Optimization on Lie Manifolds and Projective Tracking

2008 International Conference on Computer Science and Software Engineering ◽

10.1109/csse.2008.1241 ◽

2008 ◽

Author(s):

Li Guangwei ◽

Liu Yunpeng ◽

Yin Jian ◽

Shi Zelin

Keyword(s):

Lie Manifolds

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Jump Diffusion Inference in Complex Scenes

Pattern Theory ◽

10.1093/oso/9780198505709.003.0020 ◽

2006 ◽

Author(s):

Ulf Grenander ◽

Michael I. Miller

Keyword(s):

Markov Process ◽

Parameter Space ◽

Diffusion Processes ◽

Jump Process ◽

Jump Diffusion ◽

Jump Diffusion Processes ◽

Transition Distribution ◽

Conditional Expectations ◽

Sampling Algorithms ◽

Lie Manifolds

This chapter explores random sampling algorithms introduced in for generating conditional expectations in hypothesis spaces in which there is a mixture of discrete, disconnected subsets. Random samples are generated via the direct simulation of a Markov process whose state moves through the hypothesis space with the ergodic property that the transition distribution of the Markov process converges to the posterior distribution. This allows for the empirical generation of conditional expectations under the posterior. To accommodate the connected and disconnected nature of the state spaces, the Markov process is forced to satisfy jump–diffusion dynamics. Through the connected parts of the parameter space (Lie manifolds) the algorithm searches continuously, with sample paths corresponding to solutions of standard diffusion equations. Across the disconnected parts of parameter space the jump process determines the dynamics. The infinitesimal properties of these jump–diffusion processes are selected so that various sample statistics converge to their expectation under the posterior.

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