Talk:Frame (linear algebra)/Archive 1

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Unclear sentence

Added section title. —Nils von Barth (nbarth) (talk) 18:19, 2 April 2009 (UTC)

From the section Relation to bases:

Consequently, a frame is a set of vectors which [...] cannot be any arbitrary set of vectors which spans V.

That sentence doesn't make any sense to me at all. Read on its own, it may be construed to mean that no set of vectors which spans V is a frame, which of course is wrong. (Only sets which span V can possibly be frames.) I know it wasn't meant to be read that way, but the wording seems very ambiguous and unclear to me. Does someone feel up to improving that part? Eriatarka 10:57, 28 April 2007 (UTC)

The sentence is meant to point out the fact that you cannot take an arbitrary set of vectors which span a vector space and call them a frame of that vector space, ie, a frame is not just a simple generalization of a basis where the linearly independence has been dropped. The 'frame condition' is more strict than just dropping linearly independence. The set of vectors presented before that sentence is an example of a set which span a vector space but does not satisfy the 'frame condition' and therefore is not a frame. I don't understand in what way the sentence is ambiguous, can you please give an example? --KYN 07:49, 29 April 2007 (UTC)

I read it as ambiguous in the sense that I mentioned above:

>>Read on its own, it may be construed to mean that no set of vectors which spans V is a frame, which of course is wrong.<<

I would simply formulate the statement as "Not every set of vectors which spans V is a frame of V." or "Spanning the space V is not a sufficient condition for being a frame." or maybe "There are sets of vectors spanning V which are not frames of V." These all seem much clearer to me. Eriatarka 17:53, 29 April 2007 (UTC)

The suggested formulations are all fine to me --KYN 19:32, 29 April 2007 (UTC)

The word "frame"?

I like having a sense of why mathematical terms are called what they are. I'm thinking of adding this picture and caption regarding the intuition of the word "frame".

Is that accurate? Could it be phrased better. Comments? —Ben FrantzDale 01:36, 3 May 2007 (UTC)

The term frame is used in mathematics in at least two different ways. One is the meaning related to a coordinate frame or (more commonly) a coordinate system, which is what I believe that your are thinking about here. Another is the meaning described in this article which can be described as a specific way of generalizing a basis (spans a space and is linearly independent) to something which is spans the space but is not required to be linearly independent. Instead it must satisfy the frame condition described in the article. These two concepts are quite different and should not be confused. The picture you have posted here appears to be related to the first meaning of frame but not (in a way that I understand) related to the second meaning. Personally, I try to avoid using frame in the first sense since there already is a well-established term coordinate system which can be used instead and which (I believe) is more common. --KYN 10:14, 3 May 2007 (UTC)

The term frame used in the sense described in the article way, as far as I understand it, coined by Duffin & Schaeffer in their article listed in the references. Is see now that the Frame article has changed its description of this article. I will need to change it back again. --KYN 10:14, 3 May 2007 (UTC)

OK, let me try to get an intuitive understanding. Would it be accurate to say it's called a "frame" because it is a thing on which a vector space is built (akin to the frame of a building)?

Sorry, no idea why they called this particular construction a frame. The similarity with a frame of a building sounds plausible but I have also noticed that there are other meanings of that word which may fit; frame as in boundary of something.

If the definition is basically just "it's like a basis with the possible addition of 'extra' vectors that are linear combinations of the basis vectors", then would it be reasonable to say that in the 2-D image I posted above, treating it as a 2-D image, the vectors x, y, and z form a frame for R²?

Yes, this is correct.

A non-example given is

{ (1,0) , (0,1), (0,1) , (0,1) ... }

but that's not a set. that's a multiset. I think a set that has similar properties would be

{ (1,0), (0,1), (0,1.5), (0,1.75), ... }. Am I right that that set is not a frame?

The multiset idea is, I believe, a matter of taste. As far as I understand it, you need a set of vector (elements of the vector space) which are enumerable and if some of them are identical or not is not a big deal. But you are right that this last set is not a frames since the upper frame bound B = infinity.

The multiset example should be replaced as there are many examples which can be constructed without applealing to nonstandard set theory. I think that { (1,0), (0,1), (0,1/\sqrt{2}), (0,1/\sqrt{3},...) would do the trick as the sum of inner products squared with (0,1) is the harmonic series which diverges. I'm not a set theorist but innocent looking constructions can eaisly lead to paradoxes so we must be cautious. —Preceding unsigned comment added by 68.239.85.185 (talk) 19:15, 21 July 2008 (UTC)

Would the set

{ (1,0), (0,1), (0,1/2), (0,1/4), (0,1/8),... }

be a frame for R²? Those vectors do span the space but aren't linearly independent. However, they do seem to satisfy the frame condition.

Yes, this is an example of a set of vectors which constitute a frame. But is doesn't have to be infinite. These two sets of vectors in R² are also frames

{ (1,0), (0,1), (0,1/2) }

{ (1,0), (0,1)}

Thanks. —Ben FrantzDale 12:07, 3 May 2007 (UTC)

I believe that the essential idea behind this particular generalization of bases to sets which are not linearly independent is that you want to be able to construct a dual frame, i.e., a dual set of vectors such that if you have the scalar products of a vector v relative to the frame vectors, you can reconstruct v by linear combinations of the dual frame and the scalar products. This is in analogue to the relation between a basis and its dual (or reciprocal) basis. What Duffin & Schaeffer did was to prove that the frame condition is a sufficient (I don't know about necessary) condition for a dual frame to exist. --KYN 22:00, 3 May 2007 (UTC)

As I understand any finite set which include basis is a frame. Does that mean that only infinite frames are really interesting ? Or for finite frames object of interest is exact bounds A and B ? Serg3d2 06:03, 8 July 2007 (UTC)

The practical use of frames is not limited to the infinite dimensional cases. In a sense, frames introduce redundancy, regardless of whether the dimensionality is finite or infinite, which can be used, e.g., for noise reduction in signal processing. Oversampling is a good example of how frames can be used precisely in this way and it works also for a finite set of samples. --KYN 10:25, 8 July 2007 (UTC)

Incorrect Characterization

This characterization is incorrect so I removed it. Consequently, a frame is, informally, a set of vectors which:

spans V,
are allowed to be linearly dependent
should be limited in number.

The cardnality of a frame need not be finite. —Preceding unsigned comment added by 68.239.85.185 (talk) 19:38, 21 July 2008 (UTC)

Confusion

The article seems to suggest that there are two definitions for a frame. The first is any ordered basis of a vector space. The second is any set which satisfies the frame condition (which is automatically satisfied by any basis). The first definition means that all frames are bases (but not all bases are frames) while the second definition implies that all bases are frames (but not all frames are bases)! Do I have this right?! 203.167.251.186 (talk) 02:39, 5 January 2009 (UTC) DRB

You are right as long as we are talking about two different definitions of two different concepts which, unfortunately, have been given the same name. I'v tried to make this more clear in the intro, and also more clear that the article is about the "second" definition. --KYN (talk) 23:04, 5 January 2009 (UTC)

About the recent edits

Recently, User:Nbarth did some edits to extend on the second definition of frame of a vector space in terms of ordered sets of linearly independent vectors. Altough this was with good intent, I don't believe that the result came out well: (1) the new content came "on top" of the rest which was the original article on the extention ogf bases to linearly dependent vectors and (2) there is no clear transisiton between the two defintions in the text. The previos version mentiones the second defintion and the new edits don't provide much more insight to this concept to motive the changes. Therere, I propose to revert to the 2009-02-11T21:07:50 version of the article, with some newer edits added. One possible solution would be to separate the two defintions (which are clearly distinct and unrelated) into separate articles. --KYN (talk) 08:40, 2 April 2009 (UTC)

Hi KYN – sorry about the mess.

I’ve split it the other definition off into a new page, k-frame, and the current “Frame of a vector space” page is essentially what it was at the revision you mention, plus an “otheruses” link and “See also”.

I’ve also added k-frame to the disambiguation page, and a discussion of the distinction at Frame (linear algebra), as it bears mention but is too long for a disambiguation page, and fixed links to the page (to point to k-frame if appropriate).

By way of context, I was writing about the Stiefel manifold, hence wanted to link to k-frames, and found this page.

Hope this fixes everything!

—Nils von Barth (nbarth) (talk) 18:47, 2 April 2009 (UTC)

Thanks, this solution works fine. --KYN (talk) 10:05, 13 April 2009 (UTC)

Decomposition to a frame

I'm a bit hazy about how one takes a vector and decomposes it using a frame. In as much as a frame could be a non-orthogonal basis for the space, it seems like you'd need to take the inner product of your vector with the duals of the basis vectors. It seems like having more than a complete basis may make things even more complicated.

If I have a frame for a high-dimensional vector space, how would I go about producing the appropriate dual frame? In applications like data compression with wavelets or curvelets, one step is to throw out low-valued coefficients; if you don't have an orthogonal basis but rather have a general frame, wouldn't you sometimes wind up wanting to move weight from one vector in your frame to another non-orthogonal vector? For example, suppose you had a frame consisting of {(1/2,1), (1,0), (1,1)}. First, how would you decompose the vector, say (1,1.1) into that frame. Second, since the frame has non-orthogonal vectors, if I have three coefficients corresponding to the three frame vectors and replace the smallest coefficient with zero, won't I wind up reconstructing a vector that is fairly far from (1,1.1) whereas I clearly could set one coefficient to zero and adjust the other two coefficients to reproduce (1,1.1) exactly.

Feel free to correct me if my terminology is a little off. :-) —Ben FrantzDale (talk) 02:12, 3 April 2009 (UTC)

There is now a whole section on the construction of the dual frame. --KYN (talk) 21:12, 18 April 2009 (UTC)

That was very helpful. Thanks! I really like that builds on "The frame condition is both sufficient and necessary for allowing the construction of a dual or conjugate frame", which helps motivate the frame condition itself.

I would still be interested in an answer to the question of dimensionality reduction in the case of a frame with more DoFs than the space it spans... —Ben FrantzDale (talk) 21:07, 19 April 2009 (UTC)

Exactly what question of dimensionality reduction do you mean? --KYN (talk) 08:11, 20 April 2009 (UTC)

I think I understand this better now. I had somehow thought that, in the context of signal compression, there was some nonlinear step of decomposing a vector in which you picked a basis from the frame where the basis was somehow the most efficient representation of the signal. So for example, if I wanted to compress a 2D signal where the signal always had much higher frequency in one direction than the other, a conventional wavelet basis wouldn't be optimal, but I could imagine some stretched basis that could efficiently capture that. I see now that I was overthinking it: decomposing a vector, v into a frame is just a matter of taking the inner product with all of the dual frame vectors to get the components. I assume, then, compression would be done by skipping low-energy components—simple, just like using a basis. So then the frame vectors and dual frame vectors are a direct replacement for the basis and dual basis. Simple! —Ben FrantzDale (talk) 13:10, 1 September 2011 (UTC)

Question +

The text says that a set X of linearly dependent vectors may loose its ability to "expand" some vector V if elements of X are removed in order to get a linearly independent set. I cannot see how this could be true. Can anyone give an explanation? —Preceding unsigned comment added by 68.124.183.73 (talk) 17:54, 1 May 2011 (UTC)

The word "expand" is not used in the article, can you be more precise about which part of the text you are refering to? --KYN (talk) 07:35, 2 May 2011 (UTC)

Big Whoops

The article misses a very important point with respect to dual frames. A given frame does not have a unique dual frame. The dual frame given in this article, known as the canonical dual frame, is an example of one dual frame corresponding to a given frame. Each dual may have special properties that may be of advantage in the synthesis of the elements of a function space. — Preceding unsigned comment added by 128.32.52.215 (talk) 23:04, 25 January 2012 (UTC)

As usual in these cases, you are free to update/extend the article to include this proposal, in particular if it can be backed up by references. --KYN (talk) 10:28, 26 January 2012 (UTC)

Necessity of the frame condition

The frame condition is not necessary for the existence of a dual frame. Actually, the example given of a set of vectors spanning R² for which the frame condition fails is also an example illustrating that it is not a necessary condition: just take $\{(1,0),(0,1),0,0,\ldots \}$ as the dual frame. I have therefore removed the adjective "necessary". In any case, there should be more references for this article.Hairer (talk) 23:38, 4 December 2011 (UTC)

Notation for inner products

The notation used in this article, $\langle a|b\rangle$ , is likely to confuse physicists. The expression $\langle v|e_{j}\rangle e_{j}$ is a good example. To a physicist, this is conjugate-linear in the vector $|v\rangle$ , linear in the vector $|e_{j}\rangle$ , and $e_{j}$ is a scalar. Physicists would write the meaning intended in the article as $|e_{j}\rangle \langle e_{j}|v\rangle$ . I think the article should replace $\langle a|b\rangle$ with $\langle a,b\rangle$ , a notation often used by mathematicians to mean what the article intends. — Preceding unsigned comment added by Thisrod (talk • contribs) 00:29, 26 September 2013 (UTC)