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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.10614 |
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| _version_ | 1866909428177960960 |
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| author | Howlett, Phil Torokhti, Anatoli |
| author_facet | Howlett, Phil Torokhti, Anatoli |
| contents | 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 $X = B^†AC^†$ and that the optimal approximation $BB^†AC^†C$ 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 $B^†$ and $C^†$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal approximation of a large matrix by a sum of projected linear mappings on prescribed subspaces Howlett, Phil Torokhti, Anatoli Numerical Analysis 15A18 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 $X = B^†AC^†$ and that the optimal approximation $BB^†AC^†C$ 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 $B^†$ and $C^†$ 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. |
| title | Optimal approximation of a large matrix by a sum of projected linear mappings on prescribed subspaces |
| topic | Numerical Analysis 15A18 |
| url | https://arxiv.org/abs/2412.10614 |