hyperplane similarity

[gusl]

In 3D, two planes are similar if their unit normal vectors are similar (i.e. make a small angle).

What about k-dimensional hyperplanes in n dimensions?

Comments

[bhudson.livejournal.com]

They have (n-k) normals; if every normal of H1 is a small angle with every normal of H2, then yes, they're similar. No, that's total bullshit. I need to think more.

[bhudson.livejournal.com]

OK, right, there's an (n-k) space that is normal to the k-plane. Assume the k-plane goes through the origin (i.e. change coordinates to humour me). Then there's a linear transformation that describes the k-plane. It's an n x n matrix, but has rank k. The nullspace describes the normal plane.

I'm not sure exactly how to define (never mind compute) a measure of similarity though.

[gustavolacerda.livejournal.com]

I'm thinking one could represent the difference between two k-planes in min(k,n-k) vectors.
.. and you would combine the angles in the different directions in a Pythagorean way to get the total angle.

[bhudson.livejournal.com]

I forget how to define the total angle, and, miraculously, the intertubes are failing me. But it usually has to do with a cone; I'm not sure how it applies in the present case.

[rdore.livejournal.com]

I would want to say the maximum angle between any vector in one hyperplane and it's projection to the other.

[gustavolacerda.livejournal.com]

Nice! Any idea how to compute that?

Numerically, I think one can do convex optimization over the (hyperspherical) space of directions.

But I feel pretty sure that there is a simple way to compute it analytically.

[rdore.livejournal.com]

Call the subspaces V and W. Let U = V intersect W. Let T be the orthoganal complement of U. If you project down to U, I feel like it shouldn't change the answer, but might make it easier since then W and V would intersect in just a point.

[gustavolacerda.livejournal.com]

Projecting to U will definitely give a different angle than projecting to V. I have a picture with 2D planes in 3D.

Do you mean projecting to T?

[rdore.livejournal.com]

Sorry, yes. I meant to say that the angle between (projection of V to T) and (projection of W to T) feels like it should be the same as the angle between V and W. And (projection of V to T) and (projection of W to T) should intersect in a single point.

[gustavolacerda.livejournal.com]

and by "the angle between V and W", surely you mean what we defined earlier: "the maximum angle between any vector in one hyperplane and its projection to the other hyperplane".

[rdore.livejournal.com]

Yes.

[gustavolacerda.livejournal.com]

http://en.wikipedia.org/wiki/Kernel_%28matrix%29
<< The row space of a matrix A is the span of the row vectors of A. By the above reasoning, the null space of A is the orthogonal complement to the row space. >>

By slight abuse of notation, let matrix U be the row space of U, and let matrix T be the row space of T. Then T = nullspace(U), correct?


Tangentially, is nullspace(V intersect W) = span( {nullspace(V), nullspace(W)} ) ?

[rdore.livejournal.com]

Your notation is really confusing and I don't feel like working out what it means. Your notation should distinguish between:

1. A subspace
2. the basis of a subspace
3. a matrix with some specific column vectors

While these are related, these are different objects. Without the distinction, it is hard to tell what you are saying means, much less whether or not it is true.

[gustavolacerda.livejournal.com]

Let k=2, n=4.

Then U has dimension 1. Thus T has dimension 3.

Which intersection is just one point?

[gustavolacerda.livejournal.com]

Very tangentially, I'd like to look at the distributions of the cosine of random angles (uniformly distributed in the hypersphere).

In 2D space, the median will be sqrt(2)/2 ~= 0.70. This is trivial.

Claim: In high-dimensional spaces, most of the mass will be close to 0.

Proof:
We work with unit vectors. WLOG, we can choose the first vector so that the first entry is 1, and all others are zero. This way, the cosine (which is equal to the dot product) is equal to simply the first entry of the second vector.
The higher the dimension, the closer each component is concentrated around zero (the squares of every still add to 1; i.e. the pie isn't growing, but the party is). In particular, this applies to the first entry of the second vector.
QED

[gustavolacerda.livejournal.com]

Now, you talk about the maximum angle between a vector in the first hyperplane and its projection to the second hyperplane. In 3D, this vector (direction) corresponds to the gradient of the first plane, after a rotation that places the second plane flat.

What is the higher-dimensional analog of "placing flat"?

[darius.livejournal.com]

Pulling out _Geometric Algebra for Computer Science_, it has a section "The Angle Between Subspaces" -- looks like what you're after. I don't understand the discussion at first glance since I haven't read the preceding 68 pages. There's a formula that's obviously a generalization of the cosine formula for dot products.

[gustavolacerda.livejournal.com]

Thanks. It looks interesting.