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Kähler information manifolds of signal processing filters in weighted Hardy spaces

Jaehyung Choi

TL;DR

The paper develops a unified framework in which linear systems analyzed in weighted Hardy spaces possess a Kähler information geometry, with the Kähler potential equal to the squared weighted Hardy norm of a transformed transfer function. By allowing arbitrary smooth transformations $\phi$ and weight sequences $\omega$, the authors generalize prior unweighted complex cepstrum geometry to a broad class of weighted Hardy settings, including polylogarithmic representations via Cepstrum coefficients. They derive explicit metric and Levi-Civita connection formulas from the Kähler potential, enabling efficient geometric computations. The framework recovers known geometries (e.g., unweighted cepstrum, mutual information between past and future) as special cases and provides concrete expressions for ARMA/ARFIMA models under various weights. This approach offers a flexible, analytically tractable tool for analyzing information geometry of signal processing filters with practical connections to time-series modeling and system identification.

Abstract

We extend the framework of Kähler information manifolds for complex-valued signal processing filters by introducing weighted Hardy spaces and smooth transformations of transfer functions. We demonstrate that the Riemannian geometry induced from weighted Hardy norms for the smooth transformations of its transfer function is a Kähler manifold. In this setting, the Kähler potential of the linear system geometry corresponds to the squared weighted Hardy norm of the composite transfer function. With the inherent structure of Kähler manifolds, geometric quantities on the manifold of linear systems in weighted Hardy spaces can be computed more efficiently and elegantly. Moreover, this generalized framework unifies a variety of well-known information manifolds within the structure of Kähler information manifolds for signal filters. Several illustrative examples from time series models are provided, wherein the metric tensor, Levi-Civita connection, and Kähler potentials are explicitly expressed in terms of polylogarithmic functions of the poles and zeros of transfer functions parameterized by weight vectors.

Kähler information manifolds of signal processing filters in weighted Hardy spaces

TL;DR

The paper develops a unified framework in which linear systems analyzed in weighted Hardy spaces possess a Kähler information geometry, with the Kähler potential equal to the squared weighted Hardy norm of a transformed transfer function. By allowing arbitrary smooth transformations and weight sequences , the authors generalize prior unweighted complex cepstrum geometry to a broad class of weighted Hardy settings, including polylogarithmic representations via Cepstrum coefficients. They derive explicit metric and Levi-Civita connection formulas from the Kähler potential, enabling efficient geometric computations. The framework recovers known geometries (e.g., unweighted cepstrum, mutual information between past and future) as special cases and provides concrete expressions for ARMA/ARFIMA models under various weights. This approach offers a flexible, analytically tractable tool for analyzing information geometry of signal processing filters with practical connections to time-series modeling and system identification.

Abstract

We extend the framework of Kähler information manifolds for complex-valued signal processing filters by introducing weighted Hardy spaces and smooth transformations of transfer functions. We demonstrate that the Riemannian geometry induced from weighted Hardy norms for the smooth transformations of its transfer function is a Kähler manifold. In this setting, the Kähler potential of the linear system geometry corresponds to the squared weighted Hardy norm of the composite transfer function. With the inherent structure of Kähler manifolds, geometric quantities on the manifold of linear systems in weighted Hardy spaces can be computed more efficiently and elegantly. Moreover, this generalized framework unifies a variety of well-known information manifolds within the structure of Kähler information manifolds for signal filters. Several illustrative examples from time series models are provided, wherein the metric tensor, Levi-Civita connection, and Kähler potentials are explicitly expressed in terms of polylogarithmic functions of the poles and zeros of transfer functions parameterized by weight vectors.

Paper Structure

This paper contains 13 sections, 4 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Theoretical backgrounds
  3. Weighted Hardy spaces
  4. Cepstrum and weighted Hardy norms
  5. Kähler geometry
  6. Geometry of linear systems in weighted Hardy spaces
  7. Example: ARMA and ARFIMA models
  8. Arbitrary $\omega_s$
  9. $\omega_s=s^m$
  10. $m=0$
  11. $m=1$
  12. $\omega_s=|\rho|^{2s}$
  13. Conclusion

Key Result

Theorem 1

Let $h(z;\boldsymbol{\xi})$ be the transfer function of a linear system, and let $\phi$ be a smooth transformation such that the composite function $f=\phi \circ h$ belongs to a weighted Hardy space. Then, the information geometry induced by the weighted Hardy norm of $\phi \circ h$ defines a Kähler

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof