Kähler信息几何在信号处理中的应用
Kählerian information geometry for signal processing
AI总结：
本文研究了信号滤波器的信息几何与Kähler流形的对应关系，证明了最小相位线性系统的信息几何为Kähler流形，并探讨了其在信号处理中的应用与优势。
摘要：
我们证明了信号滤波器的信息几何与Kähler流形之间的对应关系。最小相位线性系统的信息几何是一个Kähler流形，信号滤波器的复频谱范数的平方对应于Kähler势。Kähler流形的Hermite结构仅在脉冲响应函数中最高次$z$的项在模型参数中为常数时才会显式出现。Kähler信息几何利用了更高效的度量张量和里奇张量计算步骤。此外，几何张量的$α$-泛化在$α$上是线性的。它还对寻找贝叶斯预测先验，如超谐波先验，具有鲁棒性，因为Kähler流形上的拉普拉斯-贝尔特里算子比非Kähler流形上的形式简单得多。几种时间序列模型在Kähler信息几何中被研究。
英文摘要：
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.
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