This paper studies hidden convexity properties associated with constrained optimization
problems over the set of rotation matrices \(\text{SO}(n)\). Such problems are nonconvex due to the
constraint\(X\in\text{SO}(n)\). Nonetheless, we show that certain linear images of \(\text{SO}(n)\) are convex,
opening up the possibility for convex optimization algorithms with provable guarantees for these
problems. Our main technical contributions show that any two-dimensional image of \(\text{SO}(n)\) is
convex and that the projection of \(\text{SO}(n)\) onto its strict upper triangular entries is convex. These
results allow us to construct exact convex reformulations for constrained optimization problems
over \(\text{SO}(n)\) with a single constraint or with constraints defined by low-rank matrices. Both of
these results are optimal in a formal sense.
Hidden convexity, optimization, and algorithms on rotation matrices
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