Today’s post is about a recent paper of mine with Christopher Kennedy and Amit Singer entitled “Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups”. Earlier this week, we finished a major revision to the paper (even the title was slightly changed) and I decided to take the opportunity to blog about it (in a previous blog post I blogged about an open problem that is raised by this work).

**— Generalized orthogonal Procrustes Problem —**

In this post I will motivate the problem using the orthogonal Procrustes problem: Given point clouds in of points each, the generalized orthogonal Procrustes problem consists of finding orthogonal transformations that best simultaneously align the point clouds. If the points are represented as the columns of matrices , where then the orthogonal Procrustes problem consists of solving

Since , (1) has the same solution as the complementary version of the problem

where has blocks given by . Note that is positive semidefinite.

In fact, we will focus on problems of the form (2), for any positive definite (which encodes a few other problems, see more here).

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