Note: This question has also been posted here.
The molecular distance geometry problem (MDGP) consists of two sub-problems:
- Given observations of noisy distances between atoms in a molecule, estimate the values of the true distances.
- Given these estimated distances, compute the locations of the atoms.
More formally, the first sub-problem can be stated as:
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 $i,j \in \{1,2,...,n\}$.
The second sub-problem can then be formulated as:
Given $\hat{\mathbf{A}} = (\hat{d}_{ij})$, which is an estimate of $\mathbf{A} = (d_{ij})$, find the points $x_1,x_2,...,x_n$ such that $\lvert\lvert x_i - x_j\rvert\rvert = \hat{d}_{ij} \ \forall \ i,j$.
The second sub-problem is well-studied in the literature. If all distances $d_{ij}$ are given, and if $K=3$, then this problem can be solved using a linear order of operations [1]. Alternatively, if only a small subset of these distances are given, then it is possible to infer the rest of the unknown distances using specific geometrical constraints, such as the triangle inequality [2].
I am currently interested in the first sub-problem. More precisely, are there references that explore different noise models for the distances between atoms and references that attempt to estimate these distances?
