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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?

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    $\begingroup$ +1. Welcome to our new community, and thank you for contributing your question here!! We hope to see much more of you in the future! I had to comment out your description of the 2nd MDGP sub-problem, because your overall question has nothing to do with it, so it was just distracting. If you have questions about it, you can ask it separately (and by the way, the 2nd sub-problem can be solved by MDS and the dozens of related methods. Also there's 59 references in the PDF you gave in the question, what's wrong with those? What's your goal? $\endgroup$ Jun 28, 2021 at 3:22
  • $\begingroup$ @NikeDattani thanks for your comment. All distance geometry algorithms use a cost function. My goal is to compare the performance of these algorithms when the cost function matches the distribution of the distances, and when there is a mismatch. For example, if the distances are Gaussian distributed, then using the sum of squared errors cost function will yield optimal distance estimates, in the sense that they maximize the corresponding likelihood function. Similarly, if the distances are Laplace distributed, then using the sum of absolute errors cost function will be optimal... $\endgroup$
    – mhdadk
    Jun 28, 2021 at 11:22
  • $\begingroup$ @NikeDattani ...In the literature, I have found that most algorithms use the sum of squared errors cost function, regardless of the distribution of the distances. Has anyone explored using the appropriate cost function based on the distribution of distances before? $\endgroup$
    – mhdadk
    Jun 28, 2021 at 11:24
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    $\begingroup$ Sorry this question hasn't gotten more attention. If it continues to not pick up much traction, you may want to try on Comp Science SE. Getting back to your actual question, I'm certainly not an expert in this area, but I would think that experimentally the distribution of errors in the distances would be fairly consistent for a given type of measurement. So I would expect that either 1. The error is thought to be Gaussian or 2. It's not clear what the distribution is. In either case, I would think they would just stick with sum of squares, due it's advantages for 1 and wide use for 2. $\endgroup$
    – Tyberius
    Nov 13, 2021 at 20:06
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    $\begingroup$ @Tyberius I might be able to answer it, I'll give it a try later in December. $\endgroup$ Dec 15, 2021 at 7:13

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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.

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