This 2-page exposition provide an alternative proof of a version of the XOR Lemma (see the two paragraphs from its intoduction reproduced below).
Focusing on Thm 1, which does capture the main idea, it is instructive to comapre it to Levin's version of the isolation lemma (see [GNW95, Lem 4]), which asserts that if any circuit of size $s'$ has correlation at most $\delta'$ with $f'$ and any circuit of size $s''$ has correlation at most $\delta''$ with $f''$, then, for every slackness $\sigma$, every circuit of size $\min(\poly(\sigma/n)\cdot s',s'')$$ has correlation at most $\delta'\delta''+\sigma$ with $f'f''$.
Levin's proof actually establishes a stronger counterpositive: for every $\eps,\delta$ and $\sigma$, if a circuit $C$ has correlation $\esp$ with $f'f''$, then either for some $x$ it holds that $C''(y)=C(x,y)$ has correlation $\delta$ with $f''$ or there exists a circuit $C'$ that is at most $\poly(n/\sigma)$ factor larger than $C$ that has correlation at least $(\eps/\delta)-\sigma$ with $f'$.
We stress that the correlation bound established by Levin
is optimal (up to the said slackness).
Essentially, the main observation is that
either there exists an $x$ such that $C''=C(x,.)$
has correlation at least $\delta$ with $f''$
or the function $T(x)=E_y[C(x,y) f''(y)]/\delta \in [\pm1]$
must have correlation at least $\eps/\delta$ with $f'$.
This is the case because
$$E_{x}[T(x)f'(x)] = E_{x,y}[C(x,y) f''(y)f'(x)] \geq \eps$$
whereas the negation of the first case means that $|T(x)|\leq\delta$
for every $x$.
Note, however, that $T$ is a mental experiment.
In order to materalize it, Levin uses a sample of $y_i$'s
that are hard-wired along with the corresponding $f''(y_i)$'s.
This yields a circuit that approximates $T$ well enough,
and the slackness is rooted in this approximation.
Viola proves a quantitatively weaker (w.r.t coreelation) version. Specifically, using the same circuit $C''$, he actually proves that for some $y_1,y_2,y_3$ the circuit $C'(x)=MAJ(C(x,y_1)f''(y_1),C(x,y_2)f''(y_2),C(x,y_3)f''(y_3))$ has correlation at least $(3\eps/2)-(\delta^3/2)$ with $f'$ (provided that for every $x$ the residual circuit $C(x,.)$ had correlation at most $\delta$ with $f''$). The point is that for some choices of $\delta$ significantly larger than $\eps$, it holds that $(3\eps/2)-(\delta^3/2)$ is significantly larger that $\eps$. (Recall that, in contrast, in Levin's result we may use $\delta = \sqrt\eps = \eps/\delta$.)
I give an alternative proof of the xor lemma which may provide a simple explanation of why xor-ing decreases correlation.
The xor lemma says that if a function has small correlation with circuits of a certain size, then the xor of many independent copies of the function has much smaller correlation (with circuits of related size). The correlation between two functions $f, g : [2]^n \to {-1, 1}$ is defined as $|Ex[f(x)g(x)]|$, where $[2] = {0, 1}$. The xor lemma has had significant impact in theoretical computer science, with applications in complexity theory, pseudorandomness, and cryptography. For background we refer to [GNW95] or [Vio]. Several proofs of the xor lemma with various parameters are known. Three are given in [GNW95], two of which also appear in [Vio].
I give an alternative proof of the xor lemma which may provide a simple explanation of why xor-ing decreases correlation. Also, the new proof appears to improve parameters in some settings. For example, we can decrease the correlation by a constant factor with only a constant-factor loss in size, whereas other proofs appear to lose a factor which depends on the original correlation. To highlight the simplicity of the argument we begin with a version whose proof requires no background.
See ECCC TR26-033.