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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.

Oliver
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    I did a bit of googling, and it's not easy to find a definition of intrinsic dimension, let alone asymptotic intrinsic dimension that would apply to sets. What definition do the slides give? Also if I had to guess what the dimensions of those four sets are, they'd be 0, 5, 10, 9. But that could certainly be wrong depending on what the definition turns out to be. I've just given their dimensions as manifolds (with boundary). – jgon Jun 13 '19 at 22:59
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    I want to second @jgon. Lungs, and trees, have a fractal dimension usually between 2 and 3, but that has a very specific geometric meaning and I've certainly never heard it called 'intrinsic'. There are many different types of dimension, so the definition you're supposed to be using is important for answering the question. – Sort of Damocles Jun 13 '19 at 23:02
  • @jgon here are the slides that were provided [https://mas-dse.github.io/DSE230/pdf-slides/IntrinsicDimension.pdf]. I do understand the examples that they provide but it still somewhat obscure to me – Oliver Jun 15 '19 at 01:15

1 Answers1

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because there are no dimensions, just integers in a set

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

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

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since ||x|| = 3

Konain Niaz
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