Sahani Pathiraja

Francesca Romana Crucinio, Sahani Pathiraja

We consider the problem of sampling from a probability distribution $π$. It is well known that this can be written as an optimisation problem over the space of probability distributions in which we aim to minimise the Kullback--Leibler divergence from $π$. We consider the effect of replacing $π$ with a sequence of moving targets $(π_t)_{t\ge0}$ defined via geometric tempering on the Wasserstein and Fisher--Rao gradient flows. We show that convergence occurs exponentially in continuous time, providing novel bounds in both cases. We also consider popular time discretisations and explore their convergence properties. We show that in the Fisher--Rao case, replacing the target distribution with a geometric mixture of initial and target distribution never leads to a convergence speed up both in continuous time and in discrete time. Finally, we explore the gradient flow structure of tempered dynamics and derive novel adaptive tempering schedules.

Francesca Romana Crucinio, Sahani Pathiraja

Wasserstein-Fisher-Rao (WFR) gradient flows have been recently proposed as a powerful sampling tool that combines the advantages of pure Wasserstein (W) and pure Fisher-Rao (FR) gradient flows. Existing algorithmic developments implicitly make use of operator splitting techniques to numerically approximate the WFR partial differential equation, whereby the W flow is evaluated over a given step size and then the FR flow (or vice versa). This works investigates the impact of the order in which the W and FR operator are evaluated and aims to provide a quantitative analysis. Somewhat surprisingly, we show that with a judicious choice of step size and operator ordering, the split scheme can converge to the target distribution faster than the exact WFR flow (in terms of model time). We obtain variational formulae describing the evolution over one time step of both splitting schemes and investigate in which settings the W-FR split should be preferred to the FR-W split. As a step towards this goal we show that the WFR gradient flow preserves log-concavity and obtain the first sharp decay bound for WFR flow.

Soumick Chatterjee, Franziska Gaidzik, Alessandro Sciarra et al.

In the domain of medical imaging, many supervised learning based methods for segmentation face several challenges such as high variability in annotations from multiple experts, paucity of labelled data and class imbalanced datasets. These issues may result in segmentations that lack the requisite precision for clinical analysis and can be misleadingly overconfident without associated uncertainty quantification. This work proposes the PULASki method as a computationally efficient generative tool for biomedical image segmentation that accurately captures variability in expert annotations, even in small datasets. This approach makes use of an improved loss function based on statistical distances in a conditional variational autoencoder structure (Probabilistic UNet), which improves learning of the conditional decoder compared to the standard cross-entropy particularly in class imbalanced problems. The proposed method was analysed for two structurally different segmentation tasks (intracranial vessel and multiple sclerosis (MS) lesion) and compare our results to four well-established baselines in terms of quantitative metrics and qualitative output. These experiments involve class-imbalanced datasets characterised by challenging features, including suboptimal signal-to-noise ratios and high ambiguity. Empirical results demonstrate the PULASKi method outperforms all baselines at the 5\% significance level. Our experiments are also of the first to present a comparative study of the computationally feasible segmentation of complex geometries using 3D patches and the traditional use of 2D slices. The generated segmentations are shown to be much more anatomically plausible than in the 2D case, particularly for the vessel task.