Locality in Coding Theory II: LTCs Madhu Sudan Harvard April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Outline of this Part Three ideas in Testing: Tensor Products Composition/Recursion Symmetry (esp. affine-invariance) April 9, 2016 Skoltech: Locality in Coding Theory

Part 0 - A Definition: Robust Testing April 9, 2016 Skoltech: Locality in Coding Theory

Recursive testing and Robustness Generic test for code 𝐶⊆{𝑔:𝐷→ 𝔽 𝑞 }: Given 𝑓:𝐷→ 𝔽 𝑞 , pick some set 𝑆⊆𝐷, and verify 𝑓 ​ 𝑆 ∈𝐵. Test defined by distribution over 𝑆 and 𝐵 Often, 𝐵 itself an error-correcting code! Robust analysis (informally): 𝑓 far from 𝐶⇒ usually 𝑓 ​ 𝑆 far from 𝐵! (Stronger conclusion than just ¬ 𝑓 ​ 𝑆 ∈𝐵 ) April 9, 2016 Skoltech: Locality in Coding Theory

Recursive testing and Robustness-II Formally: 𝐶 is 𝛼-robust w.r.t. 𝐵-test if 𝔼 𝑆 𝛿 𝑓 ​ 𝑆 ,𝐵 ≥𝛼⋅𝛿(𝑓,𝐶) 𝛼-robust ⇒ 𝛼-sound 𝜖-sound ⇒ 𝜖 ℓ -robust Recursive testing: If 𝐶 is 𝛼-robust wrt 𝐵-test and 𝐵 is (𝜖,ℓ)-LTC, then 𝐶 is 𝛼⋅𝜖,ℓ -LTC Goal from now: Robust testing. April 9, 2016 Skoltech: Locality in Coding Theory

Part 1: Tensor Product Codes April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Tensor Product Codes Given 𝐵⊆ 𝑓:𝑆→ 𝔽 𝑞 and 𝐶⊆ 𝑔:𝑇→ 𝔽 𝑞 , 𝐵⊗𝐶≝ ℎ:𝑆×𝑇→ 𝔽 𝑞 | ∀𝑥, 𝑦, ℎ ⋅,𝑦 ∈𝐵 & ℎ 𝑥,⋅ ∈𝐶 Is 𝐵⊗𝐶 non-empty? Miracle of linear algebra: dim 𝐵⊗𝐶 = dim 𝐵 ⋅dim 𝐶 (Not to be dismissed lightly! Fails miserably even for additive codes.) 𝐵 ⊗𝑚 ≝𝐵⊗𝐵⊗⋯⊗𝐵 April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Tensor Product Codes - 2 Locality from tensors: If 𝑆 =𝑛 and 𝑁= 𝑛 𝑚 then Block-length 𝐵 ⊗𝑚 =𝑁; Constraints of length 𝑁 1 𝑚 (axis-parallel lines) Candidate test: “Lines test” Pick 𝑖∈[𝑚] and 𝑎 −𝑖 ∈ 𝑆 𝑚−1 uniformly. Verify 𝑓 ​ 𝑥 −𝑖 := 𝑎 −𝑖 ∈𝐹. Sound? Robust? Hope 𝛼=𝛼(𝑚,𝛿 𝐵 ). Question raised in [Ben-Sasson+S ’04]. Answered negatively by [P. Valiant ’05]. April 9, 2016 Skoltech: Locality in Coding Theory

Testing Tensor Products Proposed by [Ben-Sasson+S’04]. Many strengthenings since, with final (?) word due to [Viderman ’11] Test 1: Test 𝐵 𝑚 using 𝑚−1 -dimensional projections! Lemma: [Viderman ‘11] ∀𝑚≥3, 𝛿, ∃𝛼>0 such that if 𝛿 𝐵 ≥𝛿 then 𝐵 𝑚 is 𝛼-robust wrt 𝐵 𝑚−1 -test. Test 2: Test 𝐵 𝑚 using 2-dimensional projections! Theorem: ∀𝑚,𝛿, ∃𝛽>0 s.t. if 𝛿 𝐵 ≥𝛿 then 𝐵 𝑚 is 𝛽-robust wrt 𝐵 2 -test Proof: Let 𝛼 𝑚 be robustness of (m-1)-dimensional test from Lemma. Then 𝛽= 𝛼 𝑚 ⋅ 𝛼 𝑚−1 ⋯ 𝛼 2 . QED. April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Proof of Lemma (𝑚=3) Reinterpretation of robustness: Pick random point 𝑥∈ 𝑆 3 Pick random plane 𝑝∋𝑥 Accept if 𝑓 𝑥 agrees with best 𝐶 2 codeword on 𝑝 Analysis: 𝑥 BAD if three planes disagree. Line/plane BAD if contain many BAD points Pr[test rejects | 𝑥 BAD] ≥ 1 3 ⇒ Pr[𝑥 BAD] small 𝑥 BAD ⇒ some line containing 𝑥 BAD ⇒ some plane containing 𝑥 BAD Throw away BAD planes; Rest of 𝑆 3 is clean ⇒ consistent with some codeword of 𝐶 3 April 9, 2016 Skoltech: Locality in Coding Theory

Testing via Tensor Products Starting with base code of rate R, get Code of rate 𝑅 1 𝜖 , distance 1−𝑅 1 𝜖 of length 𝑁, testable with locality ℓ 𝑁 = 𝑁 𝜖 But central ingredient in much better constructions! E.g., Apply [KMR-ZS’16a] and get code of Rate 𝑅, distance 1−𝑅 −𝜖, length 𝑁, locality ℓ 𝑁 = 𝑁 𝑜(1) Also central ingredient in analyses of non-tensor codes! April 9, 2016 Skoltech: Locality in Coding Theory

Part 2: Testing & Composition April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Zig-Zag Approach Pioneered by [Reingold-Vadhan-Wigderson] Identify two parameters 𝑎,𝑏 : (say we want to increase both). Identify two operations: Zig: 𝑎,𝑏 → 𝐴,𝛽 [ 𝛼≤𝑎≤𝐴 ;𝛽≤𝑏≤𝐵 ] Zag: 𝑎,𝑏 →(𝛼,𝐵) If we are lucky: Zig ∘ Zag: 𝑎,𝑏 →( 𝐴 ′ , 𝐵 ′ ) Remarkably successful approach! Expanders – [RVW, CRVW] Logspace connectivity – [Reingold] PCP – [Dinur] April 9, 2016 Skoltech: Locality in Coding Theory

Zig-Zag LTC Construction [KMR-ZS’16b] Zig-Operation: Tensoring Length Squares Robustness Reduces by O(1) factor Distance Squares Zag-Operation: Alon-Luby Transform Distance Increases Robustness decreases Length no worse Combination: Distance Maintained! Robustness smaller by constant factor April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Zig-Zag LTC Theorem [KMR-ZS’16] Theorem: For every 𝑅>0, for infinitely many 𝑛 there exists codes of length 𝑛 of rate 𝑅, distance 1 −𝑅 −𝑜(1) that are log log 𝑛 𝑛 , 1 2 -LTCs April 9, 2016 Skoltech: Locality in Coding Theory

Part 3: Testing by Symmetries April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Motivation If you wish to test natural codes (linearity, low-degree, …) how to do it? What if you want code to be LTC + LDC? April 9, 2016 Skoltech: Locality in Coding Theory

Testing via Symmetries Idea/Hope: If code designed to be “symmetric” and has local constraints, then it must be locally testable. Actuality: Not true completely generally But with some mild restrictions. Weakly true for “single-orbit properties” Strongly true for “lifted codes” Corollaries: Linearity testing, Low-degree testing … April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Lifting Results Lifting: Base Code 𝐵⊆ 𝑏: 𝔽 𝑞 𝑡 → 𝔽 𝑞 𝑚-dim. Lift 𝐿 𝑚 𝐵 = 𝑓: 𝔽 𝑞 𝑚 → 𝔽 𝑞 𝑓 𝐴 ∈𝐵 ∀𝑡−dim affine 𝐴 Theorems: ∀𝑞 ∃𝜖 ∀𝐵,𝑡, 𝑚 𝐿 𝑚 𝐵 is 𝑞 𝑡 ,𝜖 -LTC ∀𝛿 ∃𝛼 ∀𝑞,𝑡, 𝑚,𝐵, if 𝛿 𝐵 ≥𝛿 then 𝐿 𝑚 𝐵 is 𝛼-robust wrt 𝐵 2𝑡 -test. April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Single-Orbit Codes ℓ-Constraint: 𝐾= 𝑆,𝑉 ; 𝑆⊆ 𝔽 𝑞 𝑚 , 𝑉⊆ 𝑏:𝑆→ 𝔽 𝑞 , 𝑆 =ℓ 𝑓: 𝔽 𝑞 𝑚 → 𝔽 𝑞 satisfies 𝐾 if 𝑓 ​ 𝑆 ∈𝑉 ℓ-single orbit: 𝐶⊆ 𝑔: 𝔽 𝑞 𝑚 → 𝔽 𝑞 is ℓ-single orbit if there exists an ℓ-constraint 𝐾 s.t. 𝑓∈𝑃⇔∀ affine 𝐴: 𝔽 𝑞 𝑚 → 𝔽 𝑞 𝑚 , 𝑓∘𝐴 satisfies 𝐾 Lifted Property is 𝑞 𝑡 -single orbit; Single-orbit with 𝑆=subspace is “Lifted”. ∃ natural properties that are single-orbit, but not Lifted: e.g. deg ≤1: (𝑆= 0, 𝑒 1 , 𝑒 2 , 𝑒 1 + 𝑒 2 , 𝑉= 𝑓:𝑓 𝑒 1 −𝑓 0 =𝑓 𝑒 1 + 𝑒 2 −𝑓( 𝑒 2 ) April 9, 2016 Skoltech: Locality in Coding Theory

Testing Single-Orbit Codes Theorem: 𝐶⊆ 𝑓: 𝔽 𝑞 𝑚 → 𝔽 𝑞 is ℓ-single-orbit ⇒ 𝐶 is ℓ, 𝑂 ℓ −2 −LTC Extremely clean and general statement Unifies many of the “first” tests of properties. [BLR, GLRSW, RS, AKKLR, KR, JPRZ]. Clean simple proof. April 9, 2016 Skoltech: Locality in Coding Theory

ℓ-single-orbit ⇒ ℓ-locally testable 𝐶⊆ 𝔽 𝑞 𝑚 → 𝔽 𝑞 given by 𝑆={ 𝛼 1 ,…, 𝛼 ℓ },𝑉 : 𝑃= 𝑓 ∀𝐴, (𝑓∘𝐴) 𝑆 ∈𝑉 “Self-correction” based-proof: Fix 𝑓 s.t 𝜌≝Pr Rejecting 𝑓 small Define 𝑔 from 𝑓 locally Prove 𝑔 close to 𝑓 Prove 𝑔 satisfies constraint ∀𝐴 Only possible 𝑔 𝑥 = argmax 𝛽 Pr 𝐴:𝐴 𝛼 1 =𝑥 𝛽, 𝑓(𝐴 𝛼 2 ) ,…,𝑓(𝐴 𝛼 ℓ ) ∈𝑉 April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Analysis (contd.) Vote 𝐴 𝑥 =𝛽 𝑠.𝑡. 𝛽,𝑓 𝐴 𝛼 2 ,…,𝑓(𝐴 𝛼 ℓ ) ∈𝑉 𝑔 𝑥 = majority 𝐴:𝐴 𝛼 1 =𝑥 Vote 𝐴 𝑥 Key Lemma: ∀𝑥, Pr 𝐴,𝐵:𝐴 𝛼 1 =𝐵 𝛼 1 =𝑥 Vote 𝐴 𝑥 = Vote 𝐵 𝑥 ≥1 −2ℓ𝜌 [BLR,GLRSW,RS,AKLLR,KR,JPRZ] Proofs: Build a miracle ℓ×ℓ matrix M: Rows indexed by 𝐴 1 =𝐴, 𝐴 2 ,…, 𝐴 ℓ Columns by 𝐵 1 =𝐵, 𝐵 2 ,…, 𝐵 ℓ 𝑀 𝑖𝑗 = 𝐴 𝑖 𝛼 𝑗 = 𝐵 𝑗 𝛼 𝑖 ∀𝑖,𝑗 Typical row/column random Why does such a matrix exist? April 9, 2016 Skoltech: Locality in Coding Theory

Matrix Magic explained Wlog 𝐿 𝛼 1 ,…,𝐶( 𝛼 𝑡 ) independent; rest determined when 𝐶 random (affine). 𝑥 𝐴 𝛼 2 …𝐴( 𝛼 𝑡 ) … 𝐴 𝛼 ℓ Determined 𝐵( 𝛼 2 ) 𝑡 ⋮ Random 𝐵( 𝛼 𝑡 ) ⋮ Overdetermined? No! Linear algebra! 𝐵( 𝛼 ℓ ) 𝑡 April 9, 2016 Skoltech: Locality in Coding Theory

Robust testing of Lifted Codes Thm [GHS’15]: ∀𝛿 ∃𝛼 s.t. if 𝐵⊆ 𝑏: 𝔽 𝑞 𝑡 → 𝔽 𝑞 is a code of distance 𝛿 then 𝐿 𝑚 𝐵 is 𝛼-robust wrt 𝐵 2 -test. Test – not most natural one! Most natural: Inspect 𝑓 ​ 𝐴 for 𝑡-dim 𝐴 Our test: Inspect 𝑓 ​ 𝐴 for 2𝑡-dim 𝐴 Based on [Raz-Safra], [BenSassonS], …, [Viderman] Need to show: ∀𝑓 𝔼 𝐴 𝛿 𝑓 ​ 𝐴 ,𝐵 ≥𝛿(𝑓, 𝐿 𝑚 𝐵 ) Not previously known even when 𝑡=1 and 𝐵= 𝑏 | deg 𝑏 ≤𝑑 with 𝑑= 1−𝜖 𝑞 April 9, 2016 Skoltech: Locality in Coding Theory

Robust Testing of Lifted Codes For simplicity 𝐵⊆ 𝑏: 𝔽 𝑞 → 𝔽 𝑞 (𝑡=1). General geometry + symmetry ⇒ Robust analysis with 𝑚=4 ⇒ All 𝑚 How to analyze robustness of the test for constant 𝑚? April 9, 2016 Skoltech: Locality in Coding Theory

Tensors: Key to understanding Lifts Insight: 𝐿 𝑚 𝐵 = ∩ 𝑇 𝑇 𝐵 ⊗𝑚 Wishful Approach: Test verifies 𝛿 𝑓, 𝑇 𝐵 ⊗𝑚 small for random 𝑇 Maybe can combine this? Approach Fails 𝛿 𝐴 𝑓 , 𝛿 𝐵 𝑓 small ≢ 𝛿 𝐴∩𝐵 𝑓 small  April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Actual Analysis Say testing 𝐿 4 (𝐵) by querying 2-d subspace. Let 𝑃 𝑎 = {𝑓 | 𝑓 ​ line ∈𝐵 for all coordinate parallel lines, and lines in direction 𝑎} 𝐿 4 (𝐵)= ∩ 𝑎 𝑃 𝑎 ; 𝑃 𝑎 not a tensor code, but modification of tensor analysis works! ∪ 𝑎 𝑃 𝑎 ⊆ 𝐵 ⊗4 is still an error-correcting code. So 𝛿 𝑃 𝑎 𝑓 , 𝛿 𝑃 𝑏 𝑓 small ⇒ 𝛿 𝑃 𝑎 ∩ 𝑃 𝑏 𝑓 small! Putting things together ⇒ Theorem. April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Wrapping up Covered: LTCs and LDCs in high-rate regime. Many new developments: Simple, but surprising! Not Covered: LTCs and LDCs in low-query regime: [Yekhanin,Efremenko,Dvir-Gopi]: 3-query 𝔽 2 , 2-query 𝔽 2 𝑡 [Dinur, Meir, Viderman]: 3-query LTCs, Rate = 1/polylog n Open: LTCs + LDCs with ℓ 𝑛 =polylog 𝑛: Can you beat multivariate polynomials? April 9, 2016 Skoltech: Locality in Coding Theory

Skoltech: Locality in Coding Theory Thank You April 9, 2016 Skoltech: Locality in Coding Theory