# Linear Unbalanced Optimal Transport

Tuesday, January 21, 2020 - 1:25pm - 2:25pm

Lind 305

Matthew Thorpe (University of Cambridge)

Optimal transport is a powerful tool for measuring the distances between

signals. However, the most common choice is to use the Wasserstein

distance where one is required to treat the signal as a probability

measure. This places restrictive conditions on the signals and although

ad-hoc renormalisation can be applied to sets of unnormalised measures

this can often dampen features of the signal. The second disadvantage is

that despite recent advances, computing optimal transport distances for

large sets is still difficult. In this talk I will focus on the

Hellinger--Kantorovich distance, which can be applied between any pair

of non-negative measures. I will describe how the distance can be

linearised and embedded into a Euclidean space. The Euclidean distance

in the embedded space is approximately the Wasserstein distance in the

original space. This method, in particular, allows for the application

of off-the-shelf data analysis tools such as principal component

analysis.

This is joint work with Bernhard Schmitzer (TU Munich).

Matthew is a research fellow in the Cantab Capital Institute for the

Mathematics of Information at the University of Cambridge. Prior to that

he was a postdoctoral associate at Carnegie Mellon University and a PhD

student at the University of Warwick. From this coming March he will be

a lecturer (US equivalent Assistant Professor) in Applied Mathematics at

the University of Manchester.

signals. However, the most common choice is to use the Wasserstein

distance where one is required to treat the signal as a probability

measure. This places restrictive conditions on the signals and although

ad-hoc renormalisation can be applied to sets of unnormalised measures

this can often dampen features of the signal. The second disadvantage is

that despite recent advances, computing optimal transport distances for

large sets is still difficult. In this talk I will focus on the

Hellinger--Kantorovich distance, which can be applied between any pair

of non-negative measures. I will describe how the distance can be

linearised and embedded into a Euclidean space. The Euclidean distance

in the embedded space is approximately the Wasserstein distance in the

original space. This method, in particular, allows for the application

of off-the-shelf data analysis tools such as principal component

analysis.

This is joint work with Bernhard Schmitzer (TU Munich).

Matthew is a research fellow in the Cantab Capital Institute for the

Mathematics of Information at the University of Cambridge. Prior to that

he was a postdoctoral associate at Carnegie Mellon University and a PhD

student at the University of Warwick. From this coming March he will be

a lecturer (US equivalent Assistant Professor) in Applied Mathematics at

the University of Manchester.