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Version 3 changes:
-----------------
-- The new Lemma 3 in version 2 unfortunately also had an error. Specifically,
in its proof, on page 16, it is written "E_{\kappa, s_g, s_h, r} [ ...] ...
which by the chain rule ... = \frac{2}{n} E_{\kappa, s_h, r}[...]". This is
incorrect, because r depends on h, and h depends on s_g. Thus one cannot
apply chain rule to remove s_g as done in the proof. We have replaced Lemma
3 with a different, correct statement. Namely rather than obtaining that
D' and D are close in KL-divergence, we show that the marginals D'[X] and
D[X] are not too far (and similarly for Y), and that the expected conditional
distributions of X|Y=y from D' and D (where y is sampled according to D')
are close (and similarly for the roles of X and Y reversed). We have adjusted
Lemma 5 to interface correctly with this new statement. Our final results are
unchanged.
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Version 2 changes:
-----------------
-- Lemma 3 in version 1 was incorrect. Specifically, for the line
"Since s_h <= n/2 and t_h > 3n/4, X_{\kappa([n/2+1,3n/4])} and O are
independent if we do not condition on W."
in its proof, we see no justification for X_{\kappa([n/2+1,3n/4])} and O
being independent if we do not condition on W. We have replaced Lemma 3 with
a different, correct statement, and we adjusted lemma 5 appropriately to be
compatible with the new lemma 3. Our final results are unchanged.