Harmonic Exponential Families on Manifolds
نویسندگان
چکیده
In a range of fields including the geosciences, molecular biology, robotics and computer vision, one encounters problems that involve random variables on manifolds. Currently, there is a lack of flexible probabilistic models on manifolds that are fast and easy to train. We define an extremely flexible class of exponential family distributions on manifolds such as the torus, sphere, and rotation groups, and show that for these distributions the gradient of the log-likelihood can be computed efficiently using a non-commutative generalization of the Fast Fourier Transform (FFT). We discuss applications to Bayesian camera motion estimation (where harmonic exponential families serve as conjugate priors), and modelling of the spatial distribution of earthquakes on the surface of the earth. Our experimental results show that harmonic densities yield a significantly higher likelihood than the best competing method, while being orders of magnitude faster to train.
منابع مشابه
Pattern Learning and Recognition on Statistical Manifolds: An Information-Geometric Review
We review the information-geometric framework for statistical pattern recognition: First, we explain the role of statistical similarity measures and distances in fundamental statistical pattern recognition problems. We then concisely review the main statistical distances and report a novel versatile family of divergences. Depending on their intrinsic complexity, the statistical patterns are lea...
متن کامل3 Flat manifolds , harmonic spinors , and eta invariants
The aim of this paper is to calculate the eta invariants and the dimensions of the spaces of harmonic spinors for two infinite families of closed flat manifolds. The first one F CHD consists of some flat manifolds M with cyclic holonomy groups. The second one F HW is the family of generalized oriented Hantzsche-Wendt manifolds. If M ∈ F HW , and M admits a spin structure, then η(M) = 0 and h(M)...
متن کاملO ct 2 00 3 Flat manifolds , harmonic spinors , and eta invariants
The aim of this paper is to calculate the eta invariants and the dimensions of the spaces of harmonic spinors for two infinite families of closed flat manifolds. The first one F CHD consists of some flat manifolds M with cyclic holonomy groups. The second one F HW is the family of generalized oriented Hantzsche-Wendt manifolds. If M ∈ F HW , and M admits a spin structure, then η(M) = 0 and h(M)...
متن کاملStrati ed Exponential Families: Graphical Models and Model Selection
We provide a classi cation of graphical models according to their representation as exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical (DAG) models and chain graphs with no hidden variables, including DAG models with several families of local distributions, are curved exponential families (CEFs) and graph...
متن کامل