Sammon's projection

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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\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.

[edit] Software

Sammon's projection is supported by R (package MASS) and by SOM toolbox, a free functional package for Matlab.

[edit] Bibliography

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