(Nearly) Efficient Algorithms for the Graph Matching Problem - - PowerPoint PPT Presentation

(Nearly) Efficient Algorithms

for the

Graph Matching Problem

Tselil Schramm

(Harvard/MIT)

with Boaz Barak, Chi-Ning Chou, Zhixian Lei & Yueqi Sheng (Harvard)

graph matching problem (approximate graph isomorphism)

𝐻0 𝐻1

goal: find permutation of vertices that maximizes # shared edges

ma𝑦

𝜌

𝐵𝐻0, 𝜌 𝐵𝐻1

input: two graphs on 𝑜 vertices

graph matching problem (approximate graph isomorphism)

𝐻0 𝐻1 ma𝑦

𝜌

𝐵𝐻0, 𝜌 𝐵𝐻1 # matched = 4

goal: find permutation of vertices that maximizes # shared edges input: two graphs on 𝑜 vertices

graph matching problem (approximate graph isomorphism)

𝐻0 𝐻1 ma𝑦

𝜌

𝐵𝐻0, 𝜌 𝐵𝐻1

goal: find permutation of vertices that maximizes # shared edges input: two graphs on 𝑜 vertices

graph matching problem (approximate graph isomorphism)

𝐻0 𝐻1 ma𝑦

𝜌

𝐵𝐻0, 𝜌 𝐵𝐻1 # matched = 5

goal: find permutation of vertices that maximizes # shared edges input: two graphs on 𝑜 vertices

computationally hard (of course)

NP-hard: reduction from quadratic assignment problem (non-simple graphs).

[Lawler’63]

also: reduction from sparse random 3-SAT to approximate version

[O’Donnell-Wright-Wu-Zhou’14]

practitioners: undeterred

 computational biology [e.g. Singh-Xu-Berger‘08]  de-anonymization [e.g. Narayanan-Shmatikov’09]  social networks [e.g. Korula-Lattanzi’14]  image alignment [e.g. Cho-Lee’12]  machine learning [e.g. Cour-Srinivasan-Shi’07]  pattern recognition, e.g. “thirty years of graph matching in pattern recognition”

[Conte-Foggia-Sansone-Vento’04]

average case: correlated random graphs

ma𝑦

𝜌

𝐵𝐻0, 𝜌 𝐵𝐻1 ≈ 𝑞𝛿2 ⋅ 𝑜 2

sample 𝐻 ∼ 𝐻(𝑜, 𝑞) subsample edges w/prob 𝛿 𝐻 𝐻0 𝐻1 𝐻0 random permutation 𝜌

𝛿 𝛿

• avg. degree 𝑞𝛿 ⋅ 𝑜

structured model

[e.g. Pedarsani-Grossglauser’11, Lyzinski-Fishkind-Priebe’14, Korula-Lattanzi’14]

“robust average-case graph isomorphism”

average case: correlated random graphs

• avg. degree 𝑞𝛿 ⋅ 𝑜

≈ 𝑞𝛿2 ⋅ 𝑜 2 “null” model

sample 𝐻 ∼ 𝐻(𝑜, 𝑞) subsample edges w/prob 𝛿 𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿

𝜌 structured model

“robust average-case graph isomorphism”

average case: correlated random graphs

ma𝑦

𝜌

𝐵𝐻0, 𝜌 𝐵𝐻1 ≈ 𝑞𝛿 2 ⋅ 𝑜 2

• avg. degree 𝑞𝛿 ⋅ 𝑜

≈ 𝑞𝛿2 ⋅ 𝑜 2 “null” model

sample 𝐻0, 𝐻1 ∼ 𝐻(𝑜, 𝑞𝛿) 𝐻0 𝐻1

• avg. degree 𝑞𝛿 ⋅ 𝑜

sample 𝐻 ∼ 𝐻(𝑜, 𝑞) subsample edges w/prob 𝛿 𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿

𝜌 structured model

“robust average-case graph isomorphism”

information theoretic limit

Iff 𝑞𝛿2 >

log 𝑜 𝑜 , with high probability 𝜌 is the unique maximizing permutation.

Theorem

𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿 𝜌 for which 𝑞, 𝛿 can we recover 𝜌?

[Cullina-Kivayash’16&17]

algorithms for robust average case?

e.g. matching local neighborhoods average-case graph isomorphism algorithms fail. match radius-𝑙 neighborhoods?

𝐻0 𝐻1

algorithms for robust average case?

𝛿 𝛿

e.g. matching local neighborhoods average-case graph isomorphism algorithms fail. match radius-𝑙 neighborhoods?

𝐻0 𝐻1

algorithms for robust average case?

average-case graph isomorphism algorithms fail. e.g. spectral algorithm unique entries in top eigenvector give isomorphism? 𝑤max 𝑤max

𝐻0 𝐻1

algorithms for robust average case?

average-case graph isomorphism algorithms fail. e.g. spectral algorithm unique entries in top eigenvector give isomorphism? 𝑤max 𝑤max

𝛿 𝛿

+ +

𝛿 𝛿

perturb eigenvectors by ≈ 𝛿

𝐻0 𝐻1

actual algorithms for robust average case?

starting from a seed

𝜌ȁ𝑇 known 𝑇 𝜌(𝑇)

𝐻0 𝐻1

starting from a seed

𝑇 𝑣 𝑤

match vertices with similar adjacency into 𝑇

𝐻0 𝐻1

starting from a seed

𝑇 𝜌 𝑣 = 𝑤

match vertices with similar adjacency into 𝑇

need 2 ෨

𝑃 𝑜𝜗 time to guess a seed.

iff seed ≥ Ω(𝑜𝜗), the seeded algorithm approximately recovers 𝜌. [Yartseva-Grossglauser’13]

𝐻0 𝐻1

• ur results

For any 𝜗 > 0, if 𝑞 ∈

𝑜𝑝(1) 𝑜

,

𝑜

1 153

𝑜

∪

𝑜

2 3

𝑜 , 𝑜1−𝜗 𝑜

and 𝛿 = Ω(1),* there is a 𝑜𝑃(log 𝑜) time algorithm that recovers 𝜌 on 𝑜 − 𝑝(𝑜) of the vertices w/prob ≥ 0.99. Theorem

𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿 𝜌

𝑜𝑝(1) 𝑜1/153 𝑜2/3 𝑜1−𝜗 𝑜 log 𝑜 average degrees: *we allow 𝛿 = Ω

1 loglog 𝑜

𝑜1/3 𝑜1/2 𝑜3/5 structured 𝐻(𝑜, 𝑞)

For any 𝜗 > 0, if 𝑞 ∈

𝑜𝑝(1) 𝑜

,

𝑜

1 153

𝑜

∪

𝑜

2 3

𝑜 , 𝑜1−𝜗 𝑜

and 𝛿 = Ω(1),* there is a 𝑜𝑃(log 𝑜) time algorithm that recovers 𝜌 on 𝑜 − 𝑝(𝑜) of the vertices w/prob ≥ 0.99.

• ur results

Theorem

*we allow 𝛿 ≥

1 log𝑝(1) 𝑜

𝐻(𝑜, 𝑞𝛿) 𝐻0 𝐻1

If 𝑞, 𝛿 are as above then there is a poly(𝑜) time distinguishing algorithm for the structured vs null distributions. Theorem

hypothesis testing structured null 𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿 𝜌

• ur approach: small subgraphs

seedless algorithms! hypothesis testing: correlation of subgraph counts recovery: match rare subgraphs

• utline

 distinguishing/hypothesis testing  recovery  concluding

• utline

 distinguishing/hypothesis testing  recovery  concluding

distinguishing/hypothesis testing

𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿 𝜌

𝐻(𝑜, 𝑞𝛿) 𝐻0 𝐻1 structured null Given 𝐻0, 𝐻1 sampled equally likely from structured or null, decide w/prob 1 − 𝑝(1) from which. 𝐻1 𝐻0

? ? ?

brute force: is there a 𝜌 with ≥ 𝑞𝛿2𝑜2 matched edges?

…counting triangles?

𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿 𝜌

𝐻(𝑜, 𝑞𝛿) 𝐻0 𝐻1 structured null 𝐻1 𝐻0

? ? ? 𝑑𝑝𝑠𝐿3 𝐻0, 𝐻1 : = # 𝐿3 𝑗𝑜 𝐻0 # K3 𝑗𝑜 𝐻1 .

…counting triangles?

𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿 𝜌

𝐻(𝑜, 𝑞𝛿) 𝐻0 𝐻1 structured null 𝐻1 𝐻0

triangle counts in 𝐻0, 𝐻1 are independent 𝔽 𝑑𝑝𝑠

𝑙3(𝐻0, 𝐻1) ≈ 𝑞𝛿𝑜 6

𝑑𝑝𝑠𝐿3 𝐻0, 𝐻1 : = # 𝐿3 𝑗𝑜 𝐻0 # K3 𝑗𝑜 𝐻1 .

…counting triangles?

𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿 𝜌

structured 𝐻1 𝐻0

triangle counts in 𝐻0, 𝐻1 are correlated

𝐻(𝑜, 𝑞𝛿) 𝐻0 𝐻1 null

𝔽 𝑑𝑝𝑠

𝐿3(𝐻0, 𝐻1) ≈ 𝑞𝛿𝑜 6 + 𝛿2𝑞𝑜 3

…counting triangles?

𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿 𝜌

structured 𝐻(𝑜, 𝑞𝛿) 𝐻0 𝐻1 null

𝔽 𝑑𝑝𝑠

𝐿3(𝐻0, 𝐻1) ≈ 𝑞𝛿𝑜 6

𝔽 𝑑𝑝𝑠

𝐿3(𝐻0, 𝐻1) ≈ 𝑞𝛿𝑜 6 + 𝛿2𝑞𝑜 3

Variance?

Optimistically, in null case,

𝕎 𝑑𝑝𝑠

𝐿3 𝐻0, 𝐻1 1/2 ≈ 𝑞𝛿𝑜 3

structured null

“independent trials”

Suppose we had 𝑈 “independent trials”:

𝔽 𝑑𝑝𝑠𝑈 𝐻0, 𝐻1 ≈ 𝑞𝛿𝑜 6 𝔽 𝑑𝑝𝑠𝑈 𝐻0, 𝐻1 ≈ 𝑞𝛿𝑜 6 + 𝛿2𝑞𝑜 3

structured null

𝕎 𝑑𝑝𝑠𝑈 𝐻0, 𝐻1

1/2 ≈ 1

𝑈 𝑞𝛿𝑜 3

if 𝑈 > 1/𝛿6, 𝑑𝑝𝑠𝑈 is a good test 𝑑𝑝𝑠𝑈 𝐻0, 𝐻1 = 1 𝑈 ෍

𝑗=1 𝑈

𝑑𝑝𝑠

𝐿3 (𝑗)(𝐻0, 𝐻1) 𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝛿 𝛿

𝜌 𝐻(𝑜, 𝑞𝛿) 𝐻0 𝐻1

“independent trials”

near-independent subgraphs

𝐼1, … , 𝐼𝑈 what properties must 𝐼1, … , 𝐼𝑈 have to be “independent”?

Suppose we had 𝑈 “independent” subgraphs: 𝑑𝑝𝑠𝑈 𝐻0, 𝐻1 = 1 𝑈 ෍

𝑗=1 𝑈

𝑑𝑝𝑠

𝐼𝑗(𝐻0, 𝐻1)

𝔽 #𝐼 𝐻 = 5! ȁ𝑏𝑣𝑢 𝐼 ȁ ⋅ 𝑜 5 ⋅ 𝑞7

surprisingly delicate (concentration)

𝑞 = 𝑜−5/7

How many labeled copies of 𝐼 in 𝐻?

𝐻(𝑜, 𝑞)

𝐻

𝐼

≈ 𝑜5𝑞7 = Θ(1)

surprisingly delicate (concentration)

𝑞 = 𝑜−5/7

How many labeled copies of 𝐼 in 𝐻?

𝐻(𝑜, 𝑞)

𝐻

𝐼

𝔽 #𝐼 𝐻 = 5! ȁ𝑏𝑣𝑢 𝐼 ȁ ⋅ 𝑜 5 ⋅ 𝑞7 How many labeled copies of 𝐿4 in 𝐻? 𝔽 #𝐿4 𝐻 = 4! ȁ𝑏𝑣𝑢 𝐿4 ȁ ⋅ 𝑜 4 ⋅ 𝑞6 ≈ 𝑜5𝑞7 = Θ(1) ≈ 𝑜4𝑞6 = Θ(𝑜−2/7)

#𝐼(𝐻) does not concentrate!

variance of subgraph counts

𝐼

𝐻(𝑜, 𝑞)

𝐻

For a constant-sized subgraph 𝐼, Lemma

𝕎 #𝐼(𝐻) = Θ 1 ⋅ 𝔽 #𝐼 𝐻

2

min

𝐾⊂𝐼 𝔽[#𝐾 𝐻 ]

subgraph of 𝐼 with fewest expected appearances

strict balance

For a constant-sized subgraph 𝐼, Lemma

𝕎 #𝐼(𝐻) = Θ 1 ⋅ 𝔽 #𝐼 𝐻

2

min

𝐾⊂𝐼 𝔽[#𝐾 𝐻 ]

𝐼 is strictly balanced if all its strict subgraphs have edge density <

𝐹 𝐼 𝑊 𝐼 .

if 𝔽 #𝐼 𝐻 ≈ 𝑜 𝑊 𝐼 𝑞 𝐹 𝐼 = Θ(1), then 𝔽 #𝐾 𝐻 = 𝜕 1 for any 𝐾 ⊂ 𝐼. = 𝑝 1 ⋅ 𝔽 #𝐼 𝐻

concentration AND independence

If 𝐼1, … , 𝐼𝑈 are non-isomorphic strictly balanced graphs with 𝔽 #𝐼𝑗 𝐻 = Θ 1 ,

𝕎 #𝐼𝑗 𝐻 = 𝑝 1 ⋅ 𝔽 #𝐼𝑗 𝐻 𝔽 #𝐼𝑗 𝐻 ⋅ #𝐼𝑘(𝐻) = (1 + 𝑝 1 ) ⋅ 𝔽 #𝐼𝑗 𝐻 ⋅ 𝔽 #𝐼𝑘 𝐻

∀𝑗 ∈ [𝑈], ∀𝑗 ≠ 𝑘 ∈ 𝑈 ,

their counts concentrate their counts are asymptotically independent

distinguishing algorithm

For 𝑤 =

1 poly 𝛿 , design a “test set”

• f 𝑈 = 𝑤Ω(𝑓) strictly balanced graphs w/ 𝑤 vertices & 𝑓 edges.

𝐼1, … , 𝐼𝑈

set 𝑜𝑤 𝑞𝛿 𝑓 ≈ 1 𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝐻(𝑜, 𝑞𝛿) 𝐻0 𝐻1

vs.

distinguishing algorithm

For 𝑤 =

1 poly 𝛿 , design a “test set”

• f 𝑈 = 𝑤Ω(𝑓) strictly balanced graphs w/ 𝑤 vertices & 𝑓 edges.

𝐼1, … , 𝐼𝑈

𝑑𝑝𝑠𝑈 𝐻0, 𝐻1 = 1 𝑈 ෍

𝑗=1 𝑈

𝑑𝑝𝑠

𝐼𝑗(𝐻0, 𝐻1)

compute

𝔽 𝑑𝑝𝑠𝑈 𝐻0, 𝐻1 = 𝑜2𝑤(𝛿𝑞)2𝑓+𝑜𝑤 𝛿2𝑞 𝑓 set 𝑜𝑤 𝑞𝛿 𝑓 ≈ 1 𝑜2𝑤(𝛿𝑞)2𝑓 𝕎 𝑑𝑝𝑠𝑈 𝐻0, 𝐻1 = 1 𝑈 𝑜𝑤 𝛿𝑞 𝑓 < 𝑜𝑤 𝛿2𝑞 𝑓 structured null null ≥ 𝜄 < 𝜄 null structured TODO: variance in structured case. 𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

𝐻(𝑜, 𝑞𝛿) 𝐻0 𝐻1

vs.

• utline

 distinguishing/hypothesis testing  recovery  concluding

• utline

 distinguishing/hypothesis testing  test graphs  recovery  concluding

designing a “test set”

For 𝑤 =

1 poly 𝛿 , design a “test set”

• f 𝑈 = 𝑤Ω(𝑓) strictly balanced graphs w/ 𝑤 vertices & 𝑓 edges.

set 𝑜𝑤 𝑞𝛿 𝑓 ≈ 1 𝑜𝑝(1) 𝑜1/153 𝑜2/3 𝑜1−𝜗 𝑜 log 𝑜 average degree: 𝑜1/3 𝑜1/2 𝑜3/5 remember? 𝐻(𝑜, 𝑞)

𝐼1, … , 𝐼𝑈

designing a “test set”

For 𝑤 =

1 poly 𝛿 , design a “test set”

• f 𝑈 = 𝑤Ω(𝑓) strictly balanced graphs w/ 𝑤 vertices & 𝑓 edges.

set 𝑜𝑤 𝑞𝛿 𝑓 ≈ 1

𝐼1, … , 𝐼𝑈

designing a “test set”

For 𝑤 =

1 poly 𝛿 , design a “test set”

• f 𝑈 = 𝑤Ω(𝑓) strictly balanced graphs w/ 𝑤 vertices & 𝑓 edges.

set 𝑜𝑤 𝑞𝛿 𝑓 ≈ 1

𝐼1, … , 𝐼𝑈

claim: connected 𝑒-regular graphs are strictly balanced. proof: in any strict subgraph, average degree < 𝑒.

designing a “test set”

For 𝑤 =

1 poly 𝛿 , design a “test set”

• f 𝑈 = 𝑤Ω(𝑓) strictly balanced graphs w/ 𝑤 vertices & 𝑓 edges.

set 𝑜𝑤 𝑞𝛿 𝑓 ≈ 1

𝐼1, … , 𝐼𝑈

claim: connected 𝑒-regular graphs are strictly balanced. proof: in any strict subgraph, average degree < 𝑒.

𝑜2/3 𝑜 log 𝑜 average degree: 𝑜1/3 𝑜1/2 𝑜3/5 𝐻(𝑜, 𝑞)

“test set” for non-integer degrees

For 𝑤 =

1 poly 𝛿 , design a “test set”

• f 𝑈 = 𝑤Ω(𝑓) strictly balanced graphs w/ 𝑤 vertices & 𝑓 edges.

set 𝑜𝑤 𝑞𝛿 𝑓 ≈ 1

𝐼1, … , 𝐼𝑈

what if we want 2 ⋅

𝑓 𝑤 = 𝜇 ⋅ 𝑒 + 1 + 1 − 𝜇 ⋅ 𝑒? 𝑒-regular random graph on 𝑤 vertices + random matching

• n 𝜇𝑤 vertices

strict balance? expansion.

𝑜2/3 𝑜1−𝜗 𝑜 log 𝑜 average degree: 𝑜1/3 𝑜1/2 𝑜3/5 𝐻(𝑜, 𝑞)

“test set” for non-integer degrees

For 𝑤 =

1 poly 𝛿 , design a “test set”

• f 𝑈 = 𝑤Ω(𝑓) strictly balanced graphs w/ 𝑤 vertices & 𝑓 edges.

set 𝑜𝑤 𝑞𝛿 𝑓 ≈ 1

𝐼1, … , 𝐼𝑈

what if we want 2 ⋅

𝑓 𝑤 = 𝜇 ⋅ 𝑒 + 1 + 1 − 𝜇 ⋅ 𝑒? 𝑒-regular random graph on 𝑤 vertices + random matching

• n 𝜇𝑤 vertices

strict balance? expansion.

𝑒 < 3?

2-regular graphs don’t expand.

“test set” for non-integer degrees < 3

For 𝑤 =

1 poly 𝛿 , design a “test set”

• f 𝑈 = 𝑤Ω(𝑓) strictly balanced graphs w/ 𝑤 vertices & 𝑓 edges.

set 𝑜𝑤 𝑞𝛿 𝑓 ≈ 1

𝐼1, … , 𝐼𝑈

what if we want 2 ⋅

𝑓 𝑤 = 𝜇 ⋅ 3 + 1 − 𝜇 ⋅ 2? 3-regular random

graph on 𝜇𝑤 vertices subdivide edges into 𝑙 and 𝑙 + 1 - length paths 𝑣 𝑤 𝑣 𝑤

strict balance? expansion.

𝑜𝑝(1) 𝑜1/153 𝑜2/3 𝑜1−𝜗 𝑜 log 𝑜 average degree: 𝑜1/3 𝑜1/2 𝑜3/5 𝐻(𝑜, 𝑞)

designing a “test set”

For 𝑤 =

1 poly 𝛿 , design a “test set”

• f 𝑈 = 𝑤Ω(𝑓) strictly balanced graphs w/ 𝑤 vertices & 𝑓 edges.

set 𝑜𝑤 𝑞𝛿 𝑓 ≈ 1

𝐼1, … , 𝐼𝑈

Conjecture: our construction achieves all

𝑓 𝑤 𝑜1−𝜗 𝑜 log 𝑜 average degree: 𝑜1/3 𝐻(𝑜, 𝑞) 𝑒-reg +matching subdivide

+ more conditions (for recovery)

• utline

 distinguishing/hypothesis testing  test graphs  recovery  concluding

• utline

 distinguishing/hypothesis testing  test graphs  recovery  concluding

distinguishing ≠ recovery

distinguishing: subgraphs on

1 poly 𝛿 = 𝑃(1) vertices, each appearing 𝑃(1) times

• nly 𝑃(1) vertices participate in subgraphs from our test set.

distinguishing: counting subgraphs ambiguity in matching; how to conclude 𝜌 𝑣 = 𝑤?

the “black swan” approach

choose test set 𝐼1, … , 𝐼𝑈 so that 𝛿2𝑞 𝑓𝑜𝑤 ≫ 𝛿𝑞 2𝑓𝑜2𝑤, if we see 𝐼𝑗 in both graphs, it is most likely because of correlation. choose large test set 𝐼1, … , 𝐼𝑈 with 𝑤 = 𝑃(log 𝑜) vertices Ω(𝑜) vertices participate in subgraphs from our test set.

identify rare subgraphs appearing in both graphs, and match vertices.

𝐻1 𝐻0 expected number of that survive subsampling expected number of unrelated pairs of

the “black swan” approach

identify rare subgraphs appearing in both graphs, and match vertices.

𝐻1 𝐻0

Claim: there is at most one copy of each in 𝐻 with high probability

𝐻(𝑜, 𝑞)

𝐻 𝐻0 𝐻1 𝐻0

Claim: Ω(𝑜) vertices in 𝐻0 ∩ 𝐻1, appear in a surviving subsampled with high probability

proofs: second moment method

• utline

 distinguishing/hypothesis testing  test graphs  recovery  concluding

• utline

 distinguishing/hypothesis testing  test graphs  recovery  concluding

why subgraph counts/statistics?

emerging intuition/conjectures: SoS ≡𝒃𝒘𝒉 low-degree polynomials the sum-of-squares (SoS) semidefinite program is at most as powerful as “low-degree” statistics for average-case problems. known to hold for: planted clique [Barak-Hopkins-Kelner-Kothari-Moitra-Potechin’16] CSP refutation [Grigoriev’01, Schoenebeck’08, Kothari-Mori-O’Donnell-Witmer’17] tensor PCA [Hopkins-Kothari-Potechin-Raghavendra-S-Steurer’17] also known: SoS is at most as powerful as “low-degree” spectral algorithms for average-case problems [Hopkins-Kothari-Potechin-Raghavendra-S-Steurer’17]

does SoS know about the black swans?

ma𝑦

𝜌

𝐵𝐻0, 𝜌 𝐵𝐻1

does the natural SoS relaxation recover 𝜌?

• r

?

cares about can ask similar questions about other low-degree functions, e.g. non-backtracking random walk matrix.

SoS relaxation

more questions

 recovery in polynomial time?

SoS? or, many variations on our theme are possible.

 all information-theoretically possible 𝑞 ∈

log 𝑜 𝑜 , 𝑃 1 ?

 practical heuristics?

𝑜 log 𝑜 𝑜1/3

Thank you!