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Register a new userKahlerian information geometry for signal processing
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Added by
Jaehyung Choi
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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We prove the correspondence between the information geometry of a signal filter and a Kahler manifold. The information geometry of a minimum-phase linear system with a finite complex cepstrum norm is a Kahler manifold. The square of the complex cepstrum norm of the signal filter corresponds to the Kahler potential. The Hermitian structure of the Kahler manifold is explicitly emergent if and only if the impulse response function of the highest degree in $z$ is constant in model parameters. The Kahlerian information geometry takes advantage of more efficient calculation steps for the metric tensor and the Ricci tensor. Moreover, $alpha$-generalization on the geometric tensors is linear in $alpha$. It is also robust to find Bayesian predictive priors, such as superharmonic priors, because Laplace-Beltrami operators on Kahler manifolds are in much simpler forms than those of the non-Kahler manifolds. Several time series models are studied in the Kahlerian information geometry.
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We review the information geometry of linear systems and its application to Bayesian inference, and the simplification available in the Kahler manifold case. We find conditions for the information geometry of linear systems to be Kahler, and the relation of the Kahler potential to information geometric quantities such as $alpha $-divergence, information distance and the dual $alpha $-connection structure. The Kahler structure simplifies the calculation of the metric tensor, connection, Ricci tensor and scalar curvature, and the $alpha $-generalization of the geometric objects. The Laplace--Beltrami operator is also simplified in the Kahler 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 Kahler structure.
We describe and to some extent characterize a new family of Kahler spin manifolds admitting non-trivial imaginary Kahlerian Killing spinors.
A recently introduced canonical divergence $mathcal{D}$ for a dual structure $(mathrm{g}, abla, abla^*)$ is discussed in connection to other divergence functions. Finally, open problems concerning symmetry properties are outlined.
We show that a Frobenius sturcture is equivalent to a dually flat sturcture in information geometry. We define a multiplication structure on the tangent spaces of statistical manifolds, which we call the statistical product. We also define a scalar quantity, which we call the Yukawa term. By showing two examples from statistical mechanics, first the classical ideal gas, second the quantum bosonic ideal gas, we argue that the Yukawa term quantifies information generation, which resembles how mass is generated via the 3-points interaction of two fermions and a Higgs boson (Higgs mechanism). In the classical case, The Yukawa term is identically zero, whereas in the quantum case, the Yukawa term diverges as the fugacity goes to zero, which indicates the Bose-Einstein condensation.
Matrix scaling is a classical problem with a wide range of applications. It is known that the Sinkhorn algorithm for matrix scaling is interpreted as alternating e-projections from the viewpoint of classical information geometry. Recently, a generalization of matrix scaling to completely positive maps called operator scaling has been found to appear in various fields of mathematics and computer science, and the Sinkhorn algorithm has been extended to operator scaling. In this study, the operator Sinkhorn algorithm is studied from the viewpoint of quantum information geometry through the Choi representation of completely positive maps. The operator Sinkhorn algorithm is shown to coincide with alternating e-projections with respect to the symmetric logarithmic derivative metric, which is a Riemannian metric on the space of quantum states relevant to quantum estimation theory. Other types of alternating e-projections algorithms are also provided by using different information geometric structures on the positive definite cone.
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