Modeling and understanding low- and high-dimensional data is a recurring theme in graphics, optimization, learning, and vision. Abstracting away application domains reveals common threads using geometric constructs like distances, similarities, and curvatures. This shared structure suggests the possibility of developing geometric data processing as a discipline in itself.
To this end, I will introduce optimal transportation (OT) as a versatile component of the geometric data processing toolkit. Originally proposed for minimizing the cost of shipping products from producers to consumers, OT links probability and geometry using distributions to encode geometric features and developing metric machinery to quantify their relationships.
To transition OT from theory to practice, I will show how to solve previously intractable OT problems efficiently on discretized domains and demonstrate a wide range of applications enabled by this new machinery. I will illustrate the advantages and challenges of OT for geometric data processing by outlining my recent work in geometry processing, computer graphics, and machine learning. In each case, I will consider optimization aspects of the OT problem for relevant geometric domains---including triangulated surfaces, graphs, and subsets of Euclidean space---and then show how the resulting machinery can be used to approach outstanding problems in surface correspondence, modeling, and semi-supervised learning.
Justin Solomon is a PhD candidate and teaching fellow in the Geometric Computing Group at Stanford University studying problems in shape analysis, machine learning, and graphics from a geometric perspective. His work is supported by the Hertz Foundation Fellowship, the NSF Graduate Research Fellowship, and the NDSEG Fellowship. Justin holds bachelors degrees in mathematics and computer science and an MS in computer science from Stanford. He has served as the lecturer for courses in graphics, differential geometry, and numerical methods; his forthcoming textbook entitled Numerical Algorithms focuses on applications of numerical methods across modern computer science. Before his graduate studies, Justin was a member of Pixar's Tools Research group. He is a pianist, cellist, and amateur musicologist with award-winning research on early recordings of the Elgar Cello Concerto.
WEB SITE: http://www.stanford.edu/~justso1/ (includes PDFs of all papers discussed in the talk)