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

arXiv:1304.1245

Hing Yin Tsang1, Chung Hoi Wong1, Ning Xie2, Shengyu Zhang1

1. The Chinese University of Hong Kong

2. Florida International University

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Motivation 1: Fourier analysis

Bool Fourier

Parseval: If , then .Spectral norm: . Fourier sparsity:

Qustion: What can we say about Boolean with small or ? Characterization?

𝑓 : {0,1 }𝑛→ℝ �̂� : {0,1 }𝑛→ℝ

(sparsified)

{±1 }

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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.

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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? ---

𝑥 𝑦

*1. Yao. STOC, 1979.

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Log-rank conjecture• 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.

≤ log𝑂 (1 )𝑟𝑎𝑛𝑘 (𝑀𝐹 )

𝑀𝐹≝ [𝐹 (𝑥 , 𝑦 ) ]𝑥 ,𝑦

Log Rank Conjecture*2

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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.

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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.

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Previous work

• Special cases for .– : Symmetric *1

– : LTF *2

– : monotone *2

– : *3

• Hard case: much larger than – not touched yet.

log‖ �̂�‖0=Ω (𝑛)

deg ( 𝑓 )=~𝑂 (log‖𝑓‖0 )

*1. Zhang and Shi. Quantum Information & Computation, 2009.*2. Montanaro and Osborne. arXiv:0909.3392v2, 2010.*3. Kulkarni and Santha. CIAC, 2013.

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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.

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Our results: constant degree

• Theorem 1. For with : 1) Log-rank conjecture holds for . Dependence on : “only” singly exponential.

2) Fourier sparse short -DT

3) depends only on linear functions of input variables.

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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.

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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] Shpilka and Volk. ECCC, 2013.

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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.

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Our results: small spectral norm

• [Gro97]. , • Corollary 5. ,

[Gro97] Grolmusz. Theoretical Computer Science, 1997.

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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 .

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TechniquesResults Techniques Low degree:

• Fourier sparse short -DT• Green-Sanders for low-degree

Main PDT Algorithm + bound by degree reduction

Small spectral normMain PDT Algorithm

+ greedy folding

Light tail

• Close to sparse Log-rank Chang’s lemma *1

+ a rounding lemma *2

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

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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.

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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 .

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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

ℓ1 (𝑥 )ℓ1 (𝑦 )⋮

ℓ𝑟 (𝑥 )ℓ𝑟 ( 𝑦 )

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Main conjecture

• Of course, communication cost depends on how large - is.

• Conjecture 1. Boolean , -

• Conjecture 1 Log-rank Conj. for all XOR fn’s. • Most our results obtained by bounding -.• One simple bound: -– Decreasing degree is easier than .

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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. .

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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.

𝐹 (𝑥)= (−1 ) 𝑓 (𝑥 )

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f Results

Low degree • Fourier sparse short -DT• Green-Sanders: doubly-exp quasi-poly

Small spectral normLight tail • Close to sparse Log-rank

Summary

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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

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