In the context of dimensionality reduction.
Why approximate pairwise distances only over lower triangle of distance matrix?
$$\min_{\{\hat{x_i}\}} I = \sum_{i <j} ({ \hat{ d_{ij} } - d_{ij} })^2$$
(also known as raw stress)
Where $d_{ij}$ are distances of original data points and $\hat{d_{ij}}$ contain new coordinates that would approximate pairwise distances.
(as my notes give)
Is it perhaps that the upper triangle is a duplicate of the lower, since surely $d_{ij}=d_{ji}$ ($i \not= j$)?
$\require{cancel}$
You are right
\begin{eqnarray} \sum_{i,j} (\hat{d}_{ij} - d_{ij}) &=& \color{blue}{\sum_{i < j}(\hat{d}_{ij} - d_{ij})} + \color{red}{\sum_{i > j}(\hat{d}_{ij} - d_{ij})} + \color{orange}{\cancelto{0}{\sum_{i = j}(\hat{d}_{ij} - d_{ij})}} \\ &=& \color{blue}{\sum_{i < j}(\hat{d}_{ij} - d_{ji})} + \color{red}{\sum_{i > j}(\hat{d}_{ji} - d_{ji})} \\ &=& \color{blue}{\sum_{i < j}(\hat{d}_{ij} - d_{ji})} + \color{red}{\sum_{j < i}(\hat{d}_{ji} - d_{ji})} \\ &=& 2\color{blue}{\sum_{i < j}(\hat{d}_{ij} - d_{ji})} \end{eqnarray}
