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Register a new userApplication of Kahler manifold to signal processing and Bayesian inference
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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 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.
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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.
In this paper, we consider a natural map from the Kahler cone to the balanced cone of a Kahler manifold. We study its injectivity and surjecticity. We also give an analytic characterization theorem on a nef class being Kahler.
We generalize Kahler information manifolds of complex-valued signal processing filters by introducing weighted Hardy spaces and generic composite functions of transfer functions. We prove that the Riemannian geometry induced from weighted Hardy norms for composite functions of its transfer function is the Kahler manifold. Additionally, the Kahler potential of the linear system geometry corresponds to the square of the weighted Hardy norms for composite functions of its transfer function. By using the properties of Kahler manifolds, it is possible to compute various geometric objects on the manifolds from arbitrary weight vectors in much simpler ways. Additionally, Kahler information manifolds of signal filters in weighted Hardy spaces can generate various information manifolds such as Kahlerian information geometries from the unweighted complex cepstrum or the unweighted power cepstrum, the geometry of the weighted stationarity filters, and mutual information geometry under the unified framework. We also cover several examples from time series models of which metric tensor, Levi-Civita connection, and Kahler potentials are represented with polylogarithm of poles and zeros from the transfer functions when the weight vectors are in terms of polynomials.
For a compact Lie group G we define a regularized version of the Dolbeault cohomology of a G-equivariant holomorphic vector bundles over non-compact Kahler manifolds. The new cohomology is infinite-dimensional, but as a representation of G it decomposes into a sum of irreducible components, each of which appears in it with finite multiplicity. Thus equivariant Betti numbers are well defined. We study various properties of the new cohomology and prove that it satisfies a Kodaira-type vanishing theorem.
Many inference problems, notably the stochastic block model (SBM) that generates a random graph with a hidden community structure, undergo phase transitions as a function of the signal-to-noise ratio, and can exhibit hard phases in which optimal inference is information-theoretically possible but computationally challenging. In this paper we refine this description by emphasizing the existence of more generic phase diagrams with a hybrid-hard phase in which it is computationally easy to reach a non-trivial inference accuracy, but computationally hard to match the information theoretically optimal one. We support this discussion by quantitative expansions of the functional cavity equations that describe inference problems on sparse graphs. These expansions shed light on the existence of hybrid-hard phases, for a large class of planted constraint satisfaction problems, and on the question of the tightness of the Kesten-Stigum (KS) bound for the associated tree reconstruction problem. Our results show that the instability of the trivial fixed point is not a generic evidence for the Bayes-optimality of the message passing algorithms. We clarify in particular the status of the symmetric SBM with 4 communities and of the tree reconstruction of the associated Potts model: in the assortative (ferromagnetic) case the KS bound is always tight, whereas in the disassortative (antiferromagnetic) case we exhibit an explicit criterion involving the degree distribution that separates a large degree regime where the KS bound is tight and a low degree regime where it is not. We also investigate the SBM with 2 communities of different sizes, a.k.a. the asymmetric Ising model, and describe quantitatively its computational gap as a function of its asymmetry, and a version of the SBM with 2 groups of communities. We complement this study with numerical simulations of the Belief Propagation algorithm.
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