16 research outputs found
Standard Simplices and Pluralities are Not the Most Noise Stable
The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are
two conjectures stating that certain partitions are optimal with respect to
Gaussian and discrete noise stability respectively. These two conjectures are
natural generalizations of the Gaussian noise stability result by Borell (1985)
and the Majority is Stablest Theorem (2004). Here we show that the standard
simplex is not the most stable partition in Gaussian space and that Plurality
is not the most stable low influence partition in discrete space for every
number of parts , for every value of the noise and for
every prescribed measures for the different parts as long as they are not all
equal to . Our results do not contradict the original statements of the
Plurality is Stablest and Standard Simplex Conjectures in their original
statements concerning partitions to sets of equal measure. However, they
indicate that if these conjectures are true, their veracity and their proofs
will crucially rely on assuming that the sets are of equal measures, in stark
contrast to Borell's result, the Majority is Stablest Theorem and many other
results in isoperimetric theory. Given our results it is natural to ask for
(conjectured) partitions achieving the optimum noise stability.Comment: 14 page
Noise Stability of Ranked Choice Voting
We conjecture that Borda count is the ranked choice voting method that best
preserves the outcome of an election with randomly corrupted votes, among all
fair voting methods with small influences satisfying the Condorcet Loser
Criterion. This conjecture is an adaptation of the Plurality is Stablest
Conjecture to the setting of ranked choice voting. Since the plurality function
does not satisfy the Condorcet Loser Criterion, our new conjecture is not
directly related to the Plurality is Stablest Conjecture. Nevertheless, we show
that the Plurality is Stablest Conjecture implies our new Borda count is
Stablest conjecture. We therefore deduce that Borda count is stablest for
elections with three candidates when the corrupted votes are nearly
uncorrelated with the original votes. We also adapt a dimension reduction
argument to this setting, showing that the optimal ranked choice voting method
is "low-dimensional."
The Condorcet Loser Criterion asserts that a candidate must lose an election
if each other candidate is preferred in head-to-head comparisons. Lastly, we
discuss a variant of our conjecture with the Condorcet Winner Criterion as a
constraint instead of the Condorcet Loser Criterion. In this case, we have no
guess for the most stable ranked choice voting method.Comment: 16 page
Three Candidate Plurality is Stablest for Correlations at most 1/11
We prove the three candidate Plurality is Stablest Conjecture of
Khot-Kindler-Mossel-O'Donnell from 2005 for correlations satisfying
: the Plurality function is the most noise stable three
candidate election method with small influences, when the corrupted votes have
correlation with the original votes. The previous best result
of this type only achieved positive correlations at most . Our
result follows by solving the three set Standard Simplex Conjecture of
Isaksson-Mossel from 2011 for all correlations .
The Gaussian Double Bubble Problem corresponds to the case , so
in some sense, our result is a generalization of the Gaussian Double Bubble
Problem. Our result is also notable since it is the first result for any
, which is the only relevant case for computational hardness of
MAX-3-CUT. As an additional corollary, we conclude that three candidate Borda
Count is stablest for all .Comment: 38 page
Hyperstable Sets with Voting and Algorithmic Hardness Applications
The noise stability of a Euclidean set with correlation is the
probability that , where are standard Gaussian random
vectors with correlation . It is well-known that a Euclidean set
of fixed Gaussian volume that maximizes noise stability must be a half space.
For a partition of Euclidean space into parts each of Gaussian measure
, it is still unknown what sets maximize the sum of their noise
stabilities. In this work, we classify partitions maximizing noise stability
that are also critical points for the derivative of noise stability with
respect to . We call a partition satisfying these conditions hyperstable.
Uner the assumption that a maximizing partition is hyperstable, we prove:
* a (conditional) version of the Plurality is Stablest Conjecture for or
candidates.
* a (conditional) sharp Unique Games Hardness result for MAX-m-CUT for
or
* a (conditional) version of the Propeller Conjecture of Khot and Naor for
sets.
We also show that a symmetric set that is hyperstable must be star-shaped.
For partitions of Euclidean space into parts of fixed (but perhaps
unequal) Gaussian measure, the hyperstable property can only be satisfied when
all of the parts have Gaussian measure . So, as our main contribution, we
have identified a possible strategy for proving the full Plurality is Stablest
Conjecture and the full sharp hardness for MAX-m-CUT: to prove both statements,
it suffices to show that sets maximizing noise stability are hyperstable. This
last point is crucial since any proof of the Plurality is Stablest Conjecture
must use a property that is special to partitions of sets into equal measures,
since the conjecture is false in the unequal measure case.Comment: 33 pages. arXiv admin note: substantial text overlap with
arXiv:2011.0558
Dimension Reduction for Polynomials over Gaussian Space and Applications
We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016].
Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest