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

Fourier sparsity , spectral norm , and the Log-rank conjecture. arXiv :1304.1245 Hing Yin Tsang 1 , Chung Hoi Wong 1 , Ning Xie 2 , Shengyu Zhang 1. The Chinese University of Hong Kong Florida International University. Motivation 1: Fourier analysis. Bool. Fourier.

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

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  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 • Not only interesting on its own, but also important because of numerous applications. • to prove lower bounds. • Question: How to lower bound communication complexity itself? • Communication matrix

  6. Log-rank conjecture Log Rank Conjecture*2 • Rank lower bound*1 • The rank lower bound is tight. • 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.

  7. Log-rank conjecture for XOR functions • Since Log-rank conjecture appears too hard in its full generality,… • let’s try some special class of functions. • XOR functions: . --- • The linear composition of and . • Include important functions such as Equality, Hamming Distance, Gap Hamming Distance. • Connection to Fourier: .

  8. Log-rank conjecture for XOR functions • Goal: • Even this special case seems very hard. • One approach*: • : Parity decision tree complexity. (DT with queries like “”) • simulating 1 -query by 2 bits of communication. *. Zhang and Shi. Theoretical Computer Science, 2009.

  9. 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.

  10. 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.

  11. 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.

  12. 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.

  13. 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.

  14. 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.

  15. Our results: small spectral norm • Theorem 3. … • Corollary 4. • Recall: before our work, even slightly sublinear bound is conditional.

  16. Our results: small spectral norm • [Gro97]. , • Corollary 5. , • Win-win: , • Either: Cor5 improves Grolmusz’s almost quadratically, by a deterministic protocol. • Or : Log-rank conj. holds for ! [Gro97] Grolmusz. Theoretical Computer Science, 1997.

  17. 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 .

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

  19. 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.

  20. 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 .

  21. 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

  22. 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 -.

  23. Bounding linear rank • One simple bound: - • Decreasing degree is easier than . • : • small: all computation paths are short • Now: only need to prove one path is short. • Then - the path length • Next: How to bound .

  24. Effect of a -query in Fourier domain? • Query and get answer • where is a linear function. • Denote the obtained “subfunction” by . • One can define Fourier coefficients of subfunctionss.t.

  25. Fourier coefficients of subfunctions • . • It’s like a folding over the line : and “collide” iff. • Question: Can we always find a line folding along which a good fraction of Fourier coefficients collide? • If true, we can keep folding along such . • Unfortunately, don’t know. • In general, we’d like Fourier oceff collide a lot.

  26. Btw: an ideal case • If there is a small-dim subspace , s.t. half Fourier coefficients are not “lonely” in cosets, then folding along reduces -frac. of sparsity. • However…where is the subspace? • Bad news*: Random -dim makes all isolated. * Gopalan, O'Donnell, Servedio, Shpilka, and Wimme. SIAM J. Comput., 2011

  27. Now bounding • Degree reduction again. • Induction on . Apply IH on (discrete) derivative. • Derivative: . • Fact. . • Fact. .

  28. Two interesting functions: • By IH, affine with small co-dimension and . • Define two new functions. ,. • and are non-Boolean. Range: . • , • e.g. On , , so .

  29. in Fourier domain • , . • Recall: , . • So on , half-space of the Fourier coefficients disappear. • During the linear restrictions, those collide a lot and finally all annihilate.

  30. Killing the Fourier coefficients • Formally: • Thus either or is . • Say it’s . (subfn: smaller norm) (picked for this)

  31. Finishing induction • Repeating this times reduces to , reaching a linear function. One more folding makes it constant. • So . and

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

  33. Sketch for • follows. • Greedy folding: boost as quickly as possible. • keep folding over the line , • . • Two stages. • Before : increases by . • Afterwards: drops by . • Don’t always analyze though the alg aims to boost it.

  34. Before • increases by . • . • Parseval: . • . • So increases by at least

  35. After • still increases, so it’s always . • . • . • e.g. • Recall: . • Fact. drops by . • drops .

  36. Summary: structural result

  37. Summary: Communication complexity • Main protocol. • May be already efficient: cost . • Log-rank Conj. holds for XOR functions with low-degree or small spectral norm . • Improved bound for for low-degree .

  38. Concluding Remarks • Subsequently: • Thm*1. for general Boolean . • Thm*2. • Open: • Prove -. • No counterexample even for • Other applications of -? *1. Lovett. ECCC, 2013. *2. Zhang. SODA, 2014.

  39. Thanks

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