In the last post, I stated the following generalization of the weak regularity lemma:
Theorem (Low Complexity Approximation, TTV Version) Let be a probability space, a bounded function, a collection of bounded functions , and an approximation parameter.
Then there is a function such that
- has low complexity relative to : there are functions and coefficients such that
- is -indistinguishable from by , that is,
(Last time, I mentioned that our proof handled only boolean functions ; now we can handle arbitrary bounded functions, and with an “energy-decrease” style proof, this will appear in the next online revision of the paper.)
This seems to be a useful tool, limited only by one’s creativity in choosing the functions and then making use of the properties of .
As already discussed,
- if one takes to be the edges of a complete graph, and the set of indicator variables of cuts, then the existence of gives the weak regularity lemma of Frieze and Kannan; and
- if one takes to be the set of circuits of size at most , and normalizes and to be probability distributions, one gets that for every probability distribution of high entropy there is a (non-uniformly) efficiently samplable and computable distribution that is indistinguishable from by circuits of size .
In this post I’ll show how to use it to give short proofs of the Hardcore Lemma of Impagliazzo and the Dense Model Theorem of Green, Tao and Ziegler. Both proofs also have, at least in hindsight, a sense of “inevitability,” meaning that given the Low-Complexity Approximation Theorem, and given what we want to prove, both proofs get to the point in a most economical and natural way.
- The Impagliazzo Hardcore Lemma. We have already mentioned that if is “hard-on-average” for , then cannot be an approximation in the sense of being close to on most inputs. What, then, about the points on which and differ? They form an Impagliazzo Hardcore Set for , as described next.
Let be a function that is weakly hard on average for a class of algorithms . Suppose, specifically, that for every algorithm of complexity relative to we have
and, more generally, for fractional , we have
Then, construct an approximating function of complexity
relative to and such that and are -indistinguishable by . Note that, even though is “indistinguishable” from , it is also “far” from , as in (1).
Define a probability distribution that assigns to point a probability proportional to . (If were boolean, this would be the uniform distribution over points on which and differ.) Then this distribution is -dense in the uniform distribution, meaning that every point has probability at most . Observe also that we have
for every , because and have the same sign and , so we have
and so is a hardcore distribution, because the above expression is equivalent to
- The Dense Model Theorem. Suppose that is a pseudorandom set with respect to functions that have bounded complexity relative to , and let be a dense subset of , .
To find a dense model of , we take to be the characteristic function of , and we let be the low-complexity approximation, but using the uniform distribution on as .. Now suppose for simplicity that is boolean, and that is the set of inputs of on which is 1. We want to argue that is a dense model of . By assuming without loss of generality that contains the all-one function, we get from the indistinguishability of and that
and from the pseudorandomness of we have
and so and is indeed dense.
For the indistinguishability of and , take any function , and observe that
where we use both the indistinguishability of and under distribution , and the fact that the distributions and are indistinguishable by functions of bounded complexity.
This proof is appealingly intuitive, in the sense that if we expect to be indistinguishable from a large set, then when we try to approximate the characteristic function of we will end up with a low complexity function that is spread around much of , thus defining a dense model. It also shows that “relative” versions of the Regularity Lemma, such as the Regularity Lemma for subgraphs of an expander, may be derived from the regular Lemma by the above argument. A disadvantage of the argument is that it does not establish the stronger form of the Dense Model Theorem suggested by Impagliazzo, in which there is no set , but we require to have the “pseudo-density” requirement that for every low-complexity bounded function ,
which follows immediately if has density in a pseudorandom set , but that is a seemingly weaker property. (The relative Regularity Lemma in graphs had long be known to hold under such a pseudo-density assumption.)