02 / Local angles

Expected simplex skewness

Dimension, seen from a different angle.

Preparing the geometry lab…

Behind the estimate

A little geometry. A clearer picture.

How it works

Nearby vectors become less aligned as dimension grows. ESS translates their geometric spread into a local estimate.

\hat{s} = \frac{\sum_{i < j} ‖ v_{i} \land v_{j} ‖}{\sum_{i < j} ‖ v_{i} ‖ ‖ v_{j} ‖}

The vectors are centred on the neighbourhood mean. The wedge magnitude measures a parallelogram’s area.

Centre the neighbourhood.Take a point and its k neighbours, then subtract their common centroid.
Measure each vector pair.Compare the parallelogram area with the product of the two vector lengths.
Read the calibration curve.Match the weighted pair statistic to the theoretical dimension curve.

The plot is a view of the data. Distances, angles and covariance are computed in all generated coordinates. Noise, sampling and scale can change an estimate; the underlying noiseless manifold is only a reference.