Optimal transport and applications
Abstract:
Optimal transport is a mathematical discipline with numerous interesting applications. While traditionally it has found applications in civil engineering (e.g. optimal mass transport) and economics (e.g. traveling salesman problems), optimal transport modeling techniques have recently begin being widely adopted in numerous applications including: signal/image processing, machine learning, control, data science and others. This course will introduce students to the mathematics of optimal transport and its modern applications in signal/image processing and machine learning. Specifically, we will describe both the Monge & Kantorovich formulations for the optimal transport problem, solution methods, Wasserstein distances and geometry. We will also describe the concept of transport embeddings, representations, and transforms using linearized optimal transport. Finally we will describe applications of this theory to problems related to data classification, signal estimation and image modeling. Results using real data, together with software, will be demonstrated.
Meetings: Tuesdays, 4-6:20pm, MR 5 1041
Instructors: Gustavo K. Rohde, Ivan V. Medri
Content
1. Optimal transport theory:
o measures, transport maps/plans, push forward, Monge and Kantorovich optimal transport, static & dynamic formulations, Wasserstein distances, geometry, sliced Wasserstein distances
2. Numerical methods:
o Sorting, linear programming solutions, entropic regularization, PDE solutions
3. Transport representations/embeddings:
o Mathematical transforms, linearized optimal transport (LOT), cumulative distribution transform (CDT), signed CDT, Radon CDT, extensions
4. Applications:
o Detection, estimation, classification, inverse problems, transport-based morphometry, system identification, image and signal processing, machine learning