1. of 29 12/09/2014 Invariance in Property Testing @MIT 1 Two Decades of Property Testing Madhu Sudan Microsoft Research TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A AAA

2. of 29 Kepler’s Big Data Problem Tycho Brahe (~1550-1600): Tycho Brahe (~1550-1600): Wished to measure planetary motion accurately. Wished to measure planetary motion accurately. To confirm sun revolved around earth … (+ other planets around sun) To confirm sun revolved around earth … (+ other planets around sun) Spent 10% of Danish GNP Spent 10% of Danish GNP Johannes Kepler (~1575-1625s): Johannes Kepler (~1575-1625s): Believed Copernicus’s picture: planets in circular orbits. Believed Copernicus’s picture: planets in circular orbits. Addendum: Ratio of orbits based on Löwner-John ratios of platonic solids. Addendum: Ratio of orbits based on Löwner-John ratios of platonic solids. “Stole” Brahe’s data (1601). “Stole” Brahe’s data (1601). Worked on it for nine years. Worked on it for nine years. Disproved Addendum; Confirmed Copernicus (circle -> ellipse); discovered laws of planetary motion. Disproved Addendum; Confirmed Copernicus (circle -> ellipse); discovered laws of planetary motion. Nine Years? Nine Years? 12/09/2014 Invariance in Property Testing @MIT 2 Source: Michael Fowler, “Galileo & Einstein”, U. Virginia

3. of 29 The challenge of analyzing big data Standard method: Standard method: Propose concept class. Propose concept class. LEARN (parameters of) best fitting concept in class to data in hand. LEARN (parameters of) best fitting concept in class to data in hand. TEST to see if this is a good enough fit. TEST to see if this is a good enough fit. Bottleneck Bottleneck LEARNing is expensive; wasted if TEST rejects. LEARNing is expensive; wasted if TEST rejects. Can we TEST before we LEARN? Can we TEST before we LEARN? Yes: This is PROPERTY TESTING!! Yes: This is PROPERTY TESTING!! 12/09/2014 Invariance in Property Testing @MIT 3 Don’t be Ridiculous!

4. of 29 Property Testing 12/09/2014 Invariance in Property Testing @MIT 4

5. of 29 Example 1: Polling 12/09/2014 Invariance in Property Testing @MIT 5

6. of 29 Example 2: Linearity 12/09/2014 Invariance in Property Testing @MIT 6

7. of 29 Linearity Analysis 12/09/2014 Invariance in Property Testing @MIT 7

8. of 29 A subtle change 12/09/2014 Invariance in Property Testing @MIT 8

9. of 29 12/09/2014 Invariance in Property Testing @MIT 9 History ( slightly abbreviated ) [Blum,Luby,Rubinfeld – S’90] [Blum,Luby,Rubinfeld – S’90] Linearity + application to program testing Linearity + application to program testing [Babai,Fortnow,Lund – F’90] [Babai,Fortnow,Lund – F’90] Multilinearity + application to PCPs (MIP). Multilinearity + application to PCPs (MIP). [Rubinfeld+S.] [Rubinfeld+S.] Low-degree testing + Definition Low-degree testing + Definition [Goldreich,Goldwasser,Ron] [Goldreich,Goldwasser,Ron] Graph property testing + systematic study Graph property testing + systematic study Since then … many developments Since then … many developments More graph properties, statistical properties, matrix properties, properties of Boolean functions … More graph properties, statistical properties, matrix properties, properties of Boolean functions … More algebraic properties More algebraic properties

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11. of 29 Example 4: Long code/Junta testing 12/09/2014 Invariance in Property Testing @MIT 11

12. of 29 Example 5: Distribution Testing 12/09/2014 Invariance in Property Testing @MIT 12

13. of 29 What is Property Testing? 12/09/2014 Invariance in Property Testing @MIT 13 Algebra Graphs + Regularity Statistics + CLT Matrices + Linear algebra

14. of 29 (Dense) Graph Property Testing 12/09/2014 Invariance in Property Testing @MIT 14

15. of 29 Why no unification? Contrast with Low-degree testing 12/09/2014 Invariance in Property Testing @MIT 15

16. of 29 Aside: Importance of Low-degree Testing 12/09/2014 Invariance in Property Testing @MIT 16

17. of 29 Some (introspective) questions What is qualitatively novel about linearity testing relative to classical statistics? What is qualitatively novel about linearity testing relative to classical statistics? Why are the mathematical underpinnings of different themes so different? Why are the mathematical underpinnings of different themes so different? Why is there no analog of “graph property testing” (broad class of properties, totally classified wrt testability) in algebraic world? Why is there no analog of “graph property testing” (broad class of properties, totally classified wrt testability) in algebraic world? What is the context for low-degree testing? What is the context for low-degree testing? Answer to all: Invariance! Answer to all: Invariance! 12/09/2014 Invariance in Property Testing @MIT 17

18. of 29 Invariance? 12/09/2014 Invariance in Property Testing @MIT 18

19. of 29 Invariances (contd.) 12/09/2014 Invariance in Property Testing @MIT 19

20. of 29 What is Property Testing? 12/09/2014 Invariance in Property Testing @MIT 20 Algebra=? ?

21. of 29 Abstracting algebraic properties 12/09/2014 Invariance in Property Testing @MIT 21

22. of 29 12/09/2014 Invariance in Property Testing @MIT 22 Testing Linear Properties Algebraic Property = Code! (usually) Universe Universe: {f:D  R} P Don’t care Must reject Must accept P R is a field F; P is linear!

23. of 29 Why study affine-invariance? Common abstraction of properties studied in [BLR], [RS], [ALMSS], [AKKLR], [KR], [KL], [JPRZ]. Common abstraction of properties studied in [BLR], [RS], [ALMSS], [AKKLR], [KR], [KL], [JPRZ]. (Variations on low-degree polynomials) (Variations on low-degree polynomials) Hopes Hopes Unify existing proofs Unify existing proofs Classify/characterize testability Classify/characterize testability Find new testable codes (w. novel parameters) Find new testable codes (w. novel parameters) Rest of the talk: Brief summary of findings Rest of the talk: Brief summary of findings 12/09/2014 Invariance in Property Testing @MIT 23

24. of 29 Results 1: AKKLR Conjecture 12/09/2014 Invariance in Property Testing @MIT 24

25. of 29 Results 2: Accidental +ve 12/09/2014 Invariance in Property Testing @MIT 25

26. of 29 Results 3: Lifting 12/09/2014 Invariance in Property Testing @MIT 26 Bad News + Bad News = Good News!

27. of 29 Result 3: Lifting (contd.) 12/09/2014 Invariance in Property Testing @MIT 27

28. of 29 Conclusions 12/09/2014 Invariance in Property Testing @MIT 28

29. of 29 Thank You 12/09/2014 Invariance in Property Testing @MIT 29