# Multiscale semidefinite programming approach to positioning problems with pairwise structure

arXiv: https://arxiv.org/abs/2012.10046 · HTML: https://mlsanity.com/paper/2012.10046/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2020.

## Built on by (most cited first, 1 of 1)

- [Generative Modeling via Hierarchical Tensor Sketching](https://mlsanity.com/paper/2304.05305/) (2023)

## Abstract (arXiv)

We consider the optimization of pairwise objective functions, i.e., objective functions of the form H(x) = H(x1,ldots,xN) = sum1≤ i<j ≤ N Hij(xi,xj) for xi in some continuous state spaces Xi. Global optimization in this setting is generally confounded by the possible existence of spurious local minima and the impossibility of global search due to the curse of dimensionality. In this paper, we approach such problems via convex relaxation of the marginal polytope considered in graphical modeling, proceeding in a multiscale fashion which exploits the smoothness of the cost function. We show theoretically that, compared with existing methods, such an approach is advantageous even in simple settings for sensor network localization (SNL). We successfully apply our method to SNL problems, particularly difficult instances with high noise. We also validate performance on the optimization of the Lennard-Jones potential, which is plagued by the existence of many near-optimal configurations. We demonstrate that in MMR allows us to effectively explore these configurations.