Hypercontractivity, Sum-of-Squares Proofs, and their Applications (1205.4484v3)

Abstract: 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 $G$ 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.

• The paper introduces a subexponential algorithm for approximating the 2-to-q norm, providing insights into small-set expansion and SDP hierarchy techniques.
• The paper demonstrates that the Sum-of-Squares hierarchy yields constant-factor approximations and certifies previously hard Unique Games instances.
• The paper bridges quantum information with computational complexity, leveraging hypercontractivity methods to establish strong inapproximability results under ETH.

Hypercontractivity, Sum-of-Squares Proofs, and their Applications

This paper explores the computational complexity of approximating the $2$-to-$q$ norm of linear operators and explores connections with quantum information theory and the Unique Games Conjecture (UGC). Specifically, the paper presents several theoretical results concerning these connections and demonstrates both algorithmic and hardness results relating to the $2$-to-$4$ norm.

Main Contributions and Results

The paper makes several substantial contributions in the field:

1. Subexponential Algorithms: The authors show the existence of a subexponential algorithm, which offers a good approximation for the $2$-to-$q$ norm. This extension provides insight into graphs that fail to be small-set expanders via a subspace dimension argument, similar to the Arora-Barak-Steurer (2010) algorithm for Small-Set Expansion.
2. Strong Connections with Small-Set Expansion: The work introduces a robust linkage between the $2$-to-$4$ norm and the Small-Set Expansion problem, making the computation of this norm critical for solving or approximating the latter.
3. Sum-of-Squares Hierarchy: The authors employ the Sum-of-Squares semidefinite programming hierarchy to provide polynomial-time constant-factor approximations for specific instances such as random linear operators and projections to low-degree polynomials.
4. Algorithmic Performance on Unique Games Instances: The paper breaks new ground by showing that the Sum-of-Squares hierarchy can effectively certify that certain previously proposed hard instances for Unique Games are not almost satisfiable. This result shows how SDP hierarchies applied in polynomial time may potentially refute the unique games conjecture.
5. Hardness of Approximation: Assuming the Exponential Time Hypothesis (ETH) for 3-SAT, the paper establishes hardness of approximation for the $2$-to-$4$ norm. It extends these results to demonstrate the impracticality of efficiently approximating the $2$-to-$4$ norm beyond constant factors.
6. Relation to Quantum Information: The paper showcases imaginative techniques from quantum information, particularly addressing algorithmic performance concerning separable states via Sum-of-Squares and broader SDP relaxations.

Conceptual and Theoretical Implications

The results showcase the deployment of complex mathematical tools and algorithms to tackle problems previously considered intractable. By bridging gaps between different theoretical frameworks—such as sum-of-squares proofs, hypercontractivity, and properties of quantum states—the paper opens new avenues for tackling major conjectures in theoretical computer science.

In terms of practical applicability, successful approximations of the $2$-to-$q$ norm can lead to algorithmic improvements in areas dealing with high-dimensional data, graph properties, and optimization problems under uncertainty. Furthermore, it raises the possibility of new algorithms that could break existing barriers in approximating constraint satisfaction problems and in understanding isoperimetric inequalities in graphs.

Directions for Future Research

The findings spark numerous questions for future exploration:

• Tighter Bounds and Approximation Ratios: Could further refinements of Sum-of-Squares techniques or alternative SDP hierarchies provide even closer approximations and efficient algorithms?
• New Integrality Gaps: What are the implications for newly identified integrality gaps within SDP hierarchies, and could this further illuminate on the separations between different levels of the Sum-of-Squares hierarchy and other relaxations?
• Complexity with Quantum Connections: As quantum aspects are integrated into computational problems, could the newly discovered ties help design quantum algorithms that outperform classical counterparts?
• Applications Beyond Theory: Can the insights gained be applied to other computational problems outside theoretical explorations, such as automated reasoning, data mining, and AI?

Conclusion

This paper profoundly enriches the landscape of computational complexity, touching on diverse domains such as quantum theory, graph theory, and optimization. Through rigorous theoretical analysis and novel algorithmic frameworks, the authors lay the groundwork for bridging complex theoretical problems, which could have significant implications for both theory and practice in computer science.