Block Semi-Separable Matrix (Block-SSD)

• Block-SSD matrices are structured dense matrices that admit low-rank factorizations of off-diagonal blocks, facilitating efficient matrix computations.
• They represent the level-0 instance of hierarchically block separable matrices, linking single-level compression directly to multilevel skeletonization techniques.
• Their fast solvers achieve near-linear complexity for dense operator equations, making them ideal for radial kernel approximations and equality-constrained least-squares problems.

A block semi-separable matrix (block-SSD) is a structured matrix arising as the level-0 (single-level) special case within the broader hierarchically block separable (HBS) framework. Such matrices are typically dense yet data-sparse, characterized by the property that their off-diagonal blocks admit low-rank factorizations. Block-SSD matrices and their associated fast algorithms enable efficient direct and least-squares solvers for problems exhibiting nonoscillatory, asymptotically smooth behavior, including operators derived from radial kernels. Their structure facilitates near-linear complexity in both algorithmic factorization and solution procedures in a dimensionality-dependent manner (Ho et al., 2012).

1. Mathematical Definition and Structural Properties

Let $A\in\mathbb{C}^{M\times N}$ partitioned into $p\times p$ blocks, where the $i$th row block has size $m_i$ and the $j$th column block has size $n_j$:

$$A = \begin{pmatrix} A_{11} & A_{12} & \cdots & A_{1p} \ A_{21} & A_{22} & \cdots & A_{2p} \ \vdots & \vdots & \ddots & \vdots \ A_{p1} & A_{p2} & \cdots & A_{pp} \end{pmatrix}.$$

A matrix is block semi-separable (block-SSD) if every off-diagonal block ($i \neq j$) admits a low-rank factorization:

$A_{ij} = U_i S_{ij} V_j^T,$

with $U_i\in\mathbb{C}^{m_i \times k_i^r}$, $V_j\in\mathbb{C}^{n_j\times k_j^c}$, and $$\operatorname{rank}(A_{ij})\leq \min(k_i^r,k_j^c)\ll\min(m_i,n_j)$$. Defining $D=\operatorname{diag}(A_{11},A_{22},\dots,A_{pp})$, $U=\operatorname{diag}(U_1,\dots,U_p)$, $V=\operatorname{diag}(V_1,\dots,V_p)$, and $S$ as the block-matrix with $S_{ii}=0$, the block-SSD admits the canonical single-level semi-separable form:

$A = D + U S V^T.$

This structure is precisely the level-0 realization of the multilevel HBS decomposition, in which higher levels correspond to further hierarchy and recursive partitioning.

2. Connection to Hierarchically Block Separable (HBS) Matrices

Block-SSD matrices function as a foundational element within the HBS paradigm, which embeds blockwise low-rank approximability at multiple levels of partitioning. In HBS, index sets $(I_1 = \{1,\dots,M\},\; J_1=\{1,\dots,N\})$ are recursively split and assembled in a tree $\mathcal{T}$ of depth $\lambda$. At each level $\ell$, off-diagonal blocks, indexed by the corresponding tree nodes, must admit factorizations analogous to the block-SSD form. The telescoping, multilevel decomposition is defined recursively via blockwise interpolative decompositions:

$$A \approx D^{(\lambda)} + L^{(\lambda)}\Bigl[D^{(\lambda-1)} + L^{(\lambda-1)}[\cdots D^{(1)} + L^{(1)} D^{(0)} R^{(1)} ] R^{(2)} \cdots ] R^{(\lambda)}.$$

For block-SSD, this hierarchy truncates at level-0, directly yielding $A = D + U S V^T$ without further recursion.

3. Recursive Skeletonization and Matrix Compression

The key methodology for exploiting block-SSD/HBS structure is recursive skeletonization. For general HBS matrices, skeletonization proceeds bottom-up from the leaves of the partitioning tree. At the finest level ($\ell=\lambda$), each off-diagonal block is approximated by an interpolative decomposition (ID):

$$A_{ij}\approx L_i^{(\lambda)} S_{ij}^{(\lambda)} (R_j^{(\lambda)})^T,$$

with $L_i^{(\lambda)}$ and $R_j^{(\lambda)}$ extracting row/column skeletons. The process ascends levels by forming reduced systems $S^{(\lambda)}$, repartitioning, and recompressing. For block-SSD (level-0), no recursion is needed; the compression reflects the single-level structure, and the explicit factorization $A = D + U S V^T$ suffices.

4. Equality-Constrained Least Squares Embedding

Solving least-squares problems, $\min_{x\in\mathbb{C}^N} \|Ax-b\|_2$, with block-SSD structure invokes a sparse, equality-constrained embedding. The telescoping HBS form introduces auxiliary variables at each level. Collecting these in block vectors, the constrained system is:

$$\min_{\mathbf{C}\mathbf{x}=0} \|\mathbf{E} \mathbf{x} - b\|_2,$$

where both $\mathbf{E}$ and $\mathbf{C}$ are sparse matrices derived from the telescoped ID compressions, and $\mathbf{x}$ comprises the multilevel variables. For block-SSD, the embedding reduces to:

$$\begin{pmatrix} D & U \ V^T & -I \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} b \ 0 \end{pmatrix}, \quad y = V^T x,$$

yielding a system of size $(M+K)\times(N+K)$, $K = \sum k_i \ll N$, thus maintaining sparsity and computational favorability.

5. Semi-Direct Least Squares Solver

The fast semi-direct solver operates in two phases: direct precomputation and iterative least-squares refinement.

Precomputation (Direct Phase)

1. Compress $A$ via (possibly recursive) skeletonization to precision $\epsilon$, yielding sparse $\mathbf{E},\mathbf{C}$.
2. Form the weighted matrix:

$$\mathbf{A}(\tau)=\begin{pmatrix}\mathbf{E} \ \tau\mathbf{C}\end{pmatrix},\quad \tau=\epsilon_{\text{mach}}^{-1/3}.$$

1. Compute sparse QR decomposition of $\mathbf{A}(\tau)$.

Solve Phase (Deferred-Correction Refinement)

For each right hand side $b$:

1. Solve $\min\|\mathbf{A}(\tau)x - [b;0]\|$ for $x^{(0)}$ using back-substitution.
2. Initialize residuals and Lagrange multipliers.
3. Perform up to two deferred-correction steps:
   • Augment the right-hand side and solve for correction $\Delta x$.
   • Update the solution, residuals, and multipliers.

It is shown that for well-conditioned $A$ no more than two correction steps suffice (Ho et al., 2012).

6. Computational Complexity and Dimensional Scaling

The asymptotic complexity depends on both the block ranks and the ambient dimension ($d$), with the depth $\lambda\sim (1/d)\log N$. Let $k_\ell$ denote the off-diagonal rank at level $\ell$; empirical scaling for singular kernels is:

$$k_\ell \sim \begin{cases} O(\ell), & d=1 \ O(2^{(d-1)\ell}), & d>1 \end{cases}$$

Specific complexity results are:

Phase	$d=1$	$d=2$	$d=3$
Compression, QR	$O(M+N)$	$O(M+N^{3/2})$	$O(M+N^2)$
Solve	$O(M+N)$	$O(M+N\log N)$	$O(M+N^{4/3})$

If sources and targets are well-separated, $k_\ell=O(1)$ and all complexities collapse to $O(M+N)$.

7. Specialization and Significance

For matrices that are exactly block-SSD (level-0), recursive skeletonization is unnecessary; the semi-separable $A=D+U S V^T$ representation is direct, and the equality-constrained solver operates natively. The sparse system increases only by $K=\sum k_i$ auxiliary variables, preserving scalability:

$\text{Complexity}: \quad O(M+N).$

This specialization demonstrates that the classical fast SSD direct solvers are recovered within the HBS formalism, providing a unified theoretical and algorithmic framework. The block-SSD class is thus pivotal for efficient numerical solution of dense operator equations with underlying data sparsity, especially in contexts involving radial basis function approximation, updating, and downdating (Ho et al., 2012).

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Block semi-separable matrices represent the atomic case of hierarchically structured data-sparse matrices. Their innate low-rank block structure, efficient skeletal compressions, and compatibility with modern sparse linear algebra techniques enable rapid dense linear-solving procedures with near-linear complexity scaling in problem size and spatial dimension.