From Representation to Inference: Respecting and Exploiting Mathematical Structures in Computer Vision and Machine Learning

Stochastic analysis of real-world signals consists of 3 main parts: mathematical representation; probabilistic modeling; statistical inference. For it to be effective, we need mathematically-principled and practical computational tools that take into consideration not only each of these components by itself but also their interplay. This is especially true for a large class of computer-vision and machine-learning problems that involve certain mathematical structures; the latter may be a property of the data or encoded in the representation/model to ensure mathematically-desired properties and computational tractability. For concreteness, this talk will center on structures that are geometric, hierarchical, or topological. Structures present challenges. For example, on nonlinear spaces, most statistical tools are not directly applicable, and, moreover, computations can be expensive. As another example, in mixture models, topological constraints break statistical independence. Once we overcome the difficulties, however, structures offer many benefits. For example, respecting and exploiting the structure of Riemannian manifolds and/or Lie groups yield better probabilistic models that also support consistent synthesis. The latter is crucial for the employment of analysis-by-synthesis inference methods used within, e.g., a generative Bayesian framework. Likewise, imposing a certain structure on velocity fields yields highly-expressive diffeomorphisms that are also simple and computationally tractable; particularly, this facilitates MCMC inference, traditionally viewed as too expensive in this context.