On the smooth geometry of intersections of quadrics

[Jesus Briales]

Hi everyone,

there is a particular (yet general) situation which I have been encountering for a while

in different geometric problems, so I think it might be interesting to discuss it here.

I am referring in particular to constraints on a real mxn matrix X, (X in Re^{mxn})

of the form `tr(X'*B*X)=c`, where the nxn matrix B is symmetric but not necessarily positive semidefinite.

This is quite generic, as this defines a *quadric* on the elements of X.

As such, sets of this constraints could stand for:

- A simple sphere: B = Id, c=0

- A *non-smooth* cone: B = diag(+1,-1), c=0

- A rotation matrix, considering the orthogonality constraints separately.

However, the cases I've been encountering don't seem to adjust to any of these particular geometries.

So, my main question is:

- How to check if the combination/intersection of several of these `tr(X'*B_i*X)=c_i` constraints

  for particular `B_i` and `c_i` yields a *smooth matrix manifold* whose geometry

  could be exploited for optimization in the Manopt fashion?

I assume there is no general simple answer for this,

but still any directions or comments would be highly appreciated.

The resources I've looked into or found so far are:

- The generalized Stiefel manifold, for the case B is positive semidefinite.

  However, in the cases I have encountered B is indefinite

  (often with binary signed eigenvalues 0,+1,-1).

- In general, there seems to be an notable connection with moment-angle manifolds

  and under certain conditions it seems this kind of intersections of quadrics

  should yield a smooth manifold.

  An ingredient that seems to play an important role

  is the transversality of the intersection.

  See:

  - "Moment-agle manifolds and complexes. Lecture notes"

    https://arxiv.org/abs/1008.5047v2

  - "Geometric structures on moment-angle manifolds" by Taras Panov

    https://arxiv.org/abs/1302.2463v1

- The following paper:

  "Intersections of Quadrics, Moment-angle Manifolds and Connected Sums"

  by Samuel Gitler and Santiago Lopez de Medrano

  https://arxiv.org/abs/0901.2580

  seems to provide some important and general results

  on the geometry of quadric intersections, but from

  my limited knowledge on topology I hardly understood anything.

If you think this is worth to discuss in more particular deployments,

I can include details on the different cases of interest I've encountered

here or in a different post.

Best regards,

- Jesus

[Nicolas Boumal]

Hello Jesus,

This is an interesting question, and I only have a partial answer.

If I understood correctly, you are asking: given a set of constraints trace(B XX') = c (for a collection of B's and c's), how do we know that M (the set of X's that satisfy all constraints) is a smooth manifold?

Notice that this is connected to semidefinite programming (SDP), since, if we let Y = XX', then we find that Y is positive semidefinite and obeys linear constraints trace(BY) = c. The number of columns of X further constrains (potentially) the rank of Y, which is not normally allowed in SDP's aside for quite special cases. I have a paper with Afonso and Vlad about something closely related. Basically, we say in that paper that, if the set of constraints indeed makes M into a smooth manifold, then optimization on manifolds is a good way of solving the SDP (very roughly stated):

https://papers.nips.cc/paper/6517-the-non-convex-burer-monteiro-approach-works-on-smooth-semidefinite-programs.pdf

In that paper, we briefly address that question of identifying when M is a manifold. Basically, the rule is ans follows: M is a submanifold of R^{n x m} if M can be linearized at each point X into a tangent space T_X M and all of these tangent spaces have the same dimension (in the paper, there is a precise reference to the relevant lemma or proposition in Absil's book to support that claim.)

If M = {X in R^{n x m} : trace(B_k XX') = c_k for k = 1 .. K},

then, for X in M, we can differentiate the constraints to get the tangent space:

T_X M = {U in R^{n x m} : trace(B_k(UX' + XU')) = 0 for k = 1 .. K}

           = {U in R^{n x m} : trace(B_k X U') = 0 for k = 1 .. K}

The normal space is spanned by the matrices B_k X; for example, if they are linearly independent for k = 1 .. K, then the normal space has dimension K, and the tangent space has dimension nm - K. If this is independent of X, then M is a smooth submanifold of R^{n x m}.

So, there is one answer.. but of course, in general, it may be difficult to determine whether this is the case or not.

Best,

Nicolas

[Jesus Briales]

Hello Nicolas,

Great answer as usual! :) I was aware of much of the involved materials but your comments have been the key ingredient to put it all together :)

Actually my situation arises from a SDP problem and it was precisely the will to try a "Riemannian staircase"-like approach that pushed my interest on analyzing the potential smoothness of these more general manifolds.

I will try then analyzing if the set of matrices B_k X are linearly independent and let you know if it stands for any case of interest.

Thanks again,

Jesus Briales