Sammon mapping

Sammon's projection, or Sammon's mapping is an algorithm that maps a high-dimensional space to a space of lower dimensionality (see multidimensional scaling).

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:

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

The minimization can be performed either by gradient descent, as proposed initially, or by other means.

Software

Sammon's projection is supported by R (package MASS) and by SOM toolbox, a free functional package for Matlab. An implementation in C# can be found at CodeProject.

Bibliography

• J. W. Sammon, Jr. "A nonlinear mapping for data structure analysis". IEEE Transactions on Computers, 18, pp. 401–409, 1969.