On Radial Isotropic Position: Theory and Algorithms

Abstract

We review the theory of, and develop algorithms for transforming a finite point set in ${\bf R}^d$ into a set in \emph{radial isotropic position} by a nonsingular linear transformation followed by rescaling each image point to the unit sphere. This problem arises in a wide spectrum of applications in computer science and mathematics. Our algorithms use gradient descent methods for a particular convex function $f$ whose minimum defines the transformation, and our main focus is on analyzing their performance. Although the minimum can be computed exactly, by expensive symbolic algebra techniques, gradient descent only approximates the desired minimum to any desired level of accuracy. We show that computing the gradient of $f$ amounts to computing the Singular Value Decomposition (SVD) of a certain matrix associated with the input set, making it simple to implement. We believe it to be superior to other approximate techniques (mainly the ellipsoid algorithm) used previously to find this transformation, and it should run much faster in practice. We prove that $f$ is smooth, which yields convergence rate proportional to $1/ε$, where $ε$ is the desired approximation accuracy. To complete the analysis, we provide upper bounds on the norm of the optimal solution which depend on new parameters measuring "the degeneracy" in our input. We believe that our parameters capture degeneracy better than other, seemingly weaker, parameters used in previous works. We next analyze the strong convexity of $f$, and present two worst-case lower bounds on the smallest eigenvalue of its Hessian. This gives another worst-case bound on the convergence rate of another variant of gradient decent that depends only logarithmically on $1/ε$.