Dirac Mixture Trees for Fast Suboptimal Multi-Dimensional Density Approximation

We consider the problem of approximating an arbitrary multi-dimensional probability density function by means of a Dirac mixture density. Instead of an optimal solution based on minimizing a global distance measure between the true density and its approximation, a fast suboptimal anytime procedure is proposed, which is based on sequentially partitioning the state space and component placement by local optimization. The proposed procedure adaptively covers the entire state space with a gradually increasing resolution. It can be efficiently implemented by means of a pre-allocated tree structure in a straightforward manner. The resulting computational complexity is linear in the number of components and linear in the number of dimensions. This allows a large number of components to be handled, which is especially useful in high-dimensional state spaces.