标题： 在指定子空间上由投影线性映射之和对大型矩阵的最优逼近 显示英文标题

标题： Optimal approximation of a large matrix by a sum of projected linear mappings on prescribed subspaces

摘要： 我们提出并证明了一种矩阵约简方法，用于计算观测矩阵$A \in {\mathbb C}^{m \times n}$的最优逼近，该逼近为矩阵乘积之和$\sum_{i=1}^p \sum_{j=1}^q B_iX_{ij}C_j$，其中每个$B_i \in {\mathbb C}^{m \times g_i}$和$C_j \in {\mathbb C}^{h_j \times n}$是已知的，而未知的矩阵核$X_{ij}$通过最小化误差的弗罗贝尼乌斯范数来确定。 求和可以表示为一个有界线性映射$BXC$，其未知核为$X$，从给定子空间${\mathcal T} \subseteq {\mathbb C}^n$映射到由给定矩阵$C_1,\ldots,C_q$和$B_1,\ldots,B_p$的集体定义域和值域分别确定的给定子空间${\mathcal S} \subseteq {\mathbb C}^m$。 我们证明最优核是$X = B^{\dag}AC^{\dag}$，并且最优近似$BB^{\dag}AC^{\dag}C$是将观测映射$A$投影到从${\mathcal T}$到${\mathcal S}$的映射上。 如果$A$很大，$B$和$C$也可能很大，直接计算$B^{\dag}$和$C^{\dag}$变得繁琐且低效。 {提出的方法避免了}通过将求解过程简化为查找一组较小矩阵的伪逆来解决这一困难。 这显著减少了计算负担。 显示英文摘要

摘要： We propose and justify a matrix reduction method for calculating the optimal approximation of an observed matrix $A \in {\mathbb C}^{m \times n}$ by a sum $\sum_{i=1}^p \sum_{j=1}^q B_iX_{ij}C_j$ of matrix products where each $B_i \in {\mathbb C}^{m \times g_i}$ and $C_j \in {\mathbb C}^{h_j \times n}$ is known and where the unknown matrix kernels $X_{ij}$ are determined by minimizing the Frobenius norm of the error. The sum can be represented as a bounded linear mapping $BXC$ with unknown kernel $X$ from a prescribed subspace ${\mathcal T} \subseteq {\mathbb C}^n$ onto a prescribed subspace ${\mathcal S} \subseteq {\mathbb C}^m$ defined respectively by the collective domains and ranges of the given matrices $C_1,\ldots,C_q$ and $B_1,\ldots,B_p$. We show that the optimal kernel is $X = B^{\dag}AC^{\dag}$ and that the optimal approximation $BB^{\dag}AC^{\dag}C$ is the projection of the observed mapping $A$ onto a mapping from ${\mathcal T}$ to ${\mathcal S}$. If $A$ is large $B$ and $C$ may also be large and direct calculation of $B^{\dag}$ and $C^{\dag}$ becomes unwieldy and inefficient. { The proposed method avoids} this difficulty by reducing the solution process to finding the pseudo-inverses of a collection of much smaller matrices. This significantly reduces the computational burden.