Semi-discrete optimal transport methods for the semi-geostrophic equations

We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

Minimum entropy production, detailed balance and Wasserstein distance for continuous-time Markov processes

We investigate the problem of minimizing the entropy production for a physical process that can be described in terms of a Markov jump dynamics. We show that, without any further constraints, a given time-evolution may be realized at arbitrarily small entropy production, yet at the expense of diverging activity. For a fixed activity, we find that the dynamics that minimizes the entropy production is given in terms of conservative forces. The value of the minimum entropy production is expressed in terms of the graph-distance based Wasserstein distance between the initial and final configuration. This yields a new kind of speed limit relating dissipation, the average number of transitions and the Wasserstein distance. It also allows us to formulate the optimal transport problem on a graph in term of a continuous-time interpolating dynamics, in complete analogy to the continuous space setting. We demonstrate our findings for simple state networks, a time-dependent pump and for spin flips in the Ising model.

Worst-Case Higher Moment Coherent Risk Based on Optimal Transport with Application to Distributionally Robust Portfolio Optimization

The tail risk management is of great significance in the investment process. As an extension of the asymmetric tail risk measure—Conditional Value at Risk (CVaR), higher moment coherent risk (HMCR) is compatible with the higher moment information (skewness and kurtosis) of probability distribution of the asset returns as well as capturing distributional asymmetry. In order to overcome the difficulties arising from the asymmetry and ambiguity of the underlying distribution, we propose the Wasserstein distributionally robust mean-HMCR portfolio optimization model based on the kernel smoothing method and optimal transport, where the ambiguity set is defined as a Wasserstein “ball” around the empirical distribution in the weighted kernel density estimation (KDE) distribution function family. Leveraging Fenchel’s duality theory, we obtain the computationally tractable DCP (difference-of-convex programming) reformulations and show that the ambiguity version preserves the asymmetry of the HMCR measure. Primary empirical test results for portfolio selection demonstrate the efficiency of the proposed model.

Optimal transport- and kernel-based early detection of mild cognitive impairment patients based on magnetic resonance and positron emission tomography images

Background To help clinicians provide timely treatment and delay disease progression, it is crucial to identify dementia patients during the mild cognitive impairment (MCI) stage and stratify these MCI patients into early and late MCI stages before they progress to Alzheimer’s disease (AD). In the process of diagnosing MCI and AD in living patients, brain scans are collected using neuroimaging technologies such as computed tomography (CT), magnetic resonance imaging (MRI), or positron emission tomography (PET). These brain scans measure the volume and molecular activity within the brain resulting in a very promising avenue to diagnose patients early in a minimally invasive manner. Methods We have developed an optimal transport based transfer learning model to discriminate between early and late MCI. Combing this transfer learning model with bootstrap aggregation strategy, we overcome the overfitting problem and improve model stability and prediction accuracy. Results With the transfer learning methods that we have developed, we outperform the current state of the art MCI stage classification frameworks and show that it is crucial to leverage Alzheimer’s disease and normal control subjects to accurately predict early and late stage cognitive impairment. Conclusions Our method is the current state of the art based on benchmark comparisons. This method is a necessary technological stepping stone to widespread clinical usage of MRI-based early detection of AD.

Formulation and properties of a divergence used to compare probability measures without absolute continuity

This paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and exploit a representation as an infimum convolution of optimal transport cost and relative entropy. Also included are examples of computation and approximation of the divergence, and the demonstration of properties that are useful when one quantifies model uncertainty.