The classical beltway problem entails recovering a set of points from their unordered pairwise distances on the circle. This problem can be viewed as a special case of the crystallographic phase retrieval problem of recovering a sparse signal from its periodic autocorrelation. Based on this interpretation, and motivated by cryo-electron microscopy, we suggest a natural generalization to orthogonal groups: recovering a sparse signal, up to an orthogonal transformation, from its autocorrelation over the orthogonal group. If the support of the signal is collision-free, we bound the number of solutions to the beltway problem over orthogonal groups, and prove that this bound is exactly one when the support of the signal is radially collision-free (i.e., the support points have distinct magnitudes). We also prove that if the pairwise products of the signal's weights are distinct, then the autocorrelation determines the signal uniquely, up to an orthogonal transformation. We conclude the paper by considering binary signals and show that in this case, the collision-free condition need not be sufficient to determine signals up to orthogonal transformation.

中文翻译：

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正交群上的环城公路问题


经典环城公路问题需要从圆上无序的成对距离中恢复一组点。这个问题可以看作是晶体学相位检索问题的一个特例，即从其周期性自相关中恢复稀疏信号。基于这种解释，并在冷冻电子显微镜的推动下，我们建议对正交群进行自然推广：从其在正交群上的自相关中恢复稀疏信号，直至正交变换。如果信号的支撑是无碰撞的，我们将环城公路问题解的数量限制在正交群上，并证明当信号的支持是径向无碰撞时（即支撑点具有不同的大小），这个边界正好是 1。我们还证明，如果信号权重的成对乘积是不同的，那么自相关唯一地确定信号，直到正交变换。我们通过考虑二进制信号来结束本文，并表明在这种情况下，无碰撞条件不必足以确定正交变换之前的信号。