In the geometrical interpretation of SVD, the data points that we have need to be imagined as points in high dimensional space (say $d$-dimensional space). But we need to find a hyperplane in $k-$dimensional subspace that best fits the given data points
To gain insight into the SVD, treat the rows of an $n \times d$ matrix $A$ as $n$ points in a $d$-dimensional space and consider the problem of finding the best $k$-dimensional subspace with respect to the set of points.
My doubt here is about the uniqueness of $k$. Can we do decomposition for any $k \le d$ or for only certain values of $k$ or only for an unique $k$?
The paragraph is taken from the material on Singular Value Decomposition available here.