Out-of-Sample Embedding with Proximity Data: Projection versus Restricted Reconstruction

Abstract:The problem of using proximity (similarity or dissimilarity) data for the purpose of "adding a point to a vector diagram" was first studied by J.C. Gower in 1968. Since then, a number of methods -- mostly kernel methods -- have been proposed for solving what has come to be called the problem of *out-of-sample embedding*. We survey the various kernel methods that we have encountered and show that each can be derived from one or the other of two competing strategies: *projection* or *restricted reconstruction*. Projection can be analogized to a well-known formula for adding a point to a principal component analysis. Restricted reconstruction poses a different challenge: how to best approximate redoing the entire multivariate analysis while holding fixed the vector diagram that was previously obtained. This strategy results in a nonlinear optimization problem that can be simplified to a unidimensional search. Various circumstances may warrant either projection or restricted reconstruction.

19 pages, 2 figures. This version corrects an oversight in Figure 2, replacing $w=(1,...,1,0) \in \Re^{n+1}$ with $w=(1,...,1,0)/n \in \Re^{n+1}$

Machine Learning (stat.ML); Machine Learning (cs.LG); Computation (stat.CO)

arXiv:2505.06756 [stat.ML]

(or arXiv:2505.06756v2 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2505.06756 Focus to learn more arXiv-issued DOI via DataCite

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