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Register a new userDivergence functions in Information Geometry
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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.
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In Riemannian geometry geodesics are integral curves of the Riemannian distance gradient. We extend this classical result to the framework of Information Geometry. In particular, we prove that the rays of level-sets defined by a pseudo-distance are generated by the sum of two tangent vectors. By relying on these vectors, we propose a novel definition of a canonical divergence and its dual function. We prove that the new divergence allows to recover a given dual structure $(mathrm{g}, abla, abla^*)$ of {a dually convex set on} a smooth manifold $mathrm{M}$. Additionally, we show that this divergence coincides with the canonical divergence proposed by Ay and Amari in the case of: (a) self-duality, (b) dual flatness, (c) statistical geometric analogue of the concept of symmetric spaces in Riemannian geometry. For a dually convex set, the case (c) leads to a further comparison of the new divergence with the one introduced by Henmi and Kobayashi.
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.
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.
In this paper, we introduce directed networks called `divergence network in order to perform graphical calculation of divergence functions. By using the divergence networks, we can easily understand the geometric meaning of calculation results and grasp relations among divergence functions intuitively.
Information divergences are commonly used to measure the dissimilarity of two elements on a statistical manifold. Differentiable manifolds endowed with different divergences may possess different geometric properties, which can result in totally different performances in many practical applications. In this paper, we propose a total Bregman divergence-based matrix information geometry (TBD-MIG) detector and apply it to detect targets emerged into nonhomogeneous clutter. In particular, each sample data is assumed to be modeled as a Hermitian positive-definite (HPD) matrix and the clutter covariance matrix is estimated by the TBD mean of a set of secondary HPD matrices. We then reformulate the problem of signal detection as discriminating two points on the HPD matrix manifold. Three TBD-MIG detectors, referred to as the total square loss, the total log-determinant and the total von Neumann MIG detectors, are proposed, and they can achieve great performances due to their power of discrimination and robustness to interferences. Simulations show the advantage of the proposed TBD-MIG detectors in comparison with the geometric detector using an affine invariant Riemannian metric as well as the adaptive matched filter in nonhomogeneous clutter.
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