The expected edge length of a cube is equal to $e_{D}(f) = f^{1/D}$, what is the definition of `the expected edge length`?

Question

section 1.4.3 of the book "Machine Learning - A Probabilistic Perspective" gives an example about KNN:

the input is two dimensional, we have three classes, and K = 10

here is Figure 1.15

here is Figure 1.16

the expected edge length of this cube

$e_{D}(f) = f^{1/D}$

what is the definition of the expected edge length?

Answer 1

There the fraction $f$ is ment to give the $D$-dimensional volume of some "expected hypercube", which is designed such that it contains this point subset. If $s$ is the side length of the hypercube, then its volume is given by $s^D$. When equating those 2 values and solving for $s$ you'd just get $s=f^{1/D}$.

--- rk