---- - Denote by $\mathcal{V}_{K}(\mathbb{F}^{M})$ the [[Stiefel manifold]] over $\mathbb{R}$ or $\mathbb{C}$. - Let $B \in \mathbb{F}^{M \times N}$ and $A \in \mathbb{F}^{K \times N}$. > [!theorem] Theorem. ([[semi-isometric procrustes problem]]) > The **semi-isometric procrustes problem** is a threefold generalization of the [[orthogonal procrustes problem]] wherein > - The ground field $\mathbb{F}$ may be $\mathbb{R}$ or $\mathbb{C}$ (rather than just $\mathbb{R}$); > - We allow for transformations that translate and (perhaps improperly) rotate. This is *almost* like finding the best [[euclidean isometry|euclidean isometry]], except... > - We allow for $A$ and $B$ to have different sizes, hence $Q \in \mathcal{V}_{K}(\mathbb{F}^{M})$ will be [[semi-isometric matrix|semi-isometric]]. > > In summary, the model $B_{:, n} \approx QA_{:,n}$ is replaced with the model $B_{:, n} \approx QA_{:, n}+d$ where $d \in \mathbb{F}^{M}$ is an unknown translation vector. In [[matrix]] form: $B \approx QA+ d \b 1_{N}'.$ > Thus the minimization problem is $\begin{align} > (\hat{Q}, \hat{d})= & \arg \min_{Q \in \mathcal{V}_{K}^{}(\mathbb{F}^{M})} \arg \min_{d \in \mathbb{F}^{M}}g(d,Q), \\ & g(d,Q):= \|B-(QA+d \b 1_{N}')\|_{F}^{2}. > \end{align} $ ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```