High-Dimensional Geometry: Hypercubes and Gaussian

Concentration of Volume of a Hypercube

A p-dimensional unit hypercube is the subset of defined as

• The hyper cube has vertices

• Therefore, the maximum length between any two points admits

• As p increases,dmax also increases, therefore, the corners tend to stretch
• Since the volume is unity, the rest of thehypercube should shrink to keep the volume fixed
• The volume seems to concentrate at the corners as

Concentration of Volume of a Hypercube at Its Corners

Gaussians in High Dimension

Practical Implications

1. Distance Metrics: Euclidean distances become less meaningful in high dimensions.
2. Normalization: Data should often be scaled to a unit sphere or hypercube.
3. Sampling: Random points in high dimensions are almost always near edges/shells.