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Register a new userHypercontractivity, Sum-of-Squares Proofs, and their Applications
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Aram Harrow
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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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 Khots Unique Games Conjecture (UGC). We show the following: 1. For any constant even integer q>=4, a graph $G$ is a small-set expander if and only if the projector into the span of the top eigenvectors of Gs 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.
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The degree-$4$ Sum-of-Squares (SoS) SDP relaxation is a powerful algorithm that captures the best known polynomial time algorithms for a broad range of problems including MaxCut, Sparsest Cut, all MaxCSPs and tensor PCA. Despite being an explicit algorithm with relatively low computational complexity, the limits of degree-$4$ SoS SDP are not well understood. For example, existing integrality gaps do not rule out a $(2-varepsilon)$-algorithm for Vertex Cover or a $(0.878+varepsilon)$-algorithm for MaxCut via degree-$4$ SoS SDPs, each of which would refute the notorious Unique Games Conjecture. We exhibit an explicit mapping from solutions for degree-$2$ Sum-of-Squares SDP (Goemans-Williamson SDP) to solutions for the degree-$4$ Sum-of-Squares SDP relaxation on boolean variables. By virtue of this mapping, one can lift lower bounds for degree-$2$ SoS SDP relaxation to corresponding lower bounds for degree-$4$ SoS SDPs. We use this approach to obtain degree-$4$ SoS SDP lower bounds for MaxCut on random $d$-regular graphs, Sherington-Kirkpatrick model from statistical physics and PSD Grothendieck problem. Our constructions use the idea of pseudocalibration towards candidate SDP vectors, while it was previously only used to produce the candidate matrix which one would show is PSD using much technical work. In addition, we develop a different technique to bound the spectral norms of _graphical matrices_ that arise in the context of SoS SDPs. The technique is much simpler and yields better bounds in many cases than the _trace method_ -- which was the sole technique for this purpose.
Finding cliques in random graphs and the closely related planted clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for t = Theta(sqrt(n)). Here we show that beating sqrt(n) would require substantially new algorithmic ideas, by proving a lower bound for the problem in the sum-of-squares (or Lasserre) hierarchy, the most powerful class of semi-definite programming algorithms we know of: r rounds of the sum-of-squares hierarchy can only solve the planted clique for t > sqrt(n)/(C log n)^(r^2). Previously, no nontrivial lower bounds were known. Our proof is formulated as a degree lower bound in the Positivstellensatz algebraic proof system, which is equivalent to the sum-of-squares hierarchy. The heart of our (average-case) lower bound is a proof that a certain random matrix derived from the input graph is (with high probability) positive semidefinite. Two ingredients play an important role in this proof. The first is the classical theory of association schemes, applied to the average and variance of that random matrix. The second is a new large deviation inequality for matrix-valued polynomials. Our new tail estimate seems to be of independent interest and may find other applications, as it generalizes both the estimates on real-valued polynomials and on sums of independent random matrices.
Estimation is the computational task of recovering a hidden parameter $x$ associated with a distribution $D_x$, given a measurement $y$ sampled from the distribution. High dimensional estimation problems arise naturally in statistics, machine learning, and complexity theory. Many high dimensional estimation problems can be formulated as systems of polynomial equations and inequalities, and thus give rise to natural probability distributions over polynomial systems. Sum-of-squares proofs provide a powerful framework to reason about polynomial systems, and further there exist efficient algorithms to search for low-degree sum-of-squares proofs. Understanding and characterizing the power of sum-of-squares proofs for estimation problems has been a subject of intense study in recent years. On one hand, there is a growing body of work utilizing sum-of-squares proofs for recovering solutions to polynomial systems when the system is feasible. On the other hand, a general technique referred to as pseudocalibration has been developed towards showing lower bounds on the degree of sum-of-squares proofs. Finally, the existence of sum-of-squares refutations of a polynomial system has been shown to be intimately connected to the existence of spectral algorithms. In this article we survey these developments.
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.
We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order-$3$ tensor $T$ of the form $T = tau cdot v_0^{otimes 3} + A$, where $tau geq 0$ is a signal-to-noise ratio, $v_0$ is a unit vector, and $A$ is a random noise tensor, the goal is to recover the planted vector $v_0$. For the case that $A$ has iid standard Gaussian entries, we give an efficient algorithm to recover $v_0$ whenever $tau geq omega(n^{3/4} log(n)^{1/4})$, and certify that the recovered vector is close to a maximum likelihood estimator, all with high probability over the random choice of $A$. The previous best algorithms with provable guarantees required $tau geq Omega(n)$. In the regime $tau leq o(n)$, natural tensor-unfolding-based spectral relaxations for the underlying optimization problem break down (in the sense that their integrality gap is large). To go beyond this barrier, we use convex relaxations based on the sum-of-squares method. Our recovery algorithm proceeds by rounding a degree-$4$ sum-of-squares relaxations of the maximum-likelihood-estimation problem for the statistical model. To complement our algorithmic results, we show that degree-$4$ sum-of-squares relaxations break down for $tau leq O(n^{3/4}/log(n)^{1/4})$, which demonstrates that improving our current guarantees (by more than logarithmic factors) would require new techniques or might even be intractable. Finally, we show how to exploit additional problem structure in order to solve our sum-of-squares relaxations, up to some approximation, very efficiently. Our fastest algorithm runs in nearly-linear time using shifted (matrix) power iteration and has similar guarantees as above. The analysis of this algorithm also confirms a variant of a conjecture of Montanari and Richard about singular vectors of tensor unfoldings.
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