What is the difference between Intrinsic dimension and Embedding dimension of the data I have read in this paper that there are Intrinsic dimension and the Embedding dimension of the data. 
I am not familiar with these two concepts. The only I know is that the number of variables in a dataset is referring to the dimension of the data.
I have tried to read everywhere before asking this question.
So, I would appreciate your help in easily explaining these two concepts in general.  
 A: Quoting directly from the third paragraph in the introduction of this reference paper, which explains the concept quite intuitively:
"We define the embedding dimensionality of a dataset as the number of attributes of the dataset (its address space). The intrinsic dimensionality of a phenomenon (and also of the data retrieved from it) is defined as the real number of dimensions in which the points can be embedded while preserving the distances among them. For example, a plane embedded in a 50-dimensional space has intrinsic dimension 2 and embedding dimension 50."

A: In the paper the embedding dimension is the number of input variables, whilst the intrinsic dimension is the minimum number of variables needed to represent a point in the dataset.
Eg x1=X, X2=X*X then the embedding dimension is 2, but the Intrinsic dimension is 1. Similarly could have x1=sin(theta), X2=cos(theta) ....
In addition it does not have to be a perfect mapping: they are allowing a statistical relationship between input variables (is with some noise)
