# Positive semi-definite embedding for dimensionality reduction and out-of-sample extensions

arXiv: https://arxiv.org/abs/1711.07271 · HTML: https://mlsanity.com/paper/1711.07271/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Nov 2017.

## Built on by (most cited first, 1 of 1)

- [The Hyperspherical Geometry of Community Detection: Modularity as a Distance](https://mlsanity.com/paper/2107.02645/) (2021)

## Abstract (arXiv)

In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space ℝd. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the solution of an infinite dimensional analogue of a semi-definite program. This embedding is adaptive and non-linear. We discuss this problem both with weak and strong smoothness assumptions about the learned kernel. A main feature of our approach is the existence of an out-of-sample extension formula of the embedding coordinates in both cases. This extrapolation formula yields an extension of the kernel matrix to a data-dependent Mercer kernel function. Our empirical results indicate that this embedding method is more robust with respect to the influence of outliers, compared with a spectral embedding method.