LowDegree Testing Madhu Sudan MSR Survey based on

Low-Degree Testing Madhu Sudan MSR Survey … based on many works 09/02/2015 CMSA: Low-degree Testing 1 of 25

Kepler’s Problem • Tycho Brahe (~1550 -1600): – Wished to measure planetary motion accurately. – To confirm sun revolved around earth … (+ other planets around sun) – Spent 10% of Danish GNP • Johannes Kepler (~1575 -1625 s): – Believed Copernicus’s picture: planets in circular orbits. – Addendum: Ratio of orbits based on Löwner-John ratios of platonic solids. – “Stole” Brahe’s data (1601). Source: Michael Fowler, “Galileo & Einstein”, U. Virginia – Worked on it for nine years. – Disproved Addendum; Confirmed Copernicus (circle -> ellipse); discovered laws of planetary motion. • Nine Years? – To check if data fits a low-degree polynomial? 09/02/2015 CMSA: Low-degree Testing 2 of 25

Low-degree Testing • Warning: Refinements and Variations later. 09/02/2015 CMSA: Low-degree Testing 3 of 25

This talk • Some motivations • Some results • Some proofs 09/02/2015 CMSA: Low-degree Testing 4 of 25

Why Polynomials? Robustness! • 09/02/2015 CMSA: Low-degree Testing 5 of 25

Formal Definitions and Parameters • 09/02/2015 local distance CMSA: Low-degree Testing vs. global distance 6 of 25

• 09/02/2015 CMSA: Low-degree Testing 7 of 25

Generalizing to higher d • 09/02/2015 CMSA: Low-degree Testing 8 of 25

Why Low-degree Testing? • 09/02/2015 CMSA: Low-degree Testing 9 of 25

Why low-degree testing - II • 09/02/2015 CMSA: Low-degree Testing 10 of 25

Formal Definitions and Parameters • 09/02/2015 CMSA: Low-degree Testing 11 of 25

A natural test • 09/02/2015 CMSA: Low-degree Testing 12 of 25

Locality of subspace tests • 09/02/2015 CMSA: Low-degree Testing 13 of 25

Results • 09/02/2015 CMSA: Low-degree Testing 14 of 25

Theorem 1: Context + Ideas • 09/02/2015 CMSA: Low-degree Testing 15 of 25

Theorem 1 (contd. ) • 09/02/2015 CMSA: Low-degree Testing 16 of 25

Hyperplanes • 09/02/2015 CMSA: Low-degree Testing 17 of 25

• 09/02/2015 CMSA: Low-degree Testing 18 of 25

Theorem 2: Ideas • 09/02/2015 CMSA: Low-degree Testing 19 of 25

Theorem 2 (contd. ) • 09/02/2015 CMSA: Low-degree Testing 20 of 25

Testing Lifted Codes - 1 • 09/02/2015 CMSA: Low-degree Testing 21 of 25

Tensors: Key to understanding Lifts • 09/02/2015 CMSA: Low-degree Testing 22 of 25

Actual Analysis • 09/02/2015 CMSA: Low-degree Testing 23 of 25

Wrapping up • Low-degree testing: – Basic, easy to state, problem. – Quite useful in complexity, combinatorics. – Powerful theorems known. • Other connections? 09/02/2015 CMSA: Low-degree Testing 24 of 25

Thank You! 09/02/2015 CMSA: Low-degree Testing 25 of 25

(Appendix) References • Page 7 – – • • Manuel Blum, Michael Luby, Ronitt Rubinfeld: Self-Testing/Correcting with Applications to Numerical Problems. J. Comput. Syst. Sci. 47(3): 549 -595 (1993) Mihir Bellare, Don Coppersmith, Johan Håstad, Marcos A. Kiwi, Madhu Sudan: Linearity testing in characteristic two. IEEE Transactions on Information Theory 42(6): 1781 -1795 (1996) Page 9 (See references in survey below) – Probabilistically Checkable Proofs , Madhu Sudan. Communications of the ACM, 52(3): 76 -84, March 2009. Page 14 – Optimal testing of Reed-Muller codes. Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, and David Zuckerman. Electronic Colloquium on Computational Complexity , Technical Report TR 09 -086, October 2, 2009. – Optimal testing of multivariate polynomials over small prime fields. Elad Haramaty, Amir Shpilka, and Madhu Sudan. SIAM Journal on Computing, 42(2): 536 --562, April 2013. – Absolutely Sound Testing of Lifted Codes. Elad Haramaty, Noga Ron-Zewi and Madhu Sudan. Electronic Colloquium on Computational Complexity , Technical Report TR 13 -030, February 20, 2013. – Alan Guo, Elad Haramaty, Madhu Sudan: Robust testing of lifted codes with applications to low-degree testing. Electronic Colloquium on Computational Complexity (ECCC) 22: 43 (2015) – Ran Raz, Shmuel Safra: A Sub-Constant Error-Probability Low-Degree Test, and a Sub-Constant Error-Probability PCP Characterization of NP. STOC 1997: 475 -484 09/02/2015 CMSA: Low-degree Testing 26 of 25