Abstract This article introducesFiGaRo, an algorithm for computing the upper-triangular matrix in the QR decomposition of the matrix defined by the natural join over relational data.FiGaRo’s main novelty is that it pushes the QR decomposition past the join. This leads to several desirable properties. For acyclic joins, it takes time linear in the database size and independent of the join size. Its execution is equivalent to the application of a sequence of Givens rotations proportional to the join size. Its number of rounding errors relative to the classical QR decomposition algorithms is on par with the database size relative to the join output size. The QR decomposition lies at the core of many linear algebra computations including the singular value decomposition (SVD) and the principal component analysis (PCA). We show howFiGaRocan be used to compute the orthogonal matrix in the QR decomposition, the SVD and the PCA of the join output without the need to materialize the join output. A suite of experiments validate thatFiGaRocan outperform both in runtime performance and numerical accuracy the LAPACK library Intel MKL by a factor proportional to the gap between the sizes of the join output and input.

This paper introduces Figaro, an algorithm for computing the upper-triangular matrix in the QR decomposition of the matrix defined by the natural join over a relational database. The QR decomposition lies at the core of many linear algebra techniques and their machine learning applications, including: the matrix inverse; the least squares; the singular value decomposition; eigenvalue problems; and the principal component analysis.