Skeleton and projected least-squares CUR cores

Let $A\in {\mathbb{F}}^{m\times n}$ over $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ with $m,n\ge 1$, and let $I=(i_1,\dots ,i_k)$ and $J=(j_1,\dots ,j_k)$ be ordered lists of distinct row and column indices. For $k=0$, both lists and all selected matrices have zero size, with ordinary empty-matrix multiplication. Define $C=A(:,J)\in {\mathbb{F}}^{m\times k}$, $R=A(I,:)\in {\mathbb{F}}^{k\times n}$, and the cross matrix $W=A(I,J)\in {\mathbb{F}}^{k\times k}$, using the same orders in $I,J,W$.

The skeleton CUR core is $U_{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{l}}=W^{†}$ and its reconstruction is ${\widehat{A}}_{\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{l}}=C{W}^{†}R$. The projected least-squares CUR core is $U_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}=C^{†}A{R}^{†}$ and its reconstruction is ${\widehat{A}}_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}=C(C^{†}A{R}^{†})R=(C{C}^{†})A(R^{†}R)$.

For $k=0$, both reconstructions are the zero $m\times n$ matrix.

The Moore-Penrose theorem in the dependencies guarantees that these formulas are single-valued for singular as well as nonsingular $C,R,W$. The two core constructions may yield the same matrix: for $A=[1]$ and $I=J=(1)$, both cores equal $[1]$. The projection residual bound on the A page applies only to ${\widehat{A}}_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}$; it is not asserted for the skeleton core $W^{†}$.