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:<math>\|x_i - x_j\| \approx d_{i,j}</math> for all <math>i,j\in {1,\dots,M}</math>,
where <math>\|\cdot\|</math> is a [[Norm (mathematics)|vector norm]]. In classical MDS, this norm is the [[Euclidean distance]], but, in a broader sense, it may be a [[metric (mathematics)|metric]] or arbitrary distance function.<ref name="Kruskal">[[Joseph Kruskal|Kruskal, J. B.]], and Wish, M. (1978), ''Multidimensional Scaling'', Sage University Paper series on Quantitative Application in the Social Sciences, 07-011. Beverly Hills and London: Sage Publications.</ref> For example, when dealing witg mixed-type data that contain numerical as well as categorical descriptors, [[Gower's distance]] is a common alternative.
In other words, MDS attempts to find a mapping from the <math>M</math> objects into <math>\mathbb{R}^N</math> such that distances are preserved. If the dimension <math>N</math> is chosen to be 2 or 3, we may plot the vectors <math>x_i</math> to obtain a visualization of the similarities between the <math>M</math> objects. Note that the vectors <math>x_i</math> are not unique: With the Euclidean distance, they may be arbitrarily translated, rotated, and reflected, since these transformations do not change the pairwise distances <math>\|x_i - x_j\|</math>.
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