Abstract
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; when they are on a loop, it is known as the beltway. We approximate the problem by discretizing the domain and representing the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> points via an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> -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 distance distribution matching approach locally converges to a global optimizer at a linear 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.
Highlights
In this paper we address the problem of reconstructing the geometry of N points from their unassigned pairwise distances in the one-dimensional case where the points lie on a line or a loop
The point locations can be extracted in the same way as in the noisy case which we describe
Given a proper initialization z0, we propose to solve (DDM-T) via the projected gradient descent method: zt1 “ PSztη ∇f pztq, (11)
Summary
In this paper we address the problem of reconstructing the geometry of N points from their unassigned pairwise distances in the one-dimensional case where the points lie on a line or a loop. D “ `dk, 1ďk ďN ̆, where dk is the distance between the k-th pair of points and could contain noise. In standard, assigned problems, every distance dk is assigned to a pair of points tum, unu from U “ pun, 1 ďnďNq. Put differently, we know an assignment map M pkq “ tm, nu such that dk “ }umun}. U1 u2 u1 u3 u2 u3 uN uN pairwise distances in the 1D case where the points could lie on a line or a loop. The correspondence between the distance dk and the pair of points pum, unq is unknown
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