Reconstructing Point Sets from Distance Distributions

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