253 research outputs found
Hypercontractivity, Sum-of-Squares Proofs, and their Applications
We study the computational complexity of approximating the 2->q norm of
linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as
connections between this question and issues arising in quantum information
theory and the study of Khot's Unique Games Conjecture (UGC). We show the
following:
1. For any constant even integer q>=4, a graph is a "small-set expander"
if and only if the projector into the span of the top eigenvectors of G's
adjacency matrix has bounded 2->q norm. As a corollary, a good approximation to
the 2->q norm will refute the Small-Set Expansion Conjecture--a close variant
of the UGC. We also show that such a good approximation can be obtained in
exp(n^(2/q)) time, thus obtaining a different proof of the known subexponential
algorithm for Small Set Expansion.
2. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy
certify an upper bound on the 2->4 norm of the projector to low-degree
polynomials over the Boolean cube, as well certify the unsatisfiability of the
"noisy cube" and "short code" based instances of Unique Games considered by
prior works. This improves on the previous upper bound of exp(poly log n)
rounds (for the "short code"), as well as separates the "Sum of
Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to
require omega(1) rounds.
3. We show reductions between computing the 2->4 norm and computing the
injective tensor norm of a tensor, a problem with connections to quantum
information theory. Three corollaries are: (i) the 2->4 norm is NP-hard to
approximate to precision inverse-polynomial in the dimension, (ii) the 2->4
norm does not have a good approximation (in the sense above) unless 3-SAT can
be solved in time exp(sqrt(n) polylog(n)), and (iii) known algorithms for the
quantum separability problem imply a non-trivial additive approximation for the
2->4 norm.Comment: v1: 52 pages. v2: 53 pages, fixed small bugs in proofs of section 6
(on UG integrality gaps) and section 7 (on 2->4 norm of random matrices).
Added comments about real-vs-complex random matrices and about the
k-extendable vs k-extendable & PPT hierarchies. v3: fixed mistakes in random
matrix section. The result now holds only for matrices with random entries
instead of random column
Sum-of-squares proofs and the quest toward optimal algorithms
In order to obtain the best-known guarantees, algorithms are traditionally
tailored to the particular problem we want to solve. Two recent developments,
the Unique Games Conjecture (UGC) and the Sum-of-Squares (SOS) method,
surprisingly suggest that this tailoring is not necessary and that a single
efficient algorithm could achieve best possible guarantees for a wide range of
different problems.
The Unique Games Conjecture (UGC) is a tantalizing conjecture in
computational complexity, which, if true, will shed light on the complexity of
a great many problems. In particular this conjecture predicts that a single
concrete algorithm provides optimal guarantees among all efficient algorithms
for a large class of computational problems.
The Sum-of-Squares (SOS) method is a general approach for solving systems of
polynomial constraints. This approach is studied in several scientific
disciplines, including real algebraic geometry, proof complexity, control
theory, and mathematical programming, and has found applications in fields as
diverse as quantum information theory, formal verification, game theory and
many others.
We survey some connections that were recently uncovered between the Unique
Games Conjecture and the Sum-of-Squares method. In particular, we discuss new
tools to rigorously bound the running time of the SOS method for obtaining
approximate solutions to hard optimization problems, and how these tools give
the potential for the sum-of-squares method to provide new guarantees for many
problems of interest, and possibly to even refute the UGC.Comment: Survey. To appear in proceedings of ICM 201
Majority is Stablest : Discrete and SoS
The Majority is Stablest Theorem has numerous applications in hardness of
approximation and social choice theory. We give a new proof of the Majority is
Stablest Theorem by induction on the dimension of the discrete cube. Unlike the
previous proof, it uses neither the "invariance principle" nor Borell's result
in Gaussian space. The new proof is general enough to include all previous
variants of majority is stablest such as "it ain't over until it's over" and
"Majority is most predictable". Moreover, the new proof allows us to derive a
proof of Majority is Stablest in a constant level of the Sum of Squares
hierarchy.This implies in particular that Khot-Vishnoi instance of Max-Cut does
not provide a gap instance for the Lasserre hierarchy
Approximability and proof complexity
This work is concerned with the proof-complexity of certifying that
optimization problems do \emph{not} have good solutions. Specifically we
consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic
proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor,
Lasserre, and Parrilo shows that this proof system is automatizable using
semidefinite programming (SDP), meaning that any -variable degree- proof
can be found in time . Furthermore, the SDP is dual to the well-known
Lasserre SDP hierarchy, meaning that the "-round Lasserre value" of an
optimization problem is equal to the best bound provable using a degree- SOS
proof. These ideas were exploited in a recent paper by Barak et al.\ (STOC
2012) which shows that the known "hard instances" for the Unique-Games problem
are in fact solved close to optimally by a constant level of the Lasserre SDP
hierarchy.
We continue the study of the power of SOS proofs in the context of difficult
optimization problems. In particular, we show that the Balanced-Separator
integrality gap instances proposed by Devanur et al.\ can have their optimal
value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of
the KKL Theorem. We also investigate the extent to which the Khot--Vishnoi
Max-Cut integrality gap instances can have their optimum value certified by an
SOS proof. We show they can be certified to within a factor .952 ()
using a constant-degree proof. These investigations also raise an interesting
mathematical question: is there a constant-degree SOS proof of the Central
Limit Theorem?Comment: 34 page
A Cheeger Inequality for Small Set Expansion
The discrete Cheeger inequality, due to Alon and Milman (J. Comb. Theory
Series B 1985), is an indispensable tool for converting the combinatorial
condition of graph expansion to an algebraic condition on the eigenvalues of
the graph adjacency matrix. We prove a generalization of Cheeger's inequality,
giving an algebraic condition equivalent to small set expansion. This algebraic
condition is the p-to-q hypercontractivity of the top eigenspace for the graph
adjacency matrix. Our result generalizes a theorem of Barak et al (STOC 2012)
to the low small set expansion regime, and has a dramatically simpler proof;
this answers a question of Barak (2014)
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