Decision trees, protocols and the entropy-influence conjecture

Wan, Andrew; Wright, John; Wu, Chenggang

doi:10.1145/2554797.2554806

Cited by 9 publications

(20 citation statements)

References 11 publications

(32 reference statements)

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“…We note that the reduction in [53,Proposition E.2] shows that removing the requirement Inf(f ) ≥ 1 from the above inequality will prove the FEI conjecture for all Boolean functions with Inf(f ) ≥ log n. Furthermore, if we could show the FEI conjecture for Boolean functions f where aUC ⊕ (f ) = ω(1) is a slow-growing function of n, again the padding argument in [53] shows that we would be able to establish the FEI conjecture for all Boolean functions.…”

Section: :11mentioning

confidence: 81%

“…Later Wan et al [53] used Shannon's source coding theorem [49] (which characterizes entropy) to establish the FEI conjecture for read-k decision trees for constant k. Using their novel interpretation of the FEI conjecture they also reproved O'Donnell-Tan's composition theorem in an elegant way. Recently, Shalev [47] improved the constant in the FEI inequality for read-k decision trees, and further verified the conjecture when either the influence is too low, or the entropy is too high.…”

Section: Prior Workmentioning

confidence: 99%

“…Employing information-theoretic techniques for the FEI conjecture seems very natural given that the conjecture seeks to bound the entropy of a distribution. Indeed, Keller et al [32,Section 3.1] envisioned a proof of the FEI conjecture itself using large deviation estimates and the tensor structure (explained below) in a stronger way, and Wan et al [53] used Shannon's source coding theorem [49] to verify the conjecture for bounded-read decision trees.…”

Section: :11mentioning

confidence: 99%

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Improved Bounds on Fourier Entropy and Min-entropy

Arunachalam¹,

Chakraborty

Koucký

et al. 2021

ACM Trans. Comput. Theory

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Given a Boolean function f:{ -1,1} ^{n}→ { -1,1, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f ˆ (S) 2 . The Fourier Entropy-influence (FEI) conjecture of Friedgut and Kalai [28] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C > 0 such that H(f ˆ2 ) ≤ C ⋅ Inf (f), where H (fˆ2) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f In this article, we present three new contributions toward the FEI conjecture: (1) Our first contribution shows that H(f ˆ2 ) ≤ 2 ⋅ aUC ⊕ (f), where aUC ⊕ (f) is the average unambiguous parity-certificate complexity of f . This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs. We also discuss how our work makes Mansour's conjecture for DNFs a natural next step toward resolution of the FEI conjecture. (2) We next consider the weaker Fourier Min-entropy-influence (FMEI) conjecture posed by O'Donnell and others [50, 53], which asks if H ∞ fˆ2) ≤ C ⋅ Inf(f), where H ∞ fˆ2) is the min-entropy of the Fourier distribution. We show H ∞ (fˆ2) ≤ 2⋅C min ⊕ (f), where C min ⊕ (f) is the minimum parity-certificate complexity of f . We also show that for all ε≥0, we have H ∞ (fˆ2) ≤2 log⁡(∥f ˆ ∥1,ε/(1−ε)), where ∥f ˆ ∥1,ε is the approximate spectral norm of f . As a corollary, we verify the FMEI conjecture for the class of read- k DNFs (for constant k ). (3) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2 ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI, and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.

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Section: :11mentioning

confidence: 81%

Section: Prior Workmentioning

confidence: 99%

Section: :11mentioning

confidence: 99%

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Improved Bounds on Fourier Entropy and Min-entropy

Arunachalam¹,

Chakraborty

Koucký

et al. 2021

ACM Trans. Comput. Theory

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“…Conjecture 1.1 was verified for various families of Boolean functions (e.g., symmetric functions [10], random functions [3], read-once formulas [1,9], decision trees of constant average depth [11], read-k decision trees for constant k [11]) but is still open for the class of general Boolean functions.…”

Section: Conjecture 11 ([4]mentioning

confidence: 99%

“…In particular, the average decision tree complexity of the sequence (F m ) m≥0 is unbounded, and thus the construction is not subject to the upper bound on constant average depth decision trees of [11]. Each F m is still computable by a read-once formula, though.…”

Section: Afterthoughtsmentioning

confidence: 99%

Improved Lower Bounds for the Fourier Entropy/Influence Conjecture via Lexicographic Functions

Hod

2017

Preprint

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Every Boolean function can be uniquely represented as a multilinear polynomial. The entropy and the total influence are two ways to measure the concentration of its Fourier coefficients, namely the monomial coefficients in this representation: the entropy roughly measures their spread, while the total influence measures their average level. The Fourier Entropy/Influence conjecture of Friedgut and Kalai from 1996 states that the entropy to influence ratio is bounded by a universal constant C.Using lexicographic Boolean functions, we present three explicit asymptotic constructions that improve upon the previously best known lower bound C > 6.278944 by O'Donnell and Tan, obtained via recursive composition. The first uses their construction with the lexicographic function ℓ 2/3 of measure 2/3 to demonstrate that C ≥ 4 + 3 log 4 3 > 6.377444. The second generalizes their construction to biased functions and obtains C > 6.413846 using ℓ Φ , where Φ is the inverse golden ratio. The third, independent, construction gives C > 6.454784, even for monotone functions.Beyond modest improvements to the value of C, our constructions shed some new light on the properties sought in potential counterexamples to the conjecture.Additionally, we prove a Lipschitz-type condition on the total influence and spectral entropy, which may be of independent interest.

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