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Tree quasi-separable matrices: a simultaneous generalization of sequentially and hierarchically semi-separable representations

Nithin Govindarajan, Shivkumar Chandrasekaran, Patrick Dewilde

TL;DR

This work introduces tree quasi-separable (TQS) matrices as a unifying generalization of sequentially and hierarchically semi-separable representations for matrices whose graph is a tree. By modeling a graph-partitioned matrix with an input–output dynamical system along the tree, the authors derive a minimal, universal construction where generator sizes are dictated by Hankel-block ranks, ensuring TQS inherits the fast matvec and direct-solve capabilities of SSS and HSS. They prove a graph-induced rank structure (GIRS) property for TQS, show equivalence with minimal Hankel-rank-based representations, and demonstrate that TQS operations (addition, multiplication, inversion) preserve low-rank structure with predictable rank growth. The paper also provides an explicit construction algorithm, numerical validation, and a framework for efficient linear-system solving on trees, suggesting TQS as a flexible drop-in replacement with potential advantages for graph-based dynamical systems and network problems.

Abstract

We present a unification and generalization of what is known in the literature as sequentially and hierarchically semi-separable (SSS and HSS) representations for matrices. Describing rank-structured representations of (inverses of) sparse matrices whose adjacency graph is a tree, it is shown that these so-called tree quasi-separable (TQS) matrices inherit all the favorable algebraic properties of SSS and HSS under addition, products, and inversion. To arrive at these properties, we prove a key result that characterizes the conversion of any dense matrix into a TQS representation. Here, we specifically show through an explicit construction procedure that the generator sizes are dictated by the ranks of certain Hankel blocks of the matrix. Analogous to SSS and HSS, TQS matrices admit fast matrix-vector products and direct solvers provided the generator sizes are small. A sketch of the associated algorithms is provided.

Tree quasi-separable matrices: a simultaneous generalization of sequentially and hierarchically semi-separable representations

TL;DR

This work introduces tree quasi-separable (TQS) matrices as a unifying generalization of sequentially and hierarchically semi-separable representations for matrices whose graph is a tree. By modeling a graph-partitioned matrix with an input–output dynamical system along the tree, the authors derive a minimal, universal construction where generator sizes are dictated by Hankel-block ranks, ensuring TQS inherits the fast matvec and direct-solve capabilities of SSS and HSS. They prove a graph-induced rank structure (GIRS) property for TQS, show equivalence with minimal Hankel-rank-based representations, and demonstrate that TQS operations (addition, multiplication, inversion) preserve low-rank structure with predictable rank growth. The paper also provides an explicit construction algorithm, numerical validation, and a framework for efficient linear-system solving on trees, suggesting TQS as a flexible drop-in replacement with potential advantages for graph-based dynamical systems and network problems.

Abstract

We present a unification and generalization of what is known in the literature as sequentially and hierarchically semi-separable (SSS and HSS) representations for matrices. Describing rank-structured representations of (inverses of) sparse matrices whose adjacency graph is a tree, it is shown that these so-called tree quasi-separable (TQS) matrices inherit all the favorable algebraic properties of SSS and HSS under addition, products, and inversion. To arrive at these properties, we prove a key result that characterizes the conversion of any dense matrix into a TQS representation. Here, we specifically show through an explicit construction procedure that the generator sizes are dictated by the ranks of certain Hankel blocks of the matrix. Analogous to SSS and HSS, TQS matrices admit fast matrix-vector products and direct solvers provided the generator sizes are small. A sketch of the associated algorithms is provided.
Paper Structure (24 sections, 6 theorems, 66 equations, 3 figures, 2 tables)

This paper contains 24 sections, 6 theorems, 66 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A brief recap on SSS and HSS matrices
  3. SSS matrices
  4. HSS matrices
  5. Algebra with SSS and HSS matrices
  6. Tree quasi-separable (TQS) matrices
  7. Tree graphs
  8. Graph-partitioned matrices
  9. Definition of TQS matrices
  10. HSS and SSS matrices as special subcases
  11. TQS representations of sparse matrices
  12. Algebraic properties of TQS matrices
  13. Graph-induced rank structure of TQS matrices
  14. Minimal TQS representations
  15. Sums, products, and inverses of TQS matrices
  16. ...and 9 more sections

Key Result

Proposition 4.2

\newlabelprop:TQSisGIRS0 A TQS matrix $\mathrm{T}\in\mathbb{F}^{M\times N}$ with rank-profile $\lbrace \rho_{e} \rbrace_{e\in \mathbb{E}}$ satisfies the GIRS-property for $c=\max_{e\in \mathbb{E}} \rho_{e}$.

Figures (3)

  • Figure 1: TQS operators associated with elements of the graph $\mathbb{G}$.
  • Figure 2: Node $k\in\mathbb{V}$ with its children $i_1,i_2,\ldots,i_p \in \mathcal{C}(k)$ and parent $j = \mathcal{P}(k)$.
  • Figure 3: A level $k=3$ binary tree with a "nested dissection"-styled ordering of the nodes.

Theorems & Definitions (14)

  • Definition 3.1: tree quasi-separable matrices
  • Definition 4.1: GIRS-property
  • Proposition 4.2
  • Proof 1
  • Definition 4.3: minimal TQS representation
  • Theorem 4.4
  • Corollary 4.5
  • Proposition 4.6: TQS addition
  • Proof 2
  • Proposition 4.7: TQS product
  • ...and 4 more