Optimal transport at finite temperature

Cohesive urban bicycle infrastructure design through optimal transport routing in multilayer networks

Bicycle infrastructure networks must meet the needs of cyclists to position cycling as a viable transportation choice in cities. In particular, protected infrastructure should be planned cohesively for the whole city and spacious enough to accommodate all cyclists safely and prevent cyclist congestion-a common problem in cycling cities like Copenhagen. Here, we devise an adaptive method for optimal bicycle network design and for evaluating congestion criticalities on bicycle paths. The method goes beyond static network measures, using computationally efficient adaptation rules inspired by optimal transport on the dynamically updating multilayer network of roads and protected bicycle lanes. Street capacities and cyclist flows reciprocally control each other to optimally accommodate cyclists on streets with one control parameter that dictates the preference of bicycle infrastructure over roads. Applying our method to Copenhagen confirms that the city's bicycle network is generally well-developed. However, we are able to identify the network's bottlenecks, and we find, at a finer scale, disparities in network accessibility and criticalities between different neighbourhoods. Our model and results are generalizable beyond this particular case study to serve as a scalable and versatile tool for aiding urban planners in designing cycling-friendly cities.

Physicist's view on the unbalanced k-cardinality assignment problem

The k-cardinality unbalanced assignment problem asks for assigning k "agents" to k "tasks" on a one-to-one basis, while minimizing the total cost associated with the assignment, with the total number of agents N and the total number of tasks M possibly different and larger than k. While many exact algorithms have been proposed to find such an optimal assignment, these methods are computationally prohibitive when the problem is large. We propose an approach to solving the k-cardinality assignment problem using techniques adapted from statistical physics. This paper provides a full description of this formalism, including all the proofs of its main claims. We derive a strongly concave free-energy function that captures the constraints of the k-assignment problem at a finite temperature. We prove that this free energy decreases monotonically as a function of β, the inverse of temperature, to the optimal assignment cost, providing a robust framework for temperature annealing. We also prove that for large enough β values the exact solution to the k-assignment problem can be derived using simple round-off to the nearest integer of the elements of the computed assignment matrix. We show that this framework can be adapted to handle degenerate k-assignment problems. We describe a computer implementation of our framework that is optimized for the GPU parallel architecture, using the library CUDA. This implementation is found to be as efficient as state-of-the-art implementations of parallel Hungarian algorithms on generic assignment problems, and orders of magnitude faster than those algorithms for pathological assignment cases.

Fast computation of exact solutions of generic and degenerate assignment problems

The linear assignment problem is a fundamental problem in combinatorial optimization with a wide range of applications, from operational research to data science. It consists of assigning "agents" to "tasks" on a one-to-one basis, while minimizing the total cost associated with the assignment. While many exact algorithms have been developed to identify such an optimal assignment, most of these methods are computationally prohibitive for large size problems. In this paper, we propose an alternative approach to solving the assignment problem using techniques adapted from statistical physics. Our first contribution is to fully describe this formalism, including all the proofs of its main claims. In particular we derive a strongly concave effective free-energy function that captures the constraints of the assignment problem at a finite temperature. We prove that this free energy decreases monotonically as a function of β, the inverse of temperature, to the optimal assignment cost, providing a robust framework for temperature annealing. We prove also that for large enough β values the exact solution to the generic assignment problem can be derived using simple roundoff to the nearest integer of the elements of the computed assignment matrix. Our second contribution is to derive a provably convergent method to handle degenerate assignment problems, with a characterization of those problems. We describe computer implementations of our framework that are optimized for parallel architectures, one based on CPU, the other based on GPU. We show that the latter enables solving large assignment problems (of the orders of a few 10 000s) in computing clock times of the orders of minutes.

Physics approach to the variable-mass optimal-transport problem

Optimal transport (OT) has become a discipline by itself that offers solutions to a wide range of theoretical problems in probability and mathematics with applications in several applied fields such as imaging sciences, machine learning, and in data sciences in general. The traditional OT problem suffers from a severe limitation: its balance condition imposes that the two distributions to be compared be normalized and have the same total mass. However, it is important for many applications to be able to relax this constraint and allow for mass creation and/or destruction. This is true, for example, in all problems requiring partial matching. In this paper, we propose an approach to solving a generalized version of the OT problem, which we refer to as the discrete variable-mass optimal-transport (VMOT) problem, using techniques adapted from statistical physics. Our first contribution is to fully describe this formalism, including all the proofs of its main claims. In particular, we derive a strongly concave effective free-energy function that captures the constraints of the VMOT problem at a finite temperature. From its maximum we derive a weak distance (i.e., a divergence) between possibly unbalanced distribution functions. The temperature-dependent OT distance decreases monotonically to the standard variable-mass OT distance, providing a robust framework for temperature annealing. Our second contribution is to show that the implementation of this formalism has the same properties as the regularized OT algorithms in time complexity, making it a competitive approach to solving the VMOT problem. We illustrate applications of the framework to the problem of partial two- and three-dimensional shape-matching problems.

Kinetic Correlation Functionals from the Entropic Regularization of the Strictly Correlated Electrons Problem

In this work, we study the entropic regularization of the strictly correlated electrons formalism, discussing the implications for density functional theory and establishing a link with earlier works on quantum kinetic energy and classical entropy. We carry out a very preliminary investigation (using simplified models) on the use of the solution of the entropic regularized problem to build approximations for the kinetic correlation functional at large coupling strengths. We also analyze lower and upper bounds to the Hohenberg-Kohn functional using the entropic regularized strictly correlated electrons problem.

Statistical Physics Approach to the Optimal Transport Problem

Originally defined for the optimal allocation of resources, optimal transport (OT) has found many theoretical and practical applications in multiple domains of science and physics. In this Letter we develop a new method for solving the discrete version of this problem using techniques derived from statistical physics. We derive a strongly concave free energy function that captures the constraints of the OT problem at a finite temperature. Its maximum defines an optimal transport plan, or registration between the two discrete probability measures that are compared, as well as a pseudodistance between those measures that satisfies the triangular inequalities. The computation of this pseudodistance is fast and numerically stable. The temperature dependent OT pseudodistance is shown to decrease monotonically with respect to the inverse of the temperature and to converge to the standard OT distance at zero temperature, providing a robust framework for temperature annealing. We illustrate applications of this framework to the problem of image comparison.