an introduction to the theory of definable parameterization of definable sets in the o-minimal context and its application to diophantine problems. I will then go on to discuss uniformity issues with particular reference to the subanalytic case. This is joint work with Jonathan Pila and Raf Cluckers

I plan to present a brief introduction to optimal control theory (no background knowledge assumed), and discuss a fascinating and oft-forgotten family of problems where the optimal control behaves very strangely; it changes state infinitely often in finite time. This causes havoc in practice, and even more so in the literature.