Measures of fit for nonlinear biplots

Biplots provide a representation of observations and variables on the same diagram. In order to ensure that this diagram is low-dimensional (typically two-dimensional) the representation is usually an approximation - the distances between points approximate the dissimilarities between corresponding observations. Thus for any given biplot it is important to know how good this approximation is. As with linear (PCA) biplots, an overall measure of quality of a nonlinear biplot is given by considering $\textrm{trace}(\bm{Y}_r^\prime \bm{Y}_r)/ textrm{trace}(\bm{Y}^\prime \bm{Y})$ where $\bm{Y}$ is an exact representation of the $n$ observations in $n-1$ dimensional space and $\bm{Y}_r$ is the representation in $r$-dimensional space. However if the overall quality is not good, it is of interest of why that might be. For example, does the form of the dissimilarity function mean that differences due an important variable is not represented well by Euclidean distances in small number of dimensions, or are simply too many variables are important to be summarised well on any $r$ dimensional plot? So, taking measures of fit for linear biplots as a starting point, ways of exploring the fit of nonlinear biplots will be explored.