The project addresses important problems having to do with inferring the position of one or several objects in space. Such problems are relevant to a wide range of important applications, such as in robotics (e.g., autonomous vehicles, self-driving cars), geospatial information systems or GIS (e.g., sensor network localization), and more. As is often the case, the literature on these topics is overwhelmingly application-specific. Contributing theory and more principled methodology can help build a foundation that enables the research community to move forward with a common, more rigorous (mathematical) language and a deeper understanding of the problem itself and its ramifications. A particular focus will be on understanding how robust state-of-the-art methods are to gross errors in the input (ubiquitous in applications) and, for some tasks, on developing new methods that are more robust than existing ones.
The problem of graph embedding is a fundamental problem at the crossroads of a number of research areas including multivariate analysis and visualization in statistics, machine learning, metric geometry, robotics, and more. The literature on the topic is vast, and yet lacks crucial elements of theory, in particular in regards to the stability to noise and robustness to outliers. Although some robust methods are available, there is potential for improvement by leveraging existing research on the problem of robust matrix completion. At an even more fundamental level, there is the problem of characterizing which graphs are uniquely embeddable. This is studied in the rigidity theory literature. The problem is largely solved and reduces to the existence of a certain matrix, called stress matrix, that includes the coordinate vectors defined by an embedding of the graph (when one exists) in its null space, and has maximum rank with that property. However, this condition is qualitative, rather than quantitative, in that the rank is numerically unstable. This calls for the development of a quantitative theory of rigidity, as well as new methodology with performance guarantees.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
