Consider two metric spaces $(X, d)$ and $(Y, f)$, and an embedding $\mu : X \rightarrow Y$. Traditional metric space embeddings measure the quality of $\mu$ as the worst-case ratio of original to final distance: $$ \rho = \max_{p,q \in X} \{ \frac{d(x,y)}{f(\mu(x), \mu(y))}, \frac{f(\mu(x), \mu(y))}{d(x,y)} \}$$
There are other measures of quality though: Dhamdhere et al study the "average" distortion: $$ \sigma = \frac{\sum d(x,y)}{\sum f(\mu(x), \mu(y))}.$$
However, the measure I'm interested in here is the one used by MDS-like methods, which looks at the average additive error: $$ \varepsilon^2 = \sum | d(x,y) - f(\mu(x), \mu(y))|^2$$
Although MDS-like methods are studied extensively outside the theoryCS community, I'm aware of only one paper (by Dhamdhere et al) that examines optimization under this measure, and that too for the limited problem of embedding onto the line ($Y = \mathbb{R}$) (side note: Tasos Sidiropoulos' 2005 MS thesis has a nice review of earlier work)
Is there any more recent work that people are aware of regarding rigorous quality analysis under this notion of error ? While these problems are generally NP-hard, what I'm more interested in are approximations of any kind.