State estimation from high-dimensional data

It is implicit in traditional discussions of linear or nonlinear state estimation filters that there is no relation specified between the dimension of the state and the observation vector dimension. If anything though, the state would often be thought to have higher dimension. But increasingly in practice problems are arising where the reverse is the case. In this paper we show that state estimation filters, such as the Kalman filter undergo a remarkable simplification in structure and computation when the observation dimension is much larger than the state dimension. Both linear and nonlinear cases (including point processes) are discussed.