Sammon mapping

Sammon projection or Sammon mapping is an algorithm that maps a high-dimensional space to a space of lower dimensionality (see multidimensional scaling) by trying to preserve the structure of inter-point distances in high-dimensional space in the lower-dimension projection. It is particularly suited for use in exploratory data analysis. The method was proposed by John W. Sammon in 1969. It is considered a non-linear approach as the projection cannot be represented as a linear combination of the original variables as possible in techniques such as principal component analysis, which also makes it more difficult to use for classification applications.

Denote the distance between ith and jth objects in the original space by $\scriptstyle d_{ij}^{*}$, and the distance between their projections by $\scriptstyle d_{ij}^{}$. Sammon's projection aims to minimize the following error function, which is often referred to as Sammon's stress:

$E={\frac {1}{\sum \limits _{i<j}d_{ij}^{*}}}\sum _{i<j}{\frac {(d_{ij}^{*}-d_{ij})^{2}}{d_{ij}^{*}}}.$

The minimization can be performed either by gradient descent, as proposed initially, or by other means, usually involving iterative methods. The number of iterations need to be experimentally determined and convergent solutions are not always guaranteed. Many implementations prefer to use the first Principal Components as a starting configuration.

External links

• HiSee - an open-source visualizer for high dimensional data
• A C# based program with code on CodeProject.
• Matlab code and method introduction
• Extending Sammon mapping with Bregman divergences

• Extending metric multidimensional scaling with Bregman divergences