Lecture 12: SOS Lower Bounds for Planted Clique Part I
Lecture Outline • Part I: Planted Clique and the Meka-Wigderson Moments • Part II: MPW Analysis Preprocessing • Part III: MPW Analysis with Graph Matrices • Part IV: The Pessimist Strikes Back
Part I: Planted Clique and the Meka-Wigderson Moments
Review: Planted Clique • Recall the planted clique problem: Given a random graph 𝐻 where a clique of size 𝑙 has been planted, can we find this planted clique? • Variant we’ll analyze: Can we use SOS to prove that a random 𝐻 𝑜, 1 2 graph has no clique of size 𝑙 where 𝑙 ≫ 2𝑚𝑝𝑜 (the expected size of the largest clique in a random graph)?
Review: Planted Clique Equations • Variable 𝑦 𝑗 for each vertex i in G. • Want 𝑦 𝑗 = 1 if i is in the clique. • Want 𝑦 𝑗 = 0 if i is not in the clique. • Equations: 2 = 𝑦 𝑗 for all i. 𝑦 𝑗 𝑦 𝑗 𝑦 𝑘 = 0 if 𝑗, 𝑘 ∉ 𝐹(𝐻) σ 𝑗 𝑦 𝑗 ≥ 𝑙
First SOS Lower Bound • Theorem [MPW15]: ∃𝐷 > 0 such that whenever 1 𝑜 𝑒 , with high probability degree 𝑙 ≤ 𝐷 𝑒 𝑚𝑝𝑜 2 𝑒 SOS cannot prove the 𝑙 -clique equations are infeasible.
Review: SOS Lower Bound Strategy • To prove an SOS lower bound: 1. Come up with pseudo-expectation values ෨ 𝐹 which obey the required linear equations 2. Show that the moment matrix 𝑁 is PSD
MW Moments • Idea: Give each 𝑒 -clique the same weight • Define 𝑦 𝐽 = ς 𝑗∈𝐽 𝑦 𝑗 • Define 𝑂 𝑒 (𝐽) to be the number of 𝑒 -cliques containing 𝐽 . 𝑙 𝑂 𝑒 (𝐽) • MW moments: take ෨ |𝐽| 𝐹 𝑦 𝐽 = ⋅ 𝑒 𝑂 𝑒 (∅) |𝐽|
Checking σ 𝑗 𝑦 𝑗 = 𝑙 𝑙 𝑂 𝑒 (𝐽) • MW moments: take ෨ |𝐽| 𝐹 𝑦 𝐽 = ⋅ 𝑒 𝑂 𝑒 (∅) |𝐽| • MW moments obey the equation σ 𝑗 𝑦 𝑗 = 𝑙 • Proof: σ 𝑗∉𝐽 𝑂 𝑒 (𝐽 ∪ 𝑗) = 𝑒 − 𝐽 𝑂 𝑒 (𝐽) as each d-clique containing 𝐽 contains 𝑒 − |𝐽| of the 𝑗 ∉ 𝐽 𝑙 𝑙 = 𝑙−|𝐽| 𝐽 +1 |𝐽| • 𝑒−|𝐽| ⋅ 𝑒 𝑒 𝐽 +1 |𝐽| • σ 𝑗 ෨ 𝐹 𝑦 𝐽∪𝑗 = 𝐽 ෨ 𝐹 𝑦 𝐽 = 𝑙 ෨ ෨ 𝐹 𝑦 𝐽 + 𝑙 − 𝐽 𝐹 𝑦 𝐽
Part II: MPW Analysis Preprocessing
Analysis Outline • For the MPW analysis, we do the following: 1. Preprocess the moment matrix 𝑁 to make it easier to analyze. More specifically, we find a matrix 𝑁′ which is easier to analyze such that if 𝑒 𝜇 min 𝑁 ′ ≥ 𝑙 2 then 𝑁 ≽ 0 with high probability 𝑒 4𝑜 2 2. Decompose 𝑁 ′ = 𝐹 𝑁 ′ + 𝑆 and show that 𝑒 𝑒 𝐹 𝑁 ′ ≽ 𝑙 2 𝑙 2 𝐽𝑒 and w.h.p., 𝑆 ≤ 𝑒 𝑒 2𝑜 4𝑜 2 2
𝑒 Restriction to Multilinear, Degree 2 • Preprocessing Step #1: As we’ve seen from the 3XOR and knapsack lower bounds, since we 2 = 𝑦 𝑗 for all 𝑗 and have the constraints that 𝑦 𝑗 σ 𝑗 𝑦 𝑗 = 𝑙 , it is sufficient to consider the submatrix of 𝑁 with multilinear, degree 𝑒 2 indices
Approximating ෨ 𝐹[𝑦 𝐽 ] • Preprocessing Step #2: Approximate ෨ 𝐹[𝑦 𝐽 ] • Intuition: One view of ෨ 𝐹[𝑦 𝐽 ] is that ෨ 𝐹[𝑦 𝐽 ] is the expected value of 𝑦 𝐽 given what we can compute. • Remark: This is connected to pseudo- calibration/moment matching which we’ll see next lecture.
Approximating ෨ 𝐹[𝑦 𝐽 ] Continued • A priori, if we choose a clique of size 𝑙 at random, |𝐽| is part of the clique with 𝑙 𝑙 |𝐽| |𝐽| ≈ probability 𝑜 |𝐽| 𝑜 |𝐽| • If 𝐽 is not a clique, ෨ 𝐹 𝑦 𝐽 = 0 . If 𝐽 is a clique, 𝐽 is |𝐽| 2 times more likely to be part of the clique. 2 |𝐽| 𝑙 |𝐽| Thus, ෨ 𝐹 𝑦 𝐽 ≈ 2 𝑜 |𝐽| if 𝐽 is a clique and is 0 2 otherwise. • See appendix for calculations confirming this.
Approximation Error • Let 𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 be the matrix where |𝐽∪𝐾| 𝑙 |𝐽∪𝐾| 𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 𝐽𝐾 = 2 𝑜 |𝐽∪𝐾| if 𝐽 ∪ 𝐾 is a clique 2 and 𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 𝐽𝐾 = 0 otherwise. • Can show that the difference Δ = M − M 𝑏𝑞𝑞𝑠𝑝𝑦 is small (see [MPW15] for details).
The matrix 𝑁′ • Preprocessing Step #3: Fill in zero rows and columns of 𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 • If 𝐽 or 𝐾 is not a clique then (𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 ) 𝐽𝐾 = 0 . • These zero rows and columns make 𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 harder to analyze. • Definition: Take 𝑁′ to be the matrix such that |𝐽∪𝐾| 𝑙 |𝐽∪𝐾| 𝑁′ 𝐽𝐾 = 2 𝑜 |𝐽∪𝐾| if all edges are present 2 between 𝐽 ∖ 𝐾 and 𝐾 ∖ 𝐽 and 𝑁′ 𝐽𝐾 = 0 otherwise
𝑁 ′ ≽ 0 ⇒ 𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 ≽ 0 • Can view 𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 as a submatrix of 𝑁′ . • This immediately implies that if 𝑁 ′ ≽ 0 then 𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 ≽ 0 • Because of the error matrix Δ = M − M 𝑏𝑞𝑞𝑠𝑝𝑦 we need the stronger statement that with high probability, 𝜇 min 𝑁 ′ is significantly bigger than 0 .
Summary 𝑒 • We want to show that w.h.p. 𝑁 ′ ≽ 𝑙 2 where 𝑒 4𝑜 2 |𝐽∪𝐾| 𝑙 |𝐽∪𝐾| 𝑁′ is the matrix such that 𝑁′ 𝐽𝐾 = 2 𝑜 |𝐽∪𝐾| if 2 all edges are present between 𝐽 ∖ 𝐾 and 𝐾 ∖ 𝐽 and 𝑁′ 𝐽𝐾 = 0 otherwise
𝑁′ Picture for d = 4 12 13 14 15 16 23 24 25 26 34 35 36 45 46 56 12 13 2𝑙 2 14 𝑁′ 𝑗,𝑘 {𝑗,𝑘} = 𝑜 2 15 16 8𝑙 3 𝑁′ 𝑗,𝑘 {𝑗,𝑙} = 23 𝑜 3 if 24 j ∼ 𝑙 and 0 25 otherwise 26 64𝑙 4 34 𝑁′ 𝑗,𝑘 {𝑙,𝑚} = 𝑜 4 if 35 𝑗 ∼ 𝑘 , 𝑗 ∼ 𝑙 , 𝑘 ∼ 𝑙 , 36 𝑘 ∼ 𝑚 and is 0 45 otherwise 46 56
Part III: MPW Analysis with Graph Matrices
Recall Definition of 𝑆 𝐼 • Definition: Definition: If 𝑊 𝐼 = 𝑉 ∪ 𝑊 then define 𝑆 𝐼 𝐵, 𝐶 = 𝜓 𝜏(𝐹(𝐼)) where 𝜏: V H → 𝑊(𝐻) is the injective map satisfying 𝜏 𝑉 = 𝐵 , 𝜏 𝑊 = 𝐶 and preserving the ordering of 𝑉, 𝑊 . • Last lecture: Did not require 𝐵, 𝐶 to be in ascending order. • This lecture: Will require 𝐵, 𝐶 to be in ascending order. • Note: This only reduces our norms, so the probabilistic norm bounds still hold.
Review: Rough Norm Bound • Theorem [MP16]: If 𝐼 has no isolated vertices then with high probability, 𝑆 𝐼 is 𝑃 𝑜 ( 𝑊 𝐼 −𝑡 𝐼 )/2 where 𝑡 𝐼 is the minimal ෨ size of a vertex separator between 𝑉 and 𝑊 (S is a vertex separator of U and V if every path from U to V intersects S) • Note: The ෨ 𝑃 contains polylog factors and constants related to the size of 𝐼 .
Decomposition of 𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 and 𝑁′ 𝑙 |𝑉∪𝑊| • Claim: 𝑁 𝑏𝑞𝑞𝑠𝑝𝑦 = σ 𝐼 𝑜 |𝑉∪𝑊| 𝑆 𝐼 where we sum over 𝐼 which have no middle vertices. |𝑉| + |𝑊| − |𝑉∩𝑊| 𝑙 |𝑉∪𝑊| • Claim: 𝑁 ′ = σ 𝐼 2 𝑜 |𝑉∪𝑊| 𝑆 𝐼 2 2 2 where we sum over 𝐼 which have no middle vertices and which have no edges within 𝑉 or within 𝑊 . |𝑉| + |𝑊| − |𝑉∩𝑊| • Idea: Each of the 2 edges 2 2 2 within 𝑉 or 𝑊 are given for free.
Entries of E[𝑁 ′ ] |𝑉| + |𝑊| − |𝑉∩𝑊| 𝑙 |𝑉∪𝑊| • 𝑁 ′ = σ 𝐼 2 𝑜 |𝑉∪𝑊| 𝑆 𝐼 where 2 2 2 we sum over 𝐼 which have no middle vertices and which have no edges within 𝑉 or within 𝑊 . |𝐽| 2 + |𝐾| 2 − |𝐽∩𝐾| 𝑙 |𝐽∪𝐾| • Claim: E 𝑁 ′ 𝐽𝐾 = 2 2 𝑜 |𝐽∪𝐾| • Idea: For any 𝐼 which has an edge, 𝐹 𝑆 𝐼 = 0 . Otherwise, 𝐹 𝑆 𝐼 = 𝑆 𝐼
𝐹[𝑁 ′ ] Picture for d = 4 12 13 14 15 16 23 24 25 26 34 35 36 45 46 56 12 13 2𝑙 2 14 𝐹[𝑁′] 𝑗,𝑘 {𝑗,𝑘} = 𝑜 2 15 16 4𝑙 3 𝐹[𝑁′] 𝑗,𝑘 {𝑗,𝑙} = 23 𝑜 3 24 4𝑙 4 25 𝐹[𝑁′] 𝑗,𝑘 {𝑙,𝑚} = 𝑜 4 26 34 35 36 45 46 56
Analysis of 𝐹[𝑁′] • 𝐹[𝑁′] belongs to the Johnson Scheme of matrices 𝐵 whose entries 𝐵 𝐽𝐾 only depend on |𝐽 ∩ 𝐾| (See Lecture 9 on SOS Lower Bounds for Knapsack) • Can decompose 𝐹 𝑁 ′ as a sum of PSD matrices, one of which is the identity matrix 𝑒 𝑙 2 which has coefficient ≥ 𝐽𝑒 . 𝑒 2𝑜 2
One Piece of 𝑁 ′ − 𝐹[𝑁′] ( 𝑒 = 4 ) 12 13 14 15 16 23 24 25 26 34 35 36 45 46 56 12 13 14 15 0 16 23 0 24 25 60𝑙 4 26 𝑜 4 if all edges 34 between 𝐽 and 𝐾 35 are present. 36 4𝑙 4 45 − 𝑜 4 otherwise 46 56
Piece of 𝑁 ′ − 𝐹[𝑁′] Decomposition 4𝑙 4 • This piece has coefficient 𝑜 4 in 𝑆 𝐼 for all 𝐼 which have the following form (and 0 for all other 𝑆 𝐼 ): 𝑣 1 𝑤 1 𝑣 2 𝑤 2 𝑉 𝑊 Where 𝐹(𝐼) is non-empty and is a subset of the dashed lines
Piece of 𝑁 ′ − 𝐹[𝑁′] Analysis • All 𝐼 here have minimum separator size 𝑡 𝐼 at least 1 . 4−1 𝑙 4 • This gives a norm bound of ෨ 𝑃 𝑜 4 ⋅ 𝑜 = 2 𝑙 2 𝑜 ⋅ 𝑙 2 ෨ 𝑃 𝑜 2 1 𝑙 2 • This is much less than 4𝑜 2 when 𝑙 ≪ 𝑜 4 .
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