New Directions in Learning Theory
Partha Niyogi, PI
Goals: The goals of this project were to pursue
new frameworks, algorithms, analyses, and
applications of machine learning. In particular,
we chose to focus on a class of problems that
involved the interplay between current learning
theory and integration, geometry, and dynamics.
Significance: Machine learning is at the interface of statistics, computer science, and mathematics with implications for a range of scientific disciplines from cognitive science, linguistics, and biology.
Accomplishments: We made progress on a number of fronts. One direction led us to pursue algorithms for high dimensional data analysis where the data was assumed to live in a low dimensional structure such as a manifold. We showed how to provably estimate various invariants of such manifolds such as their homology, operators on them such as the Laplace operator and the heat operator, and applied these ideas to design algorithms for a range of pattern recognition tasks. Another direction led us to consider a population of interacting learning agents that have been useful as models of sociolinguistic behavior as well as flocking in animals leading to a better understanding of principles of emergent behavior. A third direction led us to investigate various theoretical guarantees on learning algorithms to better understand fundamental limits. A fourth direction led us to consider various models of acoustic pattern recognition that may provide an alternative to current speech recognition technologies.
