References related to the molecular distance geometry problem (estimating true distances based on noisy distances)

Question

One aspect of the molecular distance geometry problem (MDGP) described in this PDF, can be written as follows:

"Given observations of noisy distances between atoms in a molecule, estimate the values of the true distances."

More formally: Given the datasets $\mathcal{D}_1,\mathcal{D}_2,\dots,\mathcal{D}_n$ of noisy distances for the atoms defined by the points $\mathcal{S} = \{x_1,x_2,\dots,x_n\}$, estimate the $n \times n$ symmetric distance matrix $\mathbf{A} = (d_{ij})$, where $d_{ij} = \lvert\lvert x_i - x_j\rvert\rvert$ and $x_i \in \mathbb{R}^K$ for $i,j \in \{1,2,...,n\}$.

Are there references that explore different noise models for the distances between atoms and references that attempt to estimate these distances?

Answer 1

I'll answer my own question.

I recently co-authored a paper titled Mismatched Estimation in the Distance Geometry Problem that tackles this exact topic. This paper was recently accepted for publication at the 56th Asilomar Conference on Signals, Systems, and Computers. Here is the abstract:

We investigate mismatched estimation in the context of the distance geometry problem (DGP). In the DGP, for a set of points, we are given noisy measurements of pairwise distances between the points, and our objective is to determine the geometric locations of the points. A common approach to deal with noisy measurements of pairwise distances is to compute least-squares estimates of the locations of the points. However, these least-squares estimates are likely to be suboptimal, because they do not necessarily maximize the correct likelihood function. In this paper, we argue that more accurate estimates can be obtained when an estimation procedure using the correct likelihood function of noisy measurements is performed. Our numerical results demonstrate that least-squares estimates can be suboptimal by several dB.