Local Decoding and Testing Polynomials over Grids

Transcription

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 1 of 12

2 DeMillo-Lipton-Schwarz-Zippel Notation: FF = field, FF(dd, nn) = set of deg. dd, nn var. poly over FF δδ ff, gg = normalized Hamming distance δδ dd ff = min gg FF dd,nn δδ(ff, gg) DLSZ Lemma: Let SS FF, with 2 SS <. If ff, gg: SS nn FF satisfy ff gg FF dd, nn then δδ ff, gg 2 dd. Strengthens to δδ ff, gg 1 dd/ SS if dd < SS Holds even if SS = {0,1} For this talk: dd = constant DLSZ Lemma converts polynomials into error-correcting codes. With bonus: decodability, locality so δδ ff, gg bounded away from 0. January 11, 2018 ITCS: Polynomials over Grids 2 of 12

3 Local Decoding and Testing Informally: Given oracle access to ff: SS nn FF Testing: determine if δδ dd ff εε Decoding: If δδ dd ff δδ 0 and gg is nearest poly then determine value of gg(aa) given aa SS nn Parameters: Query complexity = # of oracle queries Testing accuracy Pr rejection δδ dd ff Decoding distance (δδ 0 above) Ideal: Query complexity OO dd (1), accuracy, dec. distance Ω dd (1) Terminology FF(dd, nn) testable (decodable) over SS if ideal achievable. Thms: For finite FF, FF(dd, nn) decodable over FF (folklore) and testable over FF ([Rubinfeld+S. 96]. [Kaufman+Ron 04]) January 11, 2018 ITCS: Polynomials over Grids 3 of 12

4 Polynomials over Grids Testing + decoding thms hold only for ff: FF nn FF 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 ff: SS nn FF for all finite SS This talk: Testing + decoding when SS = 0,1 grid (Note: equivalent to SS 1 SS nn with SS ii = 2) January 11, 2018 ITCS: Polynomials over Grids 4 of 12

5 Obstacles to decoding/testing Standard insight behind testing/decoding dd < FF ff has deg. dd iff ff restricted to lines has dd Can test/decode function on lines. Reduces complexity from FF nn to FF. When dd > FF 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: xx ii = 0 or xx ii = 1 xx ii = xx jj or xx ii = 1 xx jj (denoted xx ii = xx jj bb ) January 11, 2018 ITCS: Polynomials over Grids 5 of 12

6 Main Results Thm 1. FF(1, nn) not locally decodable over grid if char(ff) No 2-transitivity! Serious obstacle! Thm 2. FF(dd, nn) is locally decodable over grid if char FF <. O(max dd,char FF ) Query complexity = 2 Decoding without 2-transitivity! Thm 3. FF(dd, nn) is locally testable over grid ( FF) Testing without decoding! January 11, 2018 ITCS: Polynomials over Grids 6 of 12

7 Proofs: Not LDC over Q Idea dd = 1 : Consider ff = linear function that is erased on imbalanced points (of weight nn 2 cc nn, nn 2 + cc nn ) Claim 1: No local constraints over Q on balanced points and 0 nn Claim 1.1: if ff 0 nn = l ii=1 aa ii ff xx ii ff then 0 nn = l ii=1 aa ii xx ii Claim 1.2: 1 aa ii l! Claim 1.3: If xx 1,, xx l are εε-balanced jj jj then ii aa ii xx ii is εε ii aa ii Claim 2: No local constraints Not LDC Known over finite fields [BHR 04] Not immediate for Q -balanced January 11, 2018 ITCS: Polynomials over Grids 7 of 12

8 Proofs: LDC over small characteristic Given: ff δδ-close to gg FF(dd, nn), aa 0,1 nn (char FF = pp) Pick σσ: nn kk uniformly and let xx ii = aa ii yy σσ(ii) So considering ff yy 1,, yy kk = ff aa yy σσ : ff 0 = ff aa ; gg 0 = gg(aa) Claim 1: For balanced yy, have aa yy σσ aa Claim 2: If kk 2 = pptt and kk 2 > dd then gg (0) determined by gg (yy) yy = kk 2 (Usual ingredient Lucas s Theorem xx pp + yy pp = xx + yy pp ) January 11, 2018 ITCS: Polynomials over Grids 8 of 12

9 Proofs: LTC over all fields: Test Test Pick aa 0,1 nn uniformly and σσ: nn kk randomly Verify ff yy 1,, yy kk = ff aa yy σσ is of degree dd Randomly =? Not uniformly (don t know how to analyze). Instead random union-find Initially each xx ii in iith bucket. Iterate: pick two uniformly random buckets and merge till kk buckets left. Assign yy 1,, yy kk to kk buckets randomly. Allows for proof by induction January 11, 2018 ITCS: Polynomials over Grids 9 of 12

10 Proofs: LTC over all field: Analysis Key ingredients: (mimics BKSSZ analysis for RM) Main claim: Pr test rejects ff 2 OO(dd) min 1, δδ dd (ff) Proof by induction on nn with 3 cases: Case 1: ff moderately close to deg. dd Show that eventually ff disagrees from nearby poly on exactly one of 2 kk samples. Usual proof pairwise independence Our proof hypercontractivity of sphere [Polyanskiy] + prob. fact about random union-find Case 2: Pr ii,jj,bb ff xx ii =xx jj bb very close to deg. dd small Induction Case 3: ff far from deg. dd, but Pr ii,jj,bb high Use algebra to get contradiction. (Stitch different polynomials from diff. (ii, jj, bb) s to get nearby poly to ff) January 11, 2018 ITCS: Polynomials over Grids 10 of 12

11 Conclusions + Open questions Many aspects of polynomials well-understood only over domain = FF nn Grid setting (SS nn ) far less understood. When SS = {0,1} (equivalently SS = 2) testing possible without local decoding! (novel in context of low-degree tests!) General SS open!! (even SS = 3 or even SS = 0,1,2?) Is there a gap between characterizability and testability here? January 11, 2018 ITCS: Polynomials over Grids 11 of 12

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