H-matrix block representation

Let $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, let $I,J$ be nonempty finite index sets with point assignments as in Finite cluster trees and geometric admissibility, and let $\mathcal{P}$ be the block-cluster leaves of Block-cluster leaves partition the index-pair set. Fix an integer rank cap $k\ge 1$. An H-matrix representation with cap $k$ assigns to each far block $b=(\tau ,\sigma )\in \mathcal{P}$ factors $U_b\in {\mathbb{F}}^{\tau \times r_b}$ and $V_b\in {\mathbb{F}}^{\sigma \times r_b}$ with $0\le r_b\le k$, representing that block by $U_b{V}_b^{\ast }$. Each near block is stored as its full dense matrix $D_b\in {\mathbb{F}}^{\tau \times \sigma }$. Since $\mathcal{P}$ partitions $I\times J$, these blocks determine a unique represented matrix $\tilde{A}\in {\mathbb{F}}^{I\times J}$.

The rank cap on far blocks is an assumption about the representation or the target's approximability; geometric admissibility alone does not imply that a given matrix block has low rank. Near blocks remain dense and receive no low-rank guarantee.