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

arXiv: https://arxiv.org/abs/2505.06756 · HTML: https://mlsanity.com/paper/2505.06756/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 20 references that are in the map.
- First posted on arXiv 10 May 2025.

## Builds on (most cited first, 1 of 1)

- [Landmark Diffusion Maps (L-dMaps): Accelerated manifold learning out-of-sample extension](https://mlsanity.com/paper/1706.09396/) (2017)

## Abstract (arXiv)

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.