# When the distance metric is not Euclidean, the metric Multidimensional Scaling (MDS) is nonlinear?

As it is commonly known, classic metric MDS (under Euclidean distance metric) is a linear dimension reduction method (equivalent to PCA), and it is also known to us that non-metric MDS is nonlinear, however, how about a metric MDS in which the distance metric is not defined as Euclidean (e.g., cosine distance, Manhattan distance), is it a linear embedding or nonlinear embedding ?

• What loss function do you have in mind for the non-Euclidean case? – S. Kolassa - Reinstate Monica Nov 6 '17 at 12:46
• @StephanKolassa, say the cosine distance between the output low-dim points are close to cosine distance of the original high-dim points. – lynnjohn Nov 6 '17 at 12:51
• Absolutely no. Torgerson's metric MDS aka PCoA (of which you are talking), for example, is frequently used with arbitrary dissimilarities. And that does not make it a NMDS method. As noted here, NMDS is different from metric MDS by the way of transform from dissimilarities to disparities, an not by the nature of dissimilarities or by mapping disparities on lo-dim map. – ttnphns Nov 6 '17 at 20:35
• When dissimilarity is euclidean distance or a measure directly related to it (such as cosine), metric MDS is usually a natural choice. However, when dissimilarity is not euclidean or even not metric, one has to decide if he wants metric or nonmetric analysis. Nonmetric MDS will often prove more realistic and practically valuable then. – ttnphns Nov 6 '17 at 20:43
• Next to remark, it is untrue that metric MDS = PcoA (= PCA): see stats.stackexchange.com/a/14017/3277. PCoA is only one, noniterative algorithm of it. Iterative more advanced procedures (such as PROXSCAL) perform both metric and nonmetric versions of the analysis. – ttnphns Nov 6 '17 at 20:59