Low-Degree Polynomials
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1 Low-Degree Polynomials Madhu Sudan Harvard University October 7, 2016 Avi60!! Low-degree Polynomials 1 of 27
2 October 7, 2016 Avi60!! Low-degree Polynomials 2 of 27
3 Happy Birthday, Avi!! October 7, 2016 Avi60!! Low-degree Polynomials 3 of 27
4 Happy Birthday Avi!! Polynomials Cheeseburger again! October 7, 2016 Avi60!! Low-degree Polynomials 4 of 27
5 This talk Polynomials How to correct them? How to test them? Some history (Why?)!!! Current status October 7, 2016 Avi60!! Low-degree Polynomials 5 of 27
6 GLRSW-91 October 7, 2016 Avi60!! Low-degree Polynomials 6 of 27
7 Main Results Thm 1: deg. dd poly s are OO dd, Ω 1 -resilient. Defn: Family of functions FF is kk, δδ 0 -resilient if DD s.t. ff FF, gg, xx s.t. δδ ff, gg δδ 0, DD gg xx = ff xx, and DD makes kk calls to gg. Thm 2: deg. dd polynomials are dd + 2, 1 dd 3 -robust. October 7, 2016 Avi60!! Low-degree Polynomials 7 of 27
8 Resilience (visually) xx ff FF 1 2 DD gg (xx) k gg δδ ff Defn: Family of functions FF is kk, δδ 0 -resilient if DD s.t. ff FF, gg, xx s.t. δδ ff, gg δδ 0, DD gg xx = ff xx, and DD makes kk calls to gg. October 7, 2016 Avi60!! Low-degree Polynomials 8 of 27
9 Main Results Thm 1: deg. dd poly s are OO dd, Ω 1 -resilient. Defn: Family of functions FF is kk, δδ 0 -resilient if DD s.t. ff FF, gg, xx s.t. δδ ff, gg δδ 0, DD gg xx = ff xx, and DD makes kk calls to gg. Thm 2: deg. dd polynomials are dd + 2, 1 dd 3 -robust. Defn: FF {ff: SS TT} is (kk, αα)-robust if distribution PP on SS kk s.t ff δδ llllllllll ff EE VV PP δδ ff VV, FF VV αα δδ(ff, FF) October 7, 2016 Avi60!! Low-degree Polynomials 9 of 27
10 Background/Motivation: Permanent: Worst-case Average-case Analysis: [Lipton 90]: (building on [Beaver-Feigenbaum]): PPPPPPPP AA + iiii = deg. nn poly in ii. Thm: Deg. dd poly s are OO(dd), Ω dd 1 Cor: nn nn-permanent is OO nn, Ω nn 1 -resilient. ii nn + 1, gg AA + iiii = PPPPPPPP AA + iiii GG RR ii PPPPPPPP(AA + iiii) Worst-case complexity of permanent average-case complexity of permanent. Proof of Cor: Suppose Pr M gg MM PPPPPPPP MM = δδ OO nn 1 -resilient. AA, ii Pr gg AA + iiii PPPPPPPP AA + iiii = δδ RR Pr ii nn + 1 gg AA + iiii PPPPPPPP(AA + iiii)] (nn + 1)δδ 1/3 RR Union Bound Then AA, PPPPPPPP AA = GG RR (0) w.h.p. over RR where GG RR = PPPPPPPP IIIIIIIIIIIIIIIIIIIIII ii, gg AA + iiii ii nn + 1. October 7, 2016 Avi60!! Low-degree Polynomials 10 of 27
11 Improving Resilience? Error-correcting codes (ECC) to the rescue!! [GLRSW]: (PPPPPPPP AA + iiii ii [10nn]) forms an ECC which can be decoded from Ω(1)-fraction errors efficiently! Don t need to get PPPPPPPP AA + iiii = gg AA + iiii for every ii nn + 1. Instead suffices to get equality for 90% of ii 10nn. Replace Union Bound by Markov. Reed-Solomon Codes! Resilience goes up to Ω 1 from Ω nn 1 October 7, 2016 Avi60!! Low-degree Polynomials 11 of 27
12 Error-Correcting Codes!! In TCS/1990: ECCs not widely used/understood.!!!! My first introduction to ECCs! Thanks, Avi & Dick! October 7, 2016 Avi60!! Low-degree Polynomials 12 of 27
13 Some followup I m not happy [Blum, personal comm. 90] What are BCH codes? How do you decode them? What is the Ω(1) in the resilience? Our answers: Read [17] Read [17, Chapters 3, 9, 12] Blum: Go talk to Elwyn yada yada yada [Gemmell,S. 92]: Poly s are OO dd, 1 εε -resilient! 2 Decoding Berlekamp-Welch Decoder October 7, 2016 Avi60!! Low-degree Polynomials 13 of 27
14 1991: Avi visits Berkeley! (Pete+) Me Avi: Resilience is 1 2! Avi: Great. Now tell me how to make it higher! Me: What do you want? we are optimal! (under unique decoding) Avi: Everything. You are not. What if there are two polynomial pp, qq agreeing with gg on 50% of points each? (let me tell you about list-decoding ) Btw, Sigal, Dick and Ronitt can recover both pp and qq in the above case. blah blah blah symmetric functions blah blah blah factor quadratics blah blah blah October 7, 2016 Avi60!! Low-degree Polynomials 14 of 27
15 Damage Report from Avi s visit [Ar,Lipton,Rubinfeld,S. 92]: Can recover pp xx, qq xx from gg if yy gg yy = {pp yy, qq yy } and each polynomial represented often. In retrospect, I misunderstood Avi and his claims about [Sigal, Dick, Ronitt ]. [S.96,Guruswami+S.98,S.+Trevisan+Vadhan 99] Thm 1 : Poly s are OO dd, 1 oo(1) - resilient. Can recover ff 1 xx,, ff kk xx from gg s.t if Pr xx ff xx gg xx 1 1/kk then ff ff 1,, ff kk October 7, 2016 Avi60!! Low-degree Polynomials 15 of 27
16 Concluding Part 1 Thanks to Avi for introducing me to the many wonders of polynomials! to error-correcting codes to list-decoding to (accidentally?) triggering follow up. For his curiosity and (infectious?) optimism. October 7, 2016 Avi60!! Low-degree Polynomials 16 of 27
17 Part 2: Testing Polynomials October 7, 2016 Avi60!! Low-degree Polynomials 17 of 27
18 Main Results Thm 1: deg. dd poly s are OO dd, Ω 1 -resilient. Defn: Family of functions FF is kk, δδ 0 -resilient if DD s.t. ff FF, gg, xx s.t. δδ ff, gg δδ 0, DD gg xx = ff xx, and DD makes kk calls to gg. Thm 2: deg. dd polynomials are dd + 2, 1 dd 3 -robust. Defn: FF {ff: SS TT} is (kk, αα)-robust if distribution PP on SS kk s.t ff δδ llllllllll ff EE VV PP δδ ff VV, FF VV 1 oo(1) [STV 99] X αα δδ(ff, FF) October 7, 2016 Avi60!! Low-degree Polynomials 18 of 27
19 Why test? Probabilistically Checkable Proofs! Polynomials are building blocks of complexity. Complex statements (OO nn bit statements) simple statements (OO(1) long) about complex polynomials (OO nn coefficients). Low-degree testing: Verifying truth of complex statements verifying simple statements (about complex polynomials). October 7, 2016 Avi60!! Low-degree Polynomials 19 of 27
20 An Analogy Inspecting a building: Building = OO(nn) atoms OR Building = OO(1) rooms = OO(1) walls Former view: Verifying stability takes Ω nn -checks. Latter view: Verifying stability takes OO 1 -checks + OO(1)- stability of wall-checks. Polynomials Walls! October 7, 2016 Avi60!! Low-degree Polynomials 20 of 27
21 Polynomials = Walls? A (NP-)complete statement: Graph GG 0,1 nn nn is 3-colorable. Proof: Coloring (Θ(nn)-bits). Verification: Read entire coloring. Equivalent (NP-)complete statement: Given: Φ local map from poly s to poly s poly s AA, BB, CC, DD s.t. Φ AA, BB, CC, DD 0 Verification: Step 1: Test AA, BB, CC, DD are polynomials Step 2: Verify Φ AA, BB, CC, DD rr = 0 for random rr. October 7, 2016 Avi60!! Low-degree Polynomials 21 of 27
22 Polynomials = Wall - II Reduction from 3-coloring to polynomial satisfiability [Ben-Sasson-S. 04] Φ AA, BB, CC, DD xx 0, xx, yy = Φ EE AA, BB, CC, DD [xx 0, xx, yy] = (AA xx AA xx 1 AA xx 2 BB xx Π vv VV xx vv ) + xx 0 (EE xx, yy Π ii 2, 1,1,2 AA xx AA yy ii CC xx, yy Π vv VV xx vv DD xx, yy Π vv VV yy vv ) October 7, 2016 Avi60!! Low-degree Polynomials 22 of 27
23 Low-degree Testing Better αα? Ω 1? 1 oo 1? Thm 2: deg. dd polynomials are dd + 2, 1 dd 3 -robust. Definition: FF {ff: SS TT} is (kk, αα)-robust if distribution PP on SS kk s.t ff δδ llllllllll ff EE VV PP δδ ff VV, FF VV αα δδ(ff, FF) Better testers etc. Better PCPs [ALMSS 92, PS 94,AS 98,RS 98,GS 02,BSVW 04,BGHSV 05,MR 08] Robustness: [Rubinfeld-S. 92]: αα = Ω(dd 1 ) [ALMSS 92]: αα = Ω 1 Thanks again, Avi! [Raz-Safra 98,Arora-S. 98]: αα = 1 oo(1) October 7, 2016 Avi60!! Low-degree Polynomials 23 of 27
24 Other reasons to test XOR lemma for polynomials δδ ff > 0 δδ ff tt δδ rrrrrrrrrrrr as tt Low-degree testing: δδ llllllllll ff αα δδ(ff) For some tests δδ llllllllll (ff tt ) determined completely by δδ llllllllll (ff) and tt 1 2δδ llllllllll ff tt = 1 2δδ llllllllll ff tt [Gowers]: For same tests δδ FF lower bounded by some function of δδ llllllllll FF. δδ FF 1 1 2δδ llllllllll FF [ViolaWigderson 04]: Small set expanders and Short Long Codes omitted. [BGHMRS] 2 dd October 7, 2016 Avi60!! Low-degree Polynomials 24 of 27
25 State-of-the-art: Low-degree testing Long history omitted, Variations omitted. Families of interest: PP(mm, dd, qq): mm-var. deg. dd over FF qq Benchmark: distance δδ qq,dd of code PP(mm, dd, qq) Thm. 1: δδ qq,dd 1 αα 1 Thm. 2: δδ qq,dd > 0 αα > 0 Thm. 3: δδ qq,dd = oo 1 αα = Ω(δδ qq,dd ) [Raz-Safra 98]: [GHS 15]: αα < 1 ββ > 0 s.t. PP mm, ββββ, qq is qq 2, αα -robust [BKSSZ 09, HSS 11, HRS 15]: δδ > 0 αα > 0 s.t. PP mm, 1 δδ qq, qq is (qq 2, αα)-robust qq εε > 0 s.t. PP(mm, dd, qq) is (δδ 1 qq,dd, εε δδ qq,dd )-robust October 7, 2016 Avi60!! Low-degree Polynomials 25 of 27
26 Techniques? Idea 1: Use symmetry to reduce highdimensional case to low-dimension. PP mm, qq, dd PP(cc, qq, dd) (Gets mm independence). Idea 2: Find some slightly weaker, but better understood, code that contains code of interest. Typically tensor : PP cc, qq, dd PP 1, qq, dd cc Works quite generally : need symmetry + supercode. Doesn t work when tensoring doesn t! No testing thm yet for multiplicity codes! October 7, 2016 Avi60!! Low-degree Polynomials 26 of 27
27 Summary: Thanks again Avi! For generously contributing ideas (and your students ) Many inspiring moments (about low-degree polynomials, and everything else!) October 7, 2016 Avi60!! Low-degree Polynomials 27 of 27
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