Separable Expansions and Low-Rank Matrices

Abstract

In the previous chapters we studied low-rank matrices and model formats with low-rank matrices as matrix blocks. The essential question remains whether and in which cases low-rank matrices may yield a good approximation. In many cases, this property follows from the existence of a separable expansion which is the subject of this chapter.

In the case of an integral operator with a kernel function κ, the discretisation matrix inherits properties of the function κ. In Section 4.1 we demonstrate how certain separability properties of κ can be exploited to construct approximating rank-r matrices. The following two properties will be related:

• Approximability of a submatrix M|b (b suitable block) by a rank-r matrix.

• Approximability of the function κ(x, y)—restricted to a suitable subdomain corresponding to the block b—by a separable expansion with r terms.

In Section 4.2 the basic terms are explained which are needed in the sequel. The separable expansion (§4.2.1) is the starting point. In the most favourable case, exponential convergence holds (§4.2.2). Under certain conditions on the kernel function κ, separability follows from an admissibility condition (§4.2.3) for the domain X × Y where κ(·, ·) is evaluated.

In Section 4.3 we discuss separable expansions via polynomials. The Taylor expansion (§4.3.1) is a possible tool to obtain approximating polynomials, but interpolation (§4.3.2) is the easiest method to apply. A suitable regularity condition on the kernel function κ is its asymptotic smoothness, since it ensures exponential convergence (§4.3.3). Next we discuss error estimates of the Taylor expansion (§4.3.5) and interpolation (§§4.3.6–4.3.8).

Polynomial approximation is not the only choice. In Section 4.4 we consider further techniques (§§4.4.1–4.4.5). For theoretical purposes, we introduce the optimal separable expansion by the infinite singular value decomposition (§4.4.7).

Section 4.5 reveals the crucial role of a separable expansion: The discretisation of integral kernels with separation rank r yields matrices of rank ≤ r.

Section 4.6 is devoted to the error analysis.