Private Matrix Approximation and Geometry of Unitary Orbits. (arXiv:2207.02794v1 [cs.DS])

Consider the following optimization problem: Given $n \times n$ matrices $A$
and $\Lambda$, maximize $\langle A, U\Lambda U^*\rangle$ where $U$ varies over
the unitary group $\mathrm{U}(n)$. This problem seeks to approximate $A$ by a
matrix whose spectrum is the same as $\Lambda$ and, by setting $\Lambda$ to be
appropriate diagonal matrices, one can recover matrix approximation problems
such as PCA and rank-$k$ approximation. We study the problem of designing
differentially private algorithms for this optimization problem in settings
where the …

ds geometry matrix