Living in Higher Dimensions

Sir Professor David Mackay revolutionised Machine Learning. There’s no question about it. His abundance of knowledge was clear from his research, his selflessness, and his ground breaking work on Information Theory . Both the fields of Gaussian Processes and Neural Networks owe him a lot.

During my studies I came across the following question in his book which, unbeknown to me at the time, would revolutionise my way of thinking about Machine Learning.

The question itself is nothing more than a graduate level maths question, but some things register in different ways. In particular, I always visualised probability as maybe a 2 or at most 3 dimension problem. This was wrong.

Multivariate studies are hard to understand and harder to visualise, but imagine that the density in ‘space’ or ‘object’ was of a uniform distribution. That is, that density is uniformly distributed throughout the space. From here, the problem conjects that:

Probability distributions and volumes have some unexpected properties in high-dimensional spaces.

Consider a sphere of radius r in an N -dimensional real space. Show that the fraction ( f )of the volume of the sphere that is in the surface shell lying at values of the radius between r − ϵ and r, where 0 < ϵ< r , is:

Further, evaluate the function for N = 2, 10, 1000 and for ϵ/r=0.01, 0.5.

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