Talk:Vapnik–Chervonenkis dimension

message: "A classification model f with some parameter vector \theta is said to shatter a set of data points (x_1,x_2,\ldots,x_n) if, for all assignments of labels to those points, there exists a \theta such that the model f makes no errors when evaluating that set of data points."
:To me, it seems that this is saying that f shatters X (with |X|=n), if and only if f has zero classification error (for some \theta) for every possible labeling of X.
"VC dimension of a model f is h' where h' is the maximum h such that some data point set of cardinality h can be shattered by f."
:To me, it seems like this is saying f shatters X if and only if there is SOME labeling of X for which f has no classification error. So, which is correct here? From the given example, I'd say the first definition is correct.
"For example, consider a straight line as the classification model: the model used by a perceptron. The line should separate positive data points from negative data points. There exist sets of 3 points that can indeed be shattered using this model (any 3 points that are not collinear can be shattered)."
:Wait. So the article's example of 3 points and linear classifier fails on some labelings when the points are collinear (+ - + for example). So, it can't be the first definition above. However, this means that definition 2 must be right, and therefore, the linear classifier must have infinite VC dimension, since I can construct X with arbitrarily large size, as long as I label all the points on one side of the classifier one label, and the other side another label. (Not to mention, this is very confusing when the article says that NO labeling of a 4-point set can be separated).
:Please, end the madness! Rocketman768 (talk) 13:38, 3 August 2010 (UTC)
thread_title: A classification

message: Currently shatter is kind of a trivial dicdef, almost devoid of context. I'm a set theorist and AFAIK the term isn't used (or at least not much) in set theory; Google comes up with VC theory. Please just merge it into the def of VC dimension and then change shatter to a redirect here. Unless of course the notion is used somewhere in VC theory other than in the def of VC dimension; in that case, maybe merge to the VC theory page and redirect it there. --Trovatore 06:08, 5 October 2005 (UTC)
thread_title: someone should merge [[shatter]] into this article

message: The page has been updated somewhat based upon this earlier comment:I hope this gets posted; I have had difficulty editing Wikipedia pages on mathematics.
About shattering, VC-classes, VC dimension.
1. The concept of shattering is important, and indeed should be a separate article.  In fact, although this article is an excellent starting point, it should be generalized and expanded.  
2. Moreover, there should be articles entitled "VC classes" or "Vapnik Cervonenkis classes" and also an article on "empirical processes".  Please let me explain.  Empirical processes are very important in probability and statistics; the theory of empirical processes is what we humans really use every day to form realistic and valid probabilistic judgements.  
3. Shattering and Vapnik-Cervonenkis classes have become very important not only in computer learning theory, but also, in the past 25 years, they have become a vital mathematical tool in empirical processes, which is a mathematical constuct that is so important that it requires its own article.
4. This article about shattering is quite good, but is a special case and needs to be generalized.  It involves elements of a fixed set.  We need to generalize this to the concept of collections of sets (also called classes of sets).  Please let me elaborate:  A necessary preface, to be rigorous and precise:  In set theory, one often cannot discuss "sets of sets", that is, sets whose elements are sets themselves, since this leads to logical inconsistencies and paradoxes (this topic should be an article too; it is an intriguing concept and related to shattering).  Therefore, mathematicians speak and write of "classes of sets" or "collections of sets", not "sets of sets" .   
5. The author made no error!  The author did not discuss collections of sets; the article is good, but it should be generalized and illustrated with examples.
6.  The generalization:  The article on shattering should define and discuss shattering by a class (collection) of sets, not just by elements of a set H.  For example, the class of all discs in the plane (2-space) cannot shatter every finite set F  with 4 or more points, yet the class of all convex sets in the plane can shatter any finite set whatsoever, no matter how large its cardinality -- just "connect the dots".  
6. Also, the article on shattering might be more helpful to a reader if the mathematical language is augmented with some explanation in English (or any other non-technical language) and if it would include examples.
7.  I would do this myself, but I have tried to edit pages on mathematics in the Wikipedia, and have had no success; I do not know how to edit Wikipedia pages.  I wish I did know how to do so.  In fact, I hope I have been able to edit this page successfully.
8.  See Wenocur, R.S.  and Dudley, R.M.   "Some special Vapnik-Chervonenkis classes" (1981) for more information on all this. —the preceding unsigned comment is by  151.197.61.27 (talk • contribs) in multiple edits ending  14:05, 15 December 2005
:With your elaborations, shatter may indeed deserve an article, but it needs a better name. Verbs make poor article titles; somebody needs to think of a noun. Perhaps Shattering (VC theory) ? That's the best I can come up with off the top of my head, but if the concept is used outside of VC theory, then maybe something slightly more general.
:I don't want this to sound aggressive, but I'm afraid your remarks about sets of sets are just wrong. Set theorists (I am one) talk about sets of sets all the time; they aren't the cause of the Russell paradox. The cause of the Russell paradox is the idea that every property should have an extension. However there's nothing wrong with discussing "classes" or "collections" of sets if it makes it easier for the reader to keep track of the types. So I left the language as you had it, but removed the false statement about Russell. --Trovatore 19:28, 15 December 2005 (UTC)
thread_title: Anon comment

message: ...has made two edits to the article that describe VC dimension as being the maximum cardinality *plus one*. This is inconsistent with what my lecture notes say - anybody got a definitive reference? Ubermammal (talk) 11:21, 3 June 2008 (UTC)
replies: text: I've read through the notes in the first link in the article and they agree - it's maximum cardinality, not plus one. Ubermammal (talk) 11:19, 8 June 2008 (UTC)
thread_title:  147.32.80.13... 

message: The citation for the dimension of NNs is given as: 
:  Shalev-Shwartz, Shai; Ben-David, Shai (2014). Understanding Machine Learning – from Theory to Algorithms. Cambridge University Press. ISBN 9781107057135. 
why isnt this citation "good enough"? DMH43 (talk) 20:55, 31 December 2023 (UTC)
thread_title:  Why isnt citation for NN good enough?