Randomized Compression of Rank-Structured Matrices Accelerated with Graph Coloring

TL;DR

A randomized algorithm for computing a data sparse representation of a given rank structured matrix of a given rank structured matrix (a.k.a. an $H-matrix) is presented, which is particularly effective for kernel matrices involving a set of points restricted to a lower dimensional object than the ambient space.

Abstract

A randomized algorithm for computing a data sparse representation of a given rank structured matrix $A$ (a.k.a. an $H$-matrix) is presented. The algorithm draws on the randomized singular value decomposition (RSVD), and operates under the assumption that algorithms for rapidly applying $A$ and $A^{*}$ to vectors are available. The algorithm analyzes the hierarchical tree that defines the rank structure using graph coloring algorithms to generate a set of random test vectors. The matrix is then applied to the test vectors, and in a final step the matrix itself is reconstructed by the observed input-output pairs. The method presented is an evolution of the "peeling algorithm" of L. Lin, J. Lu, and L. Ying, "Fast construction of hierarchical matrix representation from matrix-vector multiplication," JCP, 230(10), 2011. For the case of uniform trees, the new method substantially reduces the pre-factor of the original peeling algorithm. More significantly, the new technique leads to dramatic acceleration for many non-uniform trees since it constructs sample vectors that are optimized for a given tree. The algorithm is particularly effective for kernel matrices involving a set of points restricted to a lower dimensional object than the ambient space, such as a boundary integral equation defined on a surface in three dimensions.

Figures (15)

• Figure 1: An $\mathcal{H}^1$ matrix for a quadtree over a uniform grid in the plane. Dense blocks are shown in dark gray, and low-rank blocks are represented with a white background and light gray rectangles representing the shapes of the low-rank factors.
• Figure 1: A binary tree structure, where the levels of the tree represent successively refined partitions of the index vector $[1, ..., 400]$.
• Figure 1: The constraint incompatibility graph corresponding to the 18 admissible blocks belonging to level 3 of the matrix shown in Figure \ref{['fig:matrix_strong_1d_lvl3']}. Each vertex corresponds to a distinct set of sampling constraints \ref{['eq:h1constraints']}. Edges connect pairs of vertices that are incompatible. The number of vertices is less than the number of admissible blocks since some admissible blocks share the same set of sampling constraints.
• Figure 1: Number of colors for the incompatibility graphs that arise from sampling one level of admissible blocks of an $\mathcal{H}^1$ matrix based on a uniform grid along a randomly oriented line through $\lbrack 0, 1 \rbrack^d$ with added perturbation over a range of dimensions $d$.
• Figure 2: An $\mathcal{H}^1$ matrix with depth 3 based on a grid over $\lbrack 0, 1 \rbrack$. Admissible blocks are shown in blue, and inadmissible blocks are shown in red.

Theorems & Definitions (4)

• Remark 1.1: Related work
• Definition 4.1: Constraint incompatibility graph
• Remark 4.2
• Remark 5.1