Local Decoding and Testing Polynomials over Grids Madhu Sudan Harvard University Joint work with Srikanth Srinivasan (IIT Bombay) January 11, 2018 ITCS: Polynomials over Grids

DeMillo-Lipton-Schwarz-Zippel Notation: 𝔽 = field, 𝔽(𝑑,𝑛) = set of deg. ≤𝑑, 𝑛 var. poly over 𝔽 𝛿 𝑓,𝑔 = normalized Hamming distance 𝛿 𝑑 𝑓 = min 𝑔∈𝔽 𝑑,𝑛 𝛿(𝑓,𝑔) DLSZ Lemma: Let 𝑆⊆𝔽, with 2≤ 𝑆 <∞. If 𝑓,𝑔: 𝑆 𝑛 →𝔽 satisfy 𝑓≠𝑔∈𝔽 𝑑,𝑛 then 𝛿 𝑓,𝑔 ≥ 2 −𝑑 . Strengthens to 𝛿 𝑓,𝑔 ≥1−𝑑/|𝑆| if 𝑑< 𝑆 Holds even if 𝑆 = {0,1} For this talk: 𝑑 = constant DLSZ Lemma converts polynomials into error-correcting codes. With bonus: decodability, locality … so 𝛿 𝑓,𝑔 bounded away from 0. January 11, 2018 ITCS: Polynomials over Grids

Local Decoding and Testing Informally: Given oracle access to 𝑓: 𝑆 𝑛 →𝔽 … Testing: determine if 𝛿 𝑑 𝑓 ≤𝜖 Decoding: If 𝛿 𝑑 𝑓 ≤ 𝛿 0 and 𝑔 is nearest poly then determine value of 𝑔(𝑎) given 𝑎∈ 𝑆 𝑛 Parameters: Query complexity = # of oracle queries Testing accuracy ≡ Pr rejection 𝛿 𝑑 𝑓 Decoding distance ( 𝛿 0 above) Ideal: Query complexity 𝑂 𝑑 (1), accuracy, dec. distance Ω 𝑑 (1) Terminology 𝔽(𝑑,𝑛) testable (decodable) over 𝑆 if ideal achievable. Thms: For finite 𝔽, 𝔽(𝑑,𝑛) decodable over 𝔽 (folklore) and testable over 𝔽 ([Rubinfeld+S.’96] …. [Kaufman+Ron’04]) January 11, 2018 ITCS: Polynomials over Grids

Polynomials over Grids Testing + decoding thms hold only for 𝑓: 𝔽 𝑛 →𝔽 Not surprising … repeated occurrence with polynomials. Even classical decoding algorithms (non-local) work only in this case! [Kopparty-Kim’16]: Non-local Decoder (up to half the distance) for 𝑓: 𝑆 𝑛 →𝔽 for all finite 𝑆 This talk: Testing + decoding when 𝑆= 0,1 “grid” (Note: equivalent to 𝑆 1 ×⋯× 𝑆 𝑛 with 𝑆 𝑖 =2) January 11, 2018 ITCS: Polynomials over Grids

Obstacles to decoding/testing Standard insight behind testing/decoding 𝑑< 𝔽 𝑓 has deg. ≤𝑑 iff 𝑓 restricted to lines has ≤𝑑 Can test/decode function on lines. Reduces complexity from 𝔽 𝑛 to 𝔽. When 𝑑> 𝔽 use subspaces (= collection of linear restrictions) “Affine invariance” → “2 transitivity” … Problem over grids: General lines don’t stay within grid! Only linear restrictions that stay in grid: 𝑥 𝑖 =0 or 𝑥 𝑖 =1 𝑥 𝑖 = 𝑥 𝑗 or 𝑥 𝑖 =1 − 𝑥 𝑗 (denoted “ 𝑥 𝑖 = 𝑥 𝑗 ⊕𝑏”) January 11, 2018 ITCS: Polynomials over Grids

ITCS: Polynomials over Grids Main Results Thm 1. 𝔽(1,𝑛) not locally decodable over grid if char(𝔽) →∞ No 2-transitivity! Serious obstacle! Thm 2. 𝔽(𝑑,𝑛) is locally decodable over grid if char 𝔽 <∞. Query complexity = 2 O(max 𝑑,char 𝔽 ) Decoding without 2-transitivity! Thm 3. 𝔽(𝑑,𝑛) is locally testable over grid (∀𝔽) Testing without decoding! January 11, 2018 ITCS: Polynomials over Grids

ITCS: Polynomials over Grids Proofs: Not LDC over ℚ Idea 𝑑=1 : Consider 𝑓 = linear function that is erased on “imbalanced” points (of weight ∉ 𝑛 2 −𝑐 𝑛 , 𝑛 2 +𝑐 𝑛 ) Claim 1: No local constraints over ℚ on balanced points and 0 𝑛 Claim 1.1: if 𝑓 0 𝑛 = 𝑖=1 ℓ 𝑎 𝑖 𝑓 𝑥 𝑖 ∀𝑓 then 0 𝑛 = 𝑖=1 ℓ 𝑎 𝑖 𝑥 𝑖 Claim 1.2: 1≤ 𝑎 𝑖 ≤ℓ! Claim 1.3: If 𝑥 1 ,…, 𝑥 ℓ are 𝜖-balanced then 𝑖 𝑗 𝑎 𝑖 𝑥 𝑖 is 𝜖⋅ 𝑖 𝑗 𝑎 𝑖 -balanced Claim 2: No local constraints ⇒ Not LDC Known over finite fields [BHR’04] Not immediate for ℚ January 11, 2018 ITCS: Polynomials over Grids

Proofs: LDC over small characteristic Given: 𝑓 𝛿-close to 𝑔∈𝔽(𝑑,𝑛), 𝑎∈ 0,1 𝑛 (char 𝔽 =𝑝) Pick 𝜎: 𝑛 → 𝑘 uniformly and let 𝑥 𝑖 = 𝑎 𝑖 ⊕ 𝑦 𝜎(𝑖) So considering 𝑓 ′ 𝑦 1 ,…, 𝑦 𝑘 =𝑓 𝑎⊕ 𝑦 𝜎 : 𝑓 ′ 0 =𝑓 𝑎 ; 𝑔 ′ 0 =𝑔(𝑎) Claim 1: For balanced 𝑦, have 𝑎⊕ 𝑦 𝜎 ⊥𝑎 Claim 2: If 𝑘 2 = 𝑝 𝑡 and 𝑘 2 >𝑑 then 𝑔′(0) determined by 𝑔′(𝑦) | 𝑦 = 𝑘 2 (Usual ingredient “Lucas’s Theorem” 𝑥 𝑝 + 𝑦 𝑝 = 𝑥+𝑦 𝑝 ) January 11, 2018 ITCS: Polynomials over Grids

Proofs: LTC over all fields: Test Pick 𝑎∈ 0,1 𝑛 uniformly and 𝜎: 𝑛 → 𝑘 “randomly” Verify 𝑓 ′ 𝑦 1 ,…, 𝑦 𝑘 =𝑓 𝑎⊕ 𝑦 𝜎 is of degree 𝑑 Randomly = ? Not uniformly (don’t know how to analyze). Instead “random union-find” Initially each 𝑥 𝑖 in 𝑖th bucket. Iterate: pick two uniformly random buckets and merge till 𝑘 buckets left. Assign 𝑦 1 ,…, 𝑦 𝑘 to 𝑘 buckets randomly. Allows for proof by induction January 11, 2018 ITCS: Polynomials over Grids

Proofs: LTC over all field: Analysis Key ingredients: (mimics BKSSZ analysis for RM) Main claim: Pr test rejects 𝑓 ≥ 2 −𝑂(𝑑) ⋅ min 1, 𝛿 𝑑 (𝑓) Proof by induction on 𝑛 with 3 cases: Case 1: 𝑓 moderately close to deg. 𝑑 Show that eventually 𝑓 disagrees from nearby poly on exactly one of 2 𝑘 samples. Usual proof – pairwise independence Our proof – hypercontractivity of sphere [Polyanskiy] + prob. fact about random union-find Case 2: Pr 𝑖,𝑗,𝑏 𝑓 ​ 𝑥 𝑖 = 𝑥 𝑗 ⊕𝑏 very close to deg. 𝑑 small Case 3: 𝑓 far from deg. 𝑑, but Pr 𝑖,𝑗,𝑏 … high Use algebra to get contradiction. (Stitch different polynomials from diff. (𝑖,𝑗,𝑏)’s to get nearby poly to 𝑓) Induction January 11, 2018 ITCS: Polynomials over Grids

Conclusions + Open questions Many aspects of polynomials well-understood only over domain = 𝔽 𝑛 Grid setting ( 𝑆 𝑛 ) far less understood. When 𝑆={0,1} (equivalently 𝑆 =2) testing possible without local decoding! (novel in context of low-degree tests!) General 𝑆 open!! (even 𝑆 =3 or even 𝑆= 0,1,2 ?) Is there a gap between characterizability and testability here? January 11, 2018 ITCS: Polynomials over Grids

ITCS: Polynomials over Grids Thank You! January 11, 2018 ITCS: Polynomials over Grids