How to interpret results of dimensionality reduction/multidimensional scaling?

Question

I performed both a SVD decomposition and a multidimensional scaling of a 6-dimensional data matrix, in order to get a better understanding of the structure of the data.

Unfortunately, all the singular values are of the same order, implying that the dimensionality of the data is indeed 6. However, I would like to be able to interpret the values of the singular vectors. For instance, the first one seems to be more or less equal in each dimension (ie (1,1,1,1,1,1)), and the second also has an interesting structure (something like (1,-1,1,-1,-1,1)).

How could I interpret these vectors? Could you point me to some literature on the subject?

Answer 1

If the singular values are precisely equal, then the singular vectors can be just about any set of orthonormal vectors, therefore they carry no information.

Generally, if two singular values are equal, the corresponding singular vectors can be rotated in the plane defined by them, and nothing changes. It will not be possible to distinguish between direction in that plane based on the data.

To show a 2D example similar to yours, ${(1, 1), (1, -1)}$ are just two orthogonal vectors, but your numerical method could just as easily have given you ${(1,0), (0,1)}$.