Do singular values of a point set determine its shape?

Question

Suppose I have $k$ points in $d$ dimensions. Let A be a $k\times d$ matrix with $i$th row giving the coordinates of $i$th point. Do singular values of this matrix have an interpretation as some kind of geometric invariant?

I tried a set of convex point sets formed by centroids of vertices of 8 dimensional 7-simplex, and there were 49 distinct singular value sets, which is the same as the number of such point sets not equivalent under coordinate permutation, obtained by Peter Shor in earlier post, I'm wondering if it's a coincidence

Answer 1

If I understand you correctly, you have certain sets of points which you would like to count up to permutations of the coordinates. This is the same as counting lists of points up to permutations of the coordinates and the order of the list. If the set of points is written as a matrix, this is the same as counting such matrices up to permutations of the rows and columns.

The list of singular values classifies a (real) matrix up to multiplication by orthogonal matrices on each side. Permutation matrices are orthogonal, so if two matrices are the same up to permuting rows and columns then they will have the same singular values. That is to say, the converse of the title of this question is true: the shape a point set determines its singular values.

Thus if there were 49 point sets in your original collection up to permutations of the coordinates, then you would get at most 49 lists of singular values. For an arbitrary collection of point sets it is quite possible that you could get fewer: multiplying a matrix by an orthogonal matrix need not yield a permutation of the rows and columns of the original matrix.

If your collection of point sets is chosen generically then no two of them will have the same list of singular values, so you will not get such a collapse, and you'll get the same number counting either way. Without performing both computations, it may be difficult to convince yourself that a particular collection of point sets (such as a higher-dimensional generalization of the problem you present) has this genericity.

EDIT: In general this type of genericity can fail badly in combinatorial applications. In particular if we look at the collection of Laplacian matrices of directed graphs, two such are equivalent up to multiplication by permutations matrices on both sides if and only if the underlying graphs are isomorphic. We cannot always distinguish nonisomorphic graphs by spectral properties like the singular values of the Laplacian. This corresponds to the fact that the collection of point sets given by Laplacians of graphs with a given number of vertices and edges can be "highly" nongeneric.

Answer 2

The singular values of this matrix are the square roots of the eigenvalues of the Gram matrix of the vectors, and can be viewed as the dimensions of the best approximating ellipsoid to the vectors (as per principal component analysis).