1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

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