Fourier sparsity , spectral norm , and the Log-rank conjecture

1. Fourier sparsity, spectral norm, and theLog-rank conjecture arXiv:1304.1245 HingYin Tsang1, Chung Hoi Wong1, Ning Xie2, Shengyu Zhang1 The Chinese University of Hong Kong Florida International University

2. Motivation 1: Fourier analysis Bool Fourier (sparsified) • Parseval: If , then. • Spectral norm:. Fourier sparsity: Qustion: What can we say about Boolean with smallor ? Characterization?

3. Some known results • Results on learnability*1,testability*2, etc. • A structural result by Green and Sanders. • Theorem*3. can be written as , where and ’s are subspaces. • Question: Improve the doubly exponential bound? *1. Kushilevitz, Mansour, SIAM J. on Computing, 1993. *2. Gopalan, O’Donnell, Servedio, Shpilka, Wimmer, SIAM J. on Computing, 2011. *3. Green and Sanders. Geometric and Functional Analysis, 2008.

4. Motivation 2: Communication complexity • Two parties, Alice and Bob, jointly compute a function . • known only to Alice and only to Bob. • Communication complexity*1: how many bits are needed to be exchanged? --- *1. Yao. STOC, 1979.

5. Log-rank conjecture Log Rank Conjecture*2 • Rank lower bound*1 • combinatorial measure linear algebra measure. • Equivalent to a bunch of other conjectures. • related to graph theory*2; nonnegative rank*3, Boolean roots of polynomials*4, quantum sampling complexity*5. • Largest known gap*6: • Best previous upper bound*7: • Conditional*8: *1. Melhorn, Schmidt. STOC, 1982. *2. Lovász, Saks. FOCS, 1988. *3. Lovász. Book Chapter, 1990. *4. Valiant. Info. Proc. Lett., 2004. *5. Ambainis, Schulman, Ta-Shma, Vazirani, Wigderson, SICOMP 2003. *6. Nisan and Wigderson. Combinatorica, 1995. *7. Kotlov. Journal of Graph Theory, 1982. *8. Ben-Sasson, Lovett, Ron-Zewi, FOCS, 2012.

6. Special class of functions • Since Log-rank conjecture appears too hard in its full generality,… • XOR functions: . --- • Include important functions such as Equality, Hamming Distance, Gap Hamming Distance. • Connection to Fourier: . • One approach*1: • : Parity decision tree complexity. (DT with queries like “”) *1. Zhang and Shi. Theoretical Computer Science, 2009.

7. One easy case • The (total) degree of as a multi-linear polynomial over . • If , then even the standard decision tree complexity is small *1,2. • Question: Are all nonzero Fourier coefficients always located in low levels? • Answer*3: Not even after change of basis. • There are with but . *1. Nisan and Smolensky. Unpublished. *2. Midrijanis. arXiv/quant-ph/0403168, 2004. *3. Zhang and Shi. Theoretical Computer Science, 2009.

8. Previous work • Special cases for . • : Symmetric *1 • : LTF *2 • : monotone *2 • : *3 • Hard case: much larger than • not touched yet. *1. Zhang and Shi. Quantum Information & Computation, 2009. *2. Montanaro and Osborne. arXiv:0909.3392v2, 2010. *3. Kulkarni and Santha. CIAC, 2013.

9. Our results: starting point • While is not a good bridge between and , another degree may be. • : degree of as a polynomial over . • Compared to Fourier sparsity, is always small. • Fact*1. . *1. Bernasconi and Codenotti. IEEE Transactions on Computers, 1999.

10. Our results: constant degree • Theorem 1. For with : • Log-rank conjecture holds for . Dependence on : “only” singly exponential. • Fourier sparseshort -DT • depends only on linear functions of input variables.

11. Our results: constant degree • [GS08] can be written as , where and ’s are subspaces. • Theorem 2: If , then we improve doubly exponential bound to quasi-polynomial: [GS08] Green and Sanders. Geometric and Functional Analysis, 2008.

12. Our results: small spectral norm • Theorem 3. For any Boolean , • , i.e. there is a large affine subspace (co-dim: ) on which is const. • . • Independent work [SV13]: , • Our bounds are quadratically better. [SV13]Shpilkaand Volk. ECCC, 2013.

13. Our results: small spectral norm • Theorem 3. … • Corollary 4. • Recall: before our work, even slightly sublinear bound is conditional. • Later [Lov13]: for general Boolean . [Lov13]Lovett. ECCC, 2013.

14. Our results: small spectral norm • [Gro97]. , • Corollary 5. , [Gro97] Grolmusz. Theoretical Computer Science, 1997.

15. Our results: light tail • Theorem 6. If is sufficiently close to -sparse, then Log-rank Conjecture holds for . • Sufficiently close to sparse: has a light tail, in .

16. Techniques *1. Chang. Duke Mathematical Journal, 2002. *2. Gopalan, O’Donnell, Servedio, Shpilka, Wimmer. SIAM Journal on Computing, 2011.

17. Approach: Degree reduction • : min s.t. • . • Theorem*1,2. with bias and degree, . • Doesn’t work for us: . • Even worse, the dependence on is horrible. • Thus impossible to generalize to . *1. Green and Tao. Contributions to Discrete Mathematics, 2009. *2. Kaufman and Lovett. FOCS, 2008.

18. A new rank • Linear polynomial rank (-): min s.t. where is linear and has • Compared to ,defined by ,we require a special . • Thus - • But we’ll show that even this -is small. • Now given - decomposation of , let’s see how to design a protocol for .

19. Main Protocol • Linear polynomial rank (-): min s.t. where is linear and has • Main protocol: rounds; each round reduces -degree by at least 1 • regardless of values of and

20. Main conjecture • Of course, communication cost depends on how large -is. • Conjecture 1. Boolean , - • Conjecture 1Log-rank Conj. for all XOR fn’s. • Most our results obtained by bounding -. • One simple bound: - • Decreasing degree is easier than .

21. Sketch of proof of Thm 1, item (1) • Thm 1, item (1): • Sufficient to show that is small • Approach: Induction on . Apply IH on (discrete) derivative. • Derivative: . • Fact. . • Fact. .

22. Sketch of proof of Thm 1, item (1) • By IH, affine w/ small codim&. • Lemma. When restricted on , half-space of Fourier coefficients disappear. • Study and their Fourier spectra. • , . • , . • On : vanishes, so does or . • Repeating times: kill all Fourier coeff.

23. Summary

24. Open • Prove -. • No counterexample even for • Other applications of -? • 1-hour talk at tomorrow’s one-day workshop on real analysis • 11:30 at Simons

25. Thanks