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.

Paper doi abstract bibtex

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.

@InProceedings{8553436, author = {F. P. Carli and A. C. Carli}, booktitle = {2018 26th European Signal Processing Conference (EUSIPCO)}, title = {A Factorization Approach to Smoothing of Hidden Reciprocal Models}, year = {2018}, pages = {1122-1126}, abstract = {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.}, keywords = {computational complexity;Gaussian processes;iterative methods;linear systems;Markov processes;matrix algebra;signal processing;sparse matrices;sparse matrices;stationary reciprocal processes;coefficient matrix;circulant tridiagonal;circulant lower bidiagonal matrix;circulant upper bidiagonal matrix;factorization approach;acausal signals;Markov processes;conditional independence relations;rich sparsity structure;smoothing problem;linear system;cyclic block tridiagonal matrix;factorization techniques;Gaussian hidden reciprocal models;positive definite block tridiagonal matrices;Gaussian RP;Signal processing algorithms;Mathematical model;Smoothing methods;Hidden Markov models;Symmetric matrices;Signal processing;Markov processes;Markov processes;acausal models;reciprocal processes;hidden Markov models;inference and learning;signal processing}, doi = {10.23919/EUSIPCO.2018.8553436}, issn = {2076-1465}, month = {Sep.}, url = {https://www.eurasip.org/proceedings/eusipco/eusipco2018/papers/1570439452.pdf}, }

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