A Kaczmarz-Inspired Method for Orthogonalization

This paper asks if the following iterative procedure approximately orthogonalizes a set of $n$ linearly independent unit vectors while preserving their span: in each iteration, access a random pair of vectors and replace one with the component perpendicular to the other, renormalized to be a unit vector. We provide a positive answer: any given set of starting vectors converges almost surely to an orthonormal basis of their span. We specifically argue that the $n$-volume of the parallelepiped generated by the vectors approaches 1 (i.e. the parallelepiped approaches a hypercube). If $A$ is the matrix formed by taking these vectors as columns, this volume is simply $\det(|A|)$ where $|A|=(A^*A)^{1/2}$. We show that $O(n^2\log(1/(\det(|A|)\varepsilon)))$ iterations suffice to bring ${\det(|A|)}$ above $1-\varepsilon$ with constant probability.