论文标题
鉴定基质关节块对角线化
Identification of Matrix Joint Block Diagonalization
论文作者
论文摘要
给定一个$ \ MATHCAL {C} = \ {c_i \} _ {i = 1}^m $ of Square Matrices，矩阵盲关节块对角线化问题（BJBDP）是为了找到完整的列级矩阵$ a $，以便$ c_i = a_ia^a $ n $ c_i_ia^a $ nist $ nist $ nist $ nist $ n $ s $ s $ s $ s $ s $ IS $对角矩阵具有尽可能多的对角线块。 BJBDP在独立子空间分析（ISA）中起着重要作用。本文考虑了BJBDP的识别问题，也就是说，在什么条件下，通过什么方式，我们可以识别对角线化器$ a $ a $和$σ_i$的块对角线结构，尤其是当$ C_I $中的噪声时。在本文中，我们提出了一种``Bi-Block对角化''方法来解决BJBDP，并建立了该方法能够完成任务的足够条件。数值模拟验证了我们的理论结果。据作者所知，BJBDP的现有数值方法没有理论保证确定确切解决方案，而我们的方法则没有。
Given a set $\mathcal{C}=\{C_i\}_{i=1}^m$ of square matrices, the matrix blind joint block diagonalization problem (BJBDP) is to find a full column rank matrix $A$ such that $C_i=AΣ_iA^\text{T}$ for all $i$, where $Σ_i$'s are all block diagonal matrices with as many diagonal blocks as possible. The BJBDP plays an important role in independent subspace analysis (ISA). This paper considers the identification problem for BJBDP, that is, under what conditions and by what means, we can identify the diagonalizer $A$ and the block diagonal structure of $Σ_i$, especially when there is noise in $C_i$'s. In this paper, we propose a ``bi-block diagonalization'' method to solve BJBDP, and establish sufficient conditions under which the method is able to accomplish the task. Numerical simulations validate our theoretical results. To the best of the authors' knowledge, existing numerical methods for BJBDP have no theoretical guarantees for the identification of the exact solution, whereas our method does.
