83 research outputs found
Harmonic Exponential Families on Manifolds
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.Comment: fixed typ
Fast NFFT based summation of radial functions
This paper is concerned with the fast summation of radial functions by the fast Fourier transform for nonequispaced data. We enhance the fast summation algorithm proposed in [20] by introducing a new regularization procedure based on the two-point Taylor interpolation by algebraic polynomials and estimate the corresponding approximation error. Our error estimates are more sophisticated than those in [20]. Beyond the kernels Kβ(χ
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