The success of SoS in solving some inference problems which appeared beyond the reach of other methods,4 together with its strength as an algorithm in related domains5 generated around 2013 a great deal of interest in the question: can SoS proofs of constant degree refute the existence ofo(
√
n)-size cliques in 4 Dictionary learning and planted sparse vector recovery, for example [31,30].
5 Particularly combinatorial optimization and for problems surrounding the unique games conjecture [27].
G ∼ G(n, 1/2)?6
Prior refutation lower bounds for SoS in the CSP setting were often proved by starting with pseudodistribution constructions which successfully proved lower bounds against weaker proof systems than SoS (such as the Sherali-Adams and Lovasz-Schrijver proof systems) and strengthening their analyses. Two groups of researchers – Meka, Potechin, and Wigderson, and, separately, Deshpande and Montanari – took this route, attempting to adapt a previous lower bound due to Feige and Krauthgamer for weaker, Lovasz-Schrijver proofs [71]. They made substantial headway, but did not succeed in proving a tight lower bound:
Theorem 4.2.1 (Meka, Potechin, Wigderson [128]). For every d ∈ , with high probability over G ∼ G(n, 1/2) there is a degree-d pseudodistribution ˜ satisfying
{x2
i − xi 0, xixj 0 if xi / xjinG} with ˜
Í
i∈[n]xi > n1/d/(log n)O(1).7
This theorem rules out the possibility that constant-degree SoS can refute existence of sub-polynomial-size cliques, but it does not finish the story. Any substantial improvement toTheorem 4.2.1comes up against a major obstacle: the pseudodistribution construction of Feige and Krauthgamer fails the positivity constraint ˜ p(x)2> 0 when parameters are set so that ˜ Íxi n1/(d−1).
Breakdown of the Feige-Krauthgamer Witness Feige and Krauthgamer sug- gested the following natural potential pseudodistribution ˜, which respects all the local consequences of the constraintsxixj 0 for i / j in G. Given a graph G and a clique-size parameter k k(n), for each S ∈ 6dn
, set ˜GxS (k/n)|S| 6 Strictly speaking this question remains unresolved: even in light of our main resultTheo- rem 1.1.1it remains possible even in light of results in this thesis that constant degree SoS could refute existence of cliques of size roughly
√ n/2
√
logn
.
7 Deshpande and Montanari [63] improved this result quantitatively for the cased 4, but still came short of a tight lower bound. See more discussion in [89].
ifS is a G-clique and otherwise ˜ GxS 0. Notice that ˜ Íxi k, and clearly ˜
p(x)xixj 0 if i / j in G.
Kelner gave the following counter-example to positivity of ˜ when k n1/3 for d > 4 [104], which shows that the Feige-Krauthgamer construction fails to account for some non-local consequences of the constraints xixj 0. For simplicity we describe the counterexample ford > 6.
Fact 4.2.2 (Kelner [104]). Let G be an n-node graph. For every i ∈ [n], let ri ∈ {±1}n
be the vector withri( j) 1 if i ∼ j, ri( j) −1 if i / j, and ri(i) 1. Every degree-6 pseudoexpectation ˜ which satisfies{x2
i xi, xixj 0 if i / j} has ˜ Í i∈[n]hri, xi4 > ˜ (Í i∈[n]xi)5.
Proof. We start by considering Í
i∈[n]xihri, xi4. Expanding, Í
i∈[n]xihri, xi4
Í
i,s,t,u,v∈[n]xixsxtxuxvri(s)ri(t)ri(u)ri(v). Since ˜ satisfiesxixj 0 for i / j, for everyi, s, t, u, v which is not a G-clique, we have
˜
xixsxtxuxvri(s)ri(t)ri(u)ri(v) ˜ xixsxtxuxv.
On the other hand, if{i, s, t, u, v} is a G-clique, then ri(s)ri(t)ri(u)ri(v) 1. So all together, ˜ Õ i,s,t,u,v∈[n] xihri, xi 4 ˜ Õ i,s,t,u,v∈[n] xixsxtxuxv ˜ Õ i xi !5 .
Now, since ˜ satisfiesx2
i − xi 0, we claim that ˜ hri, xi 4 > ˜ x ihri, xi 4
for alli, since (1 − x)
x2
ixi (1 − x)
2
. This finishes the proof.
The proof ofFact 4.2.2constructs an SoS proof whose coefficients are simple statistics which derives a nontrivial relationship between two polynomials which
each involve all the variablesx1, . . . , xn. (Note that when the polynomialhri, xi4 is evaluated at the indicator vector x of a clique S, it counts the number of 4-cliques inS with all vertices adjacent to i.) The candidate pseudoexpectation construction of Feige and Krauthgamer does not account for this sort of non-local SoS proof:
Fact 4.2.3. Let ˜ G be the Feige-Krauthgamer functional for a graph G. If d 4 and
k n1/3, then G∼G(n,1/2) ˜GÍ i∈[n]hri, xi G∼G(n,1/2) ˜G Í i∈[n]xi 5 .
The proof is a straightforward calculation, so we leave it out. An intuitive explanation of the whole problem is that degree-6 SoS proves that 4-cliques which are contained in a phantomk-clique also participate, on average, in more 5-cliques than a typical 4-clique inG(n, 1/2), but the Feige-Krauthgamer construction is not accounting for this.
Together, Kelner’s observations show that the Feige-Krauthgamer functional cannot be used to prove a tight SoS lower bound for planted clique: it must fail the positivity requirement. With a little additional work, it is possible to conclude that there is a constant-degree polynomialq(x) whose coefficients are constant-degree functions of the entries of the adjacency matrix of the graphG, such that G(n,1/2) ˜ q(x)2 0.
The difficulty runs deeper than just one polynomial such asÍ
i∈[n]hri, xi4or just the planted clique problem – as one might imagine there are many other SoS-provable inequalities between nontrivial, non-local polynomials, even just for planted clique. Other dense problems – component analysis problems, for example – suffer from similar difficulties. There are several interpretations of the origin of this difficulty – see e.g. the introduction of [29]. For now we move on to the solution proposed by pseudocalibration.