When is Rotations Averaging Hard?

Rotations averaging has become a key subproblem in global Structure from Motion methods. Several solvers exist, but they do not have guarantees of correctness. They can produce high-quality results, but also sometimes fail. Our understanding of what makes rotations averaging problems easy or hard is still very limited. To investigate the difficulty of rotations averaging, we perform a local convexity analysis under an L2 cost function. Although a previous result has shown that in general, this problem is locally convex almost nowhere, we show how this negative conclusion can be reversed by considering the gauge ambiguity. Our theoretical analysis reveals the factors that determine local convexity - noise and graph structure - as well as how they interact, which we describe by a particular Laplacian matrix. Our results are useful for predicting the difficulty of problems, and we demonstrate this on practical datasets. Our work forms the basis of a deeper understanding of the key properties of rotations averaging problems, and we discuss how it can inform the design of future solvers for this important problem.