Calculation of Intrinsic dimension of datasets

Question

Currently I am following a Machine learning course and we are looking at the intrinsic dimension of datasets. The professor gave a few examples of the intrinsic dimension of some objects (ej. the lungs have an intrinsic dimension of 2.89). However now we need to calculate the intrinsic dimension of the following objects :

1)Suppose $A= \{0,1,2,3,…,25\} $ What is the asymptotic intrinsic dimension of $A$?

2) Suppose $A=[0,1]^5$ is the five-dimensional cube. What is the asymptotic intrinsic dimension of $A$?

3)Suppose $B_{10}=\{x⃗ ∈R^{10}$ such that $ ∥x⃗ ∥≤3\} $ is the 10-dimensional ball. What is the intrinsic dimension of $B_{10}$ ?

4) Suppose $S_{10}=\{x⃗ ∈R^{10}$ such that ∥x⃗ ∥=3 } is a 10-dimensional sphere (the surface of a ball}. What is the intrinsic dimension of $S^{10}$ ?

However I have read a few slides and I still don't understand how to do it. I anyone could give me a step by step explanation in very layman's terms I will thank him.

Answer 1

1. 0

because there are no dimensions, just integers in a set

2. 5

the dimension is the exponent of epsilon in the general formula for dependence of number of elements on diameter

3. 10

there are 10 different balls in R^{10}. And the dimension of the ball is 10

4. 9

since ||x|| = 3