[Paper Review] SOS Lower Bound for Exact Planted Clique

Samuel B. Hopkins|arXiv (Cornell University)|Jul 18, 2015

Complexity and Algorithms in Graphs37 references15 citations

TL;DR

This paper establishes tight lower bounds for the Sum of Squares (SoS) hierarchy in solving the planted clique problem. It proves that degree-4 SoS cannot recover planted cliques smaller than $\widetilde{O}(\sqrt{n})$, and for degree-$2d$ SoS, the bound is $\widetilde{O}(n^{1/(d+1)})$, improving prior results by refining the certifying polynomial used in earlier analyses through a novel correction mechanism.

ABSTRACT

The problem of finding large cliques in random graphs and its "planted" variant, where one wants to recover a clique of size $ω\gg \log{(n)}$ added to an \Erdos-\Renyi graph $G \sim G(n,\frac{1}{2})$, have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size $ω= Ω(\sqrt{n})$. By contrast, information theoretically, one can recover planted cliques so long as $ω\gg \log{(n)}$. In this work, we continue the investigation of algorithms from the sum of squares hierarchy for solving the planted clique problem begun by Meka, Potechin, and Wigderson (MPW, 2015) and Deshpande and Montanari (DM,2015). Our main results improve upon both these previous works by showing: 1. Degree four SoS does not recover the planted clique unless $ω\gg \sqrt n poly \log n$, improving upon the bound $ω\gg n^{1/3}$ due to DM. A similar result was obtained independently by Raghavendra and Schramm (2015). 2. For $2 < d = o(\sqrt{\log{(n)}})$, degree $2d$ SoS does not recover the planted clique unless $ω\gg n^{1/(d + 1)} /(2^d poly \log n)$, improving upon the bound due to MPW. Our proof for the second result is based on a fine spectral analysis of the certificate used in the prior works MPW,DM and Feige and Krauthgamer (2003) by decomposing it along an appropriately chosen basis. Along the way, we develop combinatorial tools to analyze the spectrum of random matrices with dependent entries and to understand the symmetries in the eigenspaces of the set symmetric matrices inspired by work of Grigoriev (2001). An argument of Kelner shows that the first result cannot be proved using the same certificate. Rather, our proof involves constructing and analyzing a new certificate that yields the nearly tight lower bound by "correcting" the certificate of previous works.

Motivation & Objective

• To close the gap between the information-theoretic threshold ($\omega \gg \log n$) and the algorithmic threshold ($\omega = \Omega(\sqrt{n})$) for the planted clique problem.
• To analyze the limitations of the Sum of Squares (SoS) hierarchy in recovering planted cliques, particularly at low degrees.
• To improve upon prior SoS lower bounds by constructing and analyzing a corrected certifying polynomial that yields nearly tight bounds.
• To show that the MPW certifying polynomial is insufficient for proving strong SoS lower bounds beyond degree 4, necessitating more complex constructions.

Proposed method

• Constructs a new certifying polynomial for degree-4 SoS by correcting the MPW polynomial to achieve a nearly optimal lower bound.
• Analyzes the spectrum of the corrected operator using a fine spectral decomposition along a carefully chosen basis to control eigenvalues.
• Applies concentration bounds and combinatorial tools to analyze the deviation of moment expectations in random graphs with dependent entries.
• Uses a perturbation argument to show that the corrected certifier yields negative definite expectation when $\omega \ll \sqrt{n}$, implying integrality gap.
• Employs a decomposition of the certifier into symmetric components and leverages symmetries in eigenspaces inspired by Grigoriev's work.
• Proves optimality of the analysis by showing that the original MPW certifier cannot yield the same bound, necessitating the correction.

Experimental results

Research questions

• RQ1 Can the degree-4 SoS hierarchy recover planted cliques of size $\omega = \widetilde{O}(\sqrt{n})$?
• RQ2 Is the MPW certifier sufficient to prove tight lower bounds for SoS at degree 4, or must it be modified?
• RQ3 What is the tightest possible lower bound for degree-$2d$ SoS in the planted clique problem for $d = o(\sqrt{\log n})$?
• RQ4 Can the SoS hierarchy be improved beyond $\omega = \widetilde{O}(n^{1/(d+1)})$ using more complex certifiers?
• RQ5 Why does the original MPW certifier fail to yield strong bounds for SoS, and what structural features of the certifier are necessary for tight analysis?

Key findings

• The degree-4 SoS hierarchy cannot recover planted cliques of size $\omega \ll \widetilde{O}(\sqrt{n})$, improving upon the prior $n^{1/3}$ bound.
• For degree-$2d$ SoS with $d = o(\sqrt{\log n})$, the integrality gap is at least $\widetilde{O}(n^{1/(d+1)})$, matching the best known upper bounds up to polylogarithmic factors.
• A new certifying polynomial is constructed that corrects the MPW polynomial, enabling a nearly tight lower bound at degree 4.
• The analysis shows that the original MPW certifier is insufficient for proving strong SoS lower bounds, as shown by a generalization of Kelner's argument.
• The corrected certifier's spectrum is analyzed via a basis decomposition, revealing that the dominant contribution comes from components with $t = q/2$ connected components.
• The proof establishes that the variance of the corrected certifier's expectation is small, allowing the use of concentration bounds to show negative definite expectation when $\omega \ll \sqrt{n}$.