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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 1 of 12 DeMillo-Lipton-Schwarz-Zippel Notation:
= field, = set of deg. , var. poly over = normalized Hamming distance DLSZ Lemma: Let , with If satisfy then Strengthens to if Holds even if For this talk: = constant away from 0. DLSZ Lemma converts polynomials so bounded into error-correcting codes. With bonus: decodability, locality January 11, 2018 ITCS: Polynomials over Grids 2 of 12 Local Decoding and Testing
Informally: Given oracle access to Testing: determine if Decoding: If and is nearest poly then determine value of given Parameters: Query complexity = # of oracle queries Testing accuracy Decoding distance ( above) Ideal: Query complexity , accuracy, dec. distance Terminology testable (decodable) over if ideal achievable. Thms: For finite , decodable over (folklore) and testable over ([Rubinfeld+S.96] . [Kaufman+Ron04]) January 11, 2018 ITCS: Polynomials over Grids 3 of 12 Polynomials over Grids
Testing + decoding thms hold only for Not surprising repeated occurrence with polynomials. Even classical decoding algorithms (non-local) work only in this case! [Kopparty-Kim16]: Non-local Decoder (up to half the distance) for for all finite This talk: Testing + decoding when grid (Note: equivalent to with ) January 11, 2018 ITCS: Polynomials over Grids 4 of 12 Obstacles to decoding/testing Standard insight behind testing/decoding
When use subspaces has deg. iff restricted to lines has Can test/decode function on lines. Reduces complexity from to . (= collection of linear restrictions) Affine invariance 2 transitivity Problem over grids: General lines dont stay within grid! Only linear restrictions that stay in grid: or or (denoted
) January 11, 2018 ITCS: Polynomials over Grids 5 of 12 Main Results Thm 1. not locally decodable over grid if char() No 2-transitivity! Serious obstacle! Thm 2. is locally decodable over grid if . Query complexity = Decoding without 2-transitivity! Thm 3. is locally testable over grid () Testing without decoding! January 11, 2018
ITCS: Polynomials over Grids 6 of 12 Proofs: Not LDC over Idea : Consider = linear function that is erased on imbalanced points (of weight ) Claim 1: No local constraints over on balanced points and Claim 1.1: if then Claim 1.2: Claim 1.3: If are -balanced then is -balanced Claim 2: No local constraints Not LDC
Known over finite fields [BHR04] Not immediate for January 11, 2018 ITCS: Polynomials over Grids 7 of 12 Proofs: LDC over small characteristic Given: -close to , () Pick uniformly and let So considering : ; Claim 1: For balanced , have Claim 2: If and then determined by January 11, 2018
(Usual ingredient Lucass Theorem ) ITCS: Polynomials over Grids 8 of 12 Proofs: LTC over all fields: Test Test Pick uniformly and randomly Verify is of degree Randomly = ? Not uniformly (dont know how to analyze). Instead random union-find
Initially each in th bucket. Iterate: pick two uniformly random buckets and merge till buckets left. Assign to buckets randomly. Allows for proof by induction January 11, 2018 ITCS: Polynomials over Grids 9 of 12 Proofs: LTC over all field: Analysis Key ingredients: (mimics BKSSZ analysis for RM) Main claim: Proof by induction on with 3 cases: Case 1: moderately close to deg. Show that eventually disagrees from nearby poly on exactly one of samples. Usual proof pairwise independence Our proof hypercontractivity of sphere
[Polyanskiy] + prob. fact about random unionfind Induction Case 2: small Case 3: far from deg. , but high Use algebra to get contradiction. (Stitch different polynomials from diff. s to get nearby poly to ) January 11, 2018 ITCS: Polynomials over Grids 10 of 12 Conclusions + Open questions Many aspects of polynomials well-understood only over domain = Grid setting () far less understood. When (equivalently ) testing possible without
local decoding! (novel in context of low-degree tests!) General open!! (even or even ?) Is there a gap between characterizability and testability here? January 11, 2018 ITCS: Polynomials over Grids 11 of 12 Thank You! January 11, 2018 ITCS: Polynomials over Grids 12 of 12
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