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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.

## Article

View Hidden convexity, optimization, and algorithms on rotation matrices