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Pedro Lauridsen Ribeiro's user avatar
Pedro Lauridsen Ribeiro's user avatar
Pedro Lauridsen Ribeiro's user avatar
Pedro Lauridsen Ribeiro
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I want a smooth orthogonalization process
Moreover, there is no single orthonormal basis of eigenvectors of an eigenspace of a linear map since every vector in the eigenspace is an eigenvector with the same eigenvalue by definition. This of course applies to the vector subspace associated to an orthogonal projection $P$ ( = eigenspace of $P$ with eigenvalue $1$). In other words, even though the map $$k\text{-dimensional subspace } W\mapsto P_W=\text{orthogonal projection associated to }W$$ is well defined, it does not select a single orthonormal basis for any $W$ at all.
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I want a smooth orthogonalization process
I fail to understand what you're pointing at. Which map are you referring to? If you're referring to the orthogonal projection $P$ associated to a vector subspace, it's actually independent of the orthonormal basis chosen, even though you can certainly write the former in terms of the latter. Moreover, any such $P$ will act as the identity map in its image ( = vector subspace associated to $P$), so it cannot turn a non-orthogonal basis of that subspace into an orthonormal one since basis vectors of the subspace are unchanged by the action of $P$.
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I want a smooth orthogonalization process
If $\epsilon$ is large, then both bases are not close to each other and then there is no reason to expect that their GS orthonormalizations should be close to each other either. More importantly: why is this a problem to you? In either case the subspace generated by it is always the same.
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I want a smooth orthogonalization process
Just a question: is your original example actually encountered in the applications you are interested in? More precisely, do you actually expect there to occasionally flip the orientation of the basis due to e.g. rounding errors, as Ryan Budney suggests in his comment to the OP? If so, SVD methods still seem to be a good choice because they deal well with degeneracy problems such as the one encountered here thanks in large part to the distance minimization feature pointed in Vit's answer above.
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I want a smooth orthogonalization process
Your edit seems to be superfluous. By definition, basis vectors must be linearly independent and thus are nonzero. The GS procedure always takes bases to bases, but if you pick a finite set of linearly dependent vectors (particularly, if the set contains the zero vector) the GS procedure is no longer defined there because the orthogonalization part of the GS algorithm will then necessarily produce the zero vector, which cannot be normalized. Maybe this is at the root of some of your concerns in the OP...?
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I want a smooth orthogonalization process
As far as I know, SVD methods are somewhat preferred in machine learning applications, even as just part of more sophisticated methods.
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I want a smooth orthogonalization process
The Gram-Schmidt process is smooth (you can tell that just by looking at the formulae) - what is happening in your example is that the two bases in $\mathbb{R}^2$ provided in the second paragraph of the OP have different orientations and thus lie in two different connected components of the space of all bases in $\mathbb{R}^2$. In that case, the corresponding GS orthonormalizations will have opposite orientations as well (recall that the GS procedure preserves orientation) and thus don't need to be close to each other. The same is true for any finite-dimensional vector space.
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Tensor component calculation
Name spelling corrected, capital from "van der" removed, capital added in "General Relativity"
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Sobolev embedding theorems in vector bundles on non-compact manifolds
$C^l$ functions are not necessarily bounded w.r.t. $C^l$ norm w.r.t. $g$, so you're actually talking about bounded $C^l$ functions w.r.t. $g$, thus the range space $C^l_b$ when you ask for a "continuous" embedding. The space $C^l$ is not a Banach space if $M$ is not compact - you have to take the sequence of $C^l$ seminorms on a compact exhaustion of $M$ (e.g. closed balls with the same center and all natural radii if $(M,g)$ is complete) to define a (Fréchet) topology on $C^l$, which btw doesn't depend on $g$. The Sobolev embedding is continuous w.r.t. the latter if $(M,g)$ is complete.
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Sobolev embedding theorems in vector bundles on non-compact manifolds
2.) The Sobolev embedding theorem holds for all closed balls of $(M,g)$ if the latter is complete (since then any closed ball of $(M,g)$ is a compact Riemannian manifold with Lipschitz boundary, by the Hopf-Rinow theorem). This means that $H^k$ functions are $C^l$ for $k>l+\frac{\dim(M)}{2}$ but not necessarily bounded w.r.t. the global $C^l$ norm w.r.t. $g$. In that regard, recall as well that by the Nomizu-Ozeki theorem any Riemannian metric is conformal to a complete one.
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Sobolev embedding theorems in vector bundles on non-compact manifolds
I have a couple of clarifications to ask: 1.) When you say that the Sobolev embedding theorem doesn't necessarily hold for a non-compact Riemannian manifold $(M,g)$, are you referring with respect to which definition of Sobolev spaces? It most certainly holds true with respect to $H^k_0$ (i.e. the completion of $C^\infty_c$ in the $H^k$ Sobolev norm w.r.t. $g$) since one can then argue locally using charts and a finite partition of unity.
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Vector field connecting two points
visual notation improvement
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Solvability of Yang-Mills equations
I don't know if this may be of interest to the OP, but it should also be remarked that the result holds in practically${}^*$ the same generality for any globally hyperbolic 4-dim. space-time, see P. T. Chrusciel and J. Shatah, Global Existence of Solutions of the Yang-Mills Equations on Globally Hyperbolic Four-Dimensional Lorentzian Manifolds, Asian J. Math. 1 (1997) 530-548. (${}^*$ - pure Yang-Mills, with no Higgs coupling. It's also assumed that the Lie algebra of the gauge group has a faithful rep. as a matrix subalgebra)
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