Horizontal Lift and Riemannian Metric of the Quotien Manifold

[lineya...@gmail.com]

Hi Nicolas and BM,

I'm very interested in how the manifold optimization works, and when I read the book "optimization on matrix manifolds", I feel confused about some difficulties on the horizontal lift and Riemannian metric of the quotient manifold.

Let M = M_bar / ~ be the quotient manifold, and M_bar is the abstract manifold. Then I see that from the Riemannian metric g_bar of the abstract manifold, we can define a Riemannian metric  as   g_x( ξ, ζ ) =  g_bar_x( ξ_bar, ζ_bar)  and ξ_bar, ζ_bar are the horizontal lift of ξ, ζ  (equation 3.38).   However, as it shows in example 3.6.4 for Grassmann manifold. I only see the expression for g_bar, and the horizontal space is derived depending on g_bar. Then I cannot get the expression for g_x( ξ, ζ ).  Here I want to check that the Riemannian metric g_bar is actually invariant along the equivalent class [x]. The example proves that for Grassmann manifold we prove ξ_bar_{YM} = ξ_bar_{Y} · M， then 

g_bar_{YM}( ξ_bar_{YM}, ζ_bar_{YM}  ) = g_bar_{Y}( ξ_bar_{Y}, ζ_bar_{Y}).

I want to ask that:

1) The horizontal lift of  ξ is given by ξ_bar = P_x^h (ξ), i.e. the projection onto the horizontal space? Then I'm a little confused about the definition 

Recall that the horizontal lift at x ∈ π−1(x) of a tangent vector ξx ∈ TxM is the unique tangent vector ξx ∈ Hx that satisfies Dπ(x)[ξx]. [page 48,  Sec 3.6.2]

What's the difference of TxM and TxM_bar ? As far as I can see,  I think it says that any tangent vector ξ at [x] can all be represented by a unique vector ξ_bar in the horizontal space Hx and the horizontal space is irrelevant with the choice of x in the equivalent class [x].

2) For other manifold which is not the Grassmann manifold, how should we prove the the induced Riemannian metric g_x( ξ, ζ ) =  g_bar_x( ξ_bar, ζ_bar)  is invariant along the equivalent class? For example, as in the symmetricfixedrank case,   g_bar_x (ξ_bar, ζ_bar) = trace(ξ_bar^T ζ_bar), and M_bar = R_*^{n×p},   pi :  Y → YQ,  Q is orthogonal matrix (Q^TQ=I).  Or we actually needn't prove case by case? Does it always hold and we don't have to get the expression for the Riemannian metric g for M?

Sorry for that my questions are a little cumbersome, and I really appreciate for your help on understanding your brilliant works.

Thanks a lot!

Kai Yang

[Nicolas Boumal]

Hello Kai,

This is a rather wide question. Let me try to address the salient points with some pointers:

1) "the horizontal space is irrelevant with the choice of x in the equivalent class [x]." : As I understand it, this is not correct. The horizontal space is a subspace of the tangent space to the "total" manifold: that does depend on the representative you pick for the equivalent class.

2) About establishing that you pick an acceptable metric wrt the quotient, besides examples in the book, you can find two examples where this is done in detail in two of my papers:

 https://arxiv.org/abs/1211.1621  Section 3, and specifically Section 3.2 for what happens when you take the quotient

 http://www.sciencedirect.com/science/article/pii/S0024379515001342 Section 2 and a little bit of Section 3.

I hope these detailed examples can help.

Best,

Nicolas

[lineya...@gmail.com]

Hi Nicolas,

Thanks for your helpful explanations. I read the two papers you mentioned, and I find that in these papers

1) Riemannian metric on the abstract manifold is developed (or inherited from the Euclidean space). Then the Riemannian metric on the quotient manifold is defined with respect to the horizontal lift of tangent vectors, and they claim that this metric is invariant along the equivalent class.  It's the case for  

equation (3.20) in  https://arxiv.org/abs/1211.1621   "Cram er-Rao bounds for synchronization of rotations"    and   http://www.sciencedirect.com/science/article/pii/S0024379515001342 Section 2 "Low-rank matrix completion via preconditioned optimization on the Grassmann manifold"

Does it mean that if we find the correct horizontal lift operation, then this kind of definition for Riemannian metric on the quotient manifold is always invariant along the equivalent class? 

2)I also find that the retraction operation is also defined in the abstract manifold, then on the quotient manifold retraction is defined with horizontal lift and canonical projection. So when we develop an algorithm, we don't exactly need to write down the expressions for Riemannian metric and retraction for the quotient manifold M,  but to compute them using horizontal lift and Riemannian metric and retraction on the abstract manifold M_bar. Is that correct?

I bother you again since I have limited knowledge about manifold (all from the book Optimization on Matrix Manifold and your papers/examples/tutorials)  and I want to get it right. Thank you again!

Kai Yang

[Nicolas Boumal]

One small thing first: when considering M = N / ~, where ~ is an equivalence relation, I would call N the total space and M the quotient space. Typically, N would be a rather concrete space (such as a set of matrices, e.g., Stiefel), and M would be more abstract (in the sense that it is a set of equivalence classes of matrices, e.g., Grassmann). Si I might call M an abstract manifold. I suspect that what you call the "abstract manifold" is actually N, isn't it?

 

1) Does it mean that if we find the correct horizontal lift operation, then this kind of definition for Riemannian metric on the quotient manifold is always invariant along the equivalent class?

I'm not sure how to answer this question, since the "correct horizontal lift" may lead to a circular reasoning here. These papers (at least one of them) have precise references to either Boothby or Chavel (as I recall) for precise statements on that topic.

As for question 2), you may not need to write things down in the abstract space for computational purposes (since, indeed, everything computational will happen in the horizontal space), but it's still necessary to make sure that operations in the horizontal space have the proper invariances, i.e., that they amount to proper operations on the quotient space. This is what the papers references explain.

I'm sorry that this is not a lot more specific. These are technical points that require quite a bit of space and time to get right, which is why I highly recommend studying the books.

Best,

Nicolas

[lineya...@gmail.com]

Hi Nicolas,

Yes I did mean N as the "abstract manifold", and thanks for your suggestions. Actually I've been much enlighted now from what you said. I really appreciate it and I'll go on reading the part of horizontal lift and invariance following what you refered.

Best,

Kai