Reconstructing Point Sets From Distance Distributions

We address the problem of reconstructing a set points on line or loop from their unassigned noisy pairwise distances. When lie line, is known as turnpike; when they are loop, it beltway. approximate by discretizing domain and representing $N$ via an -hot encoding, which density supported discretized domain. show how distance distribution then simply collection quadratic functionals this propose to recover point locations so that estimated matches measured distribution. This can be cast constrained nonconvex optimization we solve using projected gradient descent with suitable spectral initializer. derive conditions under proposed matching approach locally converges global optimizer at linear rate. Compared conventional backtracking approach, our method jointly reconstructs all robust noise in measurements. substantiate these claims state-of-the-art performance across number numerical experiments. Our first practical large-scale beltway where loop.