Approximating Matrices with Multiple Symmetries

Abstract

If a tensor with various symmetries is properly unfolded, then the resulting matrix inherits those symmetries. As tensor computations become increasingly important it is imperative that we develop efficient structure preserving methods for matrices with multiple symmetries. In this paper we consider how to exploit and preserve structure in the pivoted Cholesky factorization when approximating a matrix $A$ that is both symmetric ($A=A^T$) and what we call perfect shuffle symmetric, or perf-symmetric. The latter property means that $A = \Pi A\Pi$, where $\Pi$ is a permutation with the property that $\Pi v = v$ if $v$ is the vec of a symmetric matrix and $\Pi v = -v$ if $v$ is the vec of a skew-symmetric matrix. Matrices with this structure can arise when an order-4 tensor $\cal A$ is unfolded and its elements satisfy ${\cal A}(i_{1},i_{2},i_{3},i_{4}) = {\cal A}(i_{2},i_{1},i_{3},i_{4}) ={\cal A}(i_{1},i_{2},i_{4},i_{3}) ={\cal A}(i_{3},i_{4},i_{1},i_{2}).$ This is the case in certain quantum chemistry applications where the tensor entries are electronic repulsion integrals. Our technique involves a closed-form block diagonalization followed by one or two half-sized pivoted Cholesky factorizations. This framework allows for a lazy evaluation feature that is important if the entries in $\cal A$ are expensive to compute. In addition to being a structure preserving rank reduction technique, we find that this approach for obtaining the Cholesky factorization reduces the work by up to a factor of 4 (see [G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013]).