A Factorization Approach to Smoothing of Hidden Reciprocal Models. Carli, F. P. & Carli, A. C. In 2018 26th European Signal Processing Conference (EUSIPCO), pages 1122-1126, Sep., 2018.
A Factorization Approach to Smoothing of Hidden Reciprocal Models [pdf]Paper  doi  abstract   bibtex   
Acausal signals are ubiquitous in science and engineering. These processes are usually indexed by space, instead of time. Similarly to Markov processes, reciprocal processes (RPs) are defined in terms of conditional independence relations, which imply a rich sparsity structure for this class of models. In particular, the smoothing problem for Gaussian RPs can be traced back to the problem of solving a linear system with a cyclic block tridiagonal matrix as coefficient matrix. In this paper we propose two factorization techniques for the solution of the smoothing problem for Gaussian hidden reciprocal models (HRMs). The first method relies on a clever split of the problem in two subsystems where the matrices to be inverted are positive definite block tridiagonal matrices. We can thus rely on the rich literature for this kind of sparse matrices to devise an iterative procedure for the solution of the problem. The second approach, applies to scalar valued stationary reciprocal processes, in which case the coefficient matrix becomes circulant tridiagonal (and symmetric), and is based on the direct factorization of the coefficient matrix into the product of a circulant lower bidiagonal matrix and a circulant upper bidiagonal matrix. The computational complexity of both algorithms scales linearly with the length of the observation interval.

Downloads: 0