*[Corrected, thanks - T.]*

I have spent a couple of months trying to understand it but it seems impregnable but at the same time very important and deep.

Regards.

]]>For the sake of clompleteness let me mention that I wrote a paper about the spectral proof of the stronng regularity lemma which clarifies the same issues that you discuss (see below). It may also contain some new interesting aspect. (For example connection to graph limits and group theory)

Limits of kernel operators and the spectral regularity lemma

http://arxiv.org/abs/1003.5588

European J. of Comb, Volume 32, Issue 7, October 2011, p. 1156-1167

*[Thanks, I'e added a link to this in the main post. Good to see that we now have spectral proofs of the weak, strong, and standard regularity lemmas in the literature - T.]*

Even with my cynics hat on, I’m more inclined towards the weaker interpretation: referees have no *strong* incentive to actually understand a paper rather than just see if anything it says looks “obviously wrong”.

*[Corrected, thanks - T.]*

PS: Minor comment. One tag is misspelled as “eigenvales”.

*[Corrected, thanks -T.]*

So if I understood correct, there should still be a way to obtain Chung’s version in a simple algorithmic way as in the graph case suggested by Frieze and Kannan.

More precisely, there should be an analog of Lemma 2 in http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.44.721 for the higher order case.

That is, something along these lines must hold:

/// Let be a dimensional Tensor , with and . If is a small positive real number then,

(a) if there exists such that and then

(b) if then there exists such that and where , is a constant. Such might be constructed in polynomial time. ///

Is this a correct way of thinking about it?

]]>To explain this, begin with the graph case. An edge set E can be viewed as an indicator function of two vertex-valued variables . Graph regularity has to do with understanding sums such as

and this can be done by using the SVD to split into finite rank components such as .

Now consider a 3-uniform hypergraph, which now comes with an indicator function . If one were interested in counting “order 1″ sums such as

then a bounded rank approximation to would be effective; this would correspond, roughly speaking, to the hypergraph regularity lemma of Chung. But in practice, this type of regularity is insufficient; one often needs to count expressions such as

(*)

for some graphs F,G,H, and bounded rank approximations can be quite terrible for this purpose. (See for instance the discussion in this paper of Gowers.) In particular, there are 3-uniform hypergraphs with no bounded rank approxmiations which have nontrivial behaviour with respect to sums such as (*). Instead one has to consider “order 2″ bounded rank approximations to , using linear combinations of functions of the form rather than . (These two-variable functions in turn then need to be approximated by “order 1″ bounded rank functions.) This can still be done, but one is now quite far from the classical theory of rank.

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