arXiv subjects

Andrew P. Mullhaupt

Publications and source records attributed to Andrew P. Mullhaupt.

5 recordsLinked to original sources

arXiv2018

Banded Matrix Fraction Representation of Triangular Input Normal Pairs

An input pair $(A,B)$ is triangular input normal if and only if $A$ is triangular and $AA^* + BB^* = I_n$, where $I_n$ is theidentity matrix. Input normal pairs generate an orthonormal basis for the impulse response. Every input pair may be transformed to a triangular input normal pair. A new system representation is given: $(A,B)$ is triangular normal and $A$ is a matrix fraction, $A=M^{-1}N$, where $M$ and $N$ are triangular matrices of low bandwidth. For single input pairs, $M$ and $N$ are bidiagonal and an explicit parameterization is given in terms of the eigenvalues of $A$. This band fraction structure allows for fast updates of state space systems and fast system identification. When A has only real eigenvalues, one state advance requires $3n$ multiplications for the single input case.

stat.ME↗

arXiv2018

Fast Adaptive Identification of Stable Innovation Filters

The adaptive identification of the impulse response of an innovation filter is considered. The impulse response is a finite sum of known basis functions with unknown coefficients. These unknown coefficients are estimated using a pseudolinear regression. This estimate is implemented using a square root algorithm based on a displacement rank structure. When the initial conditions have low displacement rank, the filter update is $O(n)$. If the filter architecture is chosen to be triangular input balanced, the estimation problem is well-conditioned and a simple, low rank initialization is available.

stat.ME↗

arXiv2015

Kählerian information geometry for signal processing

We prove the correspondence between the information geometry of a signal filter and a Kähler manifold. The information geometry of a minimum-phase linear system with a finite complex cepstrum norm is a Kähler manifold. The square of the complex cepstrum norm of the signal filter corresponds to the Kähler potential. The Hermitian structure of the Kähler manifold is explicitly emergent if and only if the impulse response function of the highest degree in $z$ is constant in model parameters. The Kählerian information geometry takes advantage of more efficient calculation steps for the metric tensor and the Ricci tensor. Moreover, $α$-generalization on the geometric tensors is linear in $α$. It is also robust to find Bayesian predictive priors, such as superharmonic priors, because Laplace-Beltrami operators on Kähler manifolds are in much simpler forms than those of the non-Kähler manifolds. Several time series models are studied in the Kählerian information geometry.

math.DG↗

arXiv2015

Geometric shrinkage priors for Kählerian signal filters

We construct geometric shrinkage priors for Kählerian signal filters. Based on the characteristics of Kähler manifolds, an efficient and robust algorithm for finding superharmonic priors which outperform the Jeffreys prior is introduced. Several ansätze for the Bayesian predictive priors are also suggested. In particular, the ansätze related to Kähler potential are geometrically intrinsic priors to the information manifold of which the geometry is derived from the potential. The implication of the algorithm to time series models is also provided.

math.ST↗

arXiv2015

Application of Kähler manifold to signal processing and Bayesian inference

We review the information geometry of linear systems and its application to Bayesian inference, and the simplification available in the Kähler manifold case. We find conditions for the information geometry of linear systems to be Kähler, and the relation of the Kähler potential to information geometric quantities such as $α$-divergence, information distance and the dual $α$-connection structure. The Kähler structure simplifies the calculation of the metric tensor, connection, Ricci tensor and scalar curvature, and the $α$-generalization of the geometric objects. The Laplace--Beltrami operator is also simplified in the Kähler geometry. One of the goals in information geometry is the construction of Bayesian priors outperforming the Jeffreys prior, which we use to demonstrate the utility of the Kähler structure.

math.DG↗