Previous talks at the SCCS Colloquium

Michael Plainer: Transport of Discontinuous Densities with Normalizing Flows

SCCS Colloquium |


Most scientific equipment is unable to fully observe all involved variables. Either because they are (inherently) obscured or due to noise introduced by the sensors themselves. Finding the relation between the input and observed targets is a crucial part to fully understand the underlying process. In the case where observations can be recorded in the form of densities, a transport to another space can reveal such a relation. Classical transportation algorithms— such as those solving the optimal transport problem—can be used when the dimension of the source and target density match. But those are not suitable to transport a density from a lower dimension into a higher dimensional space. Especially in scenarios where only part of the variables can be recorded (i.e., observing a marginal density) and the underlying manifold is higher dimensional (i.e., following a joint density) such an algorithm is needed. In this paper, we will rely on previous research suggesting that by collecting time delayed measurements, the underlying smooth manifold can be reconstructed, revealing the “true” relation. Finding a correct dimensional embedding of this manifold allows the transport into the correct—possibly higher—dimension, even with classical algorithms. We will fully parameterize the procedure and use neural spline flows for the concrete transport. Most notably, we will demonstrate how this procedure can be used to reconstruct the movement of bacteria on a simulated cell, by transporting sampled one-dimensional positions to 2D. We will extend the approach and illustrate how we can construct jointly smooth functions on the time delayed measurements. Essentially, by combining observations from different sensors the underlying shared space can be reconstructed (up to diffeomorphism) even when the process involves non-invertible functions.

Guided research presentation. Michael is advised by Dr. Felix Dietrich.