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VII Simulation Results In this section, we provide several simulation results to validate the efficiency of our model. We consider a HetNet scenario similar to the one presented in Fig. , in which there are [MATH] randomly distributed users that are covered by 9 BSs. There is one cellular SP in the center of a macro ce... |
Fig. compares the sum utility of BSs under these three scenarios while we change the number of users (load) from [MATH] to [MATH] |
When the number of users is less than 150, PT outperforms EUT in terms of SPs sum utility. The reason is that when the load is low, SPs offer higher data rates to users, hence, even with low service guarantees they can satisfy the user’s minimum data rate constraint. And due to probability weighting effect (PWE) under ... |
, sum utility of users will be increased by increasing the number of users as more users will be associated to SPs until the number of users get close to the network capacity which is 400 users in our setting. After that, the user association rate reduces again as bid selection constraints for many users cannot be met ... |
To see how much extra costs the SPs have to incur to retain their users under PT using bandwidth expansion, we compare the average bandwidth consumption of users in EUT vs PT with bandwidth expansion feature in Fig. |
When the SPs advertised service guarantees are less than [MATH] threshold, which occurs when the number of users are less than 150, the SPs are not required to consume any extra BW to retain their EUT users under PT. However, by increasing the number of users beyond 150, the number of users who receive an offer with a ... |
VIII Conclusion This paper studied the problem of user association in wireless HetNets under PT, where all covering WiFi and cellular SPs offer data services to users who are free to accept or reject any of the received offers. We modeled this problem using a Stackelberg game, and extracted all potential NEs for this g... |
# Source: arxiv 1806.03646 # Title: On the Fourier Entropy Influence Conjecture for Extremal Classes # Sections: all # Downloaded: 2026-03-03T01:45:03.903303+00:00 |
On the Fourier Entropy Influence Conjecture for Extremal Classes Abstract The Fourier Entropy-Influence (FEI) Conjecture of Friedgut and Kalai FK96 states that [MATH] holds for every Boolean function [MATH] , where [MATH] denotes the spectral entropy of [MATH] [MATH] is its total influence, and [MATH] is a universal co... |
In this work, we prove the conjecture for the extremal cases, i.e. functions with small influence and functions with high entropy. Specifically, we show that: |
FEI holds for the class of functions with [MATH] with the constant [MATH] . Furthermore, proving FEI for a class of functions with [MATH] for some [MATH] will imply FEI for the class of all Boolean functions. |
Additionally, we show that FEI holds for the class of functions with constant [MATH] , completing the results of CKLS16 that bounded the entropy of such functions. We also improve the result of WWW14 for read-k decision trees, from [MATH] to [MATH] . Finally, we suggest a direction for proving FEI for read-k DNFs, and ... |
Introduction Boolean functions [MATH] are one of the most basic objects in the theory of computer science. The Fourier analysis of Boolean functions has become prominent over the years as a powerful tool in the study of Boolean functions, with applications in many fields such as complexity theory, learning theory, soci... |
For Boolean-valued functions, by applying Parseval’s identity we have [MATH] and therefore the squared Fourier coefficients [MATH] can be viewed as a probability distribution [MATH] , named the spectral distribution of [MATH] The spectral entropy of [MATH] is defined to be the Shannon entropy of [MATH] , namely |
[MATH] . This can be intuitively thought as how "spread out" the Fourier coefficients of [MATH] are. The total influence of a function [MATH] , one of the most basic measures of a Boolean function, can be defined as [MATH] , the expected size of a subset [MATH] according to the spectral distribution, and can be intuiti... |
The Fourier Entropy Influence conjecture, posed by Friedgut and Kalai FK96 states that for any Boolean function the ratio of its spectral entropy and its total influence is upper-bounded by a universal constant. |
Conjecture 1 FK96 ) There exists a universal constant [MATH] such that for all [MATH] with influence [MATH] and spectral entropy [MATH] we have [MATH] |
The original motivation for the conjecture in FK96 emerged from studying threshold phenomena of monotone graph properties in random graphs. Specifically for a function [MATH] that represents a monotone property of a graph with [MATH] vertices (e.g. connectivity), FEI implies that [MATH] . The best known bound as of tod... |
Proving Conjecture will have other interesting applications. Probably the most important consequence of the conjecture is its implication of a variant of Mansour’s conjecture from 1995 Man95 stating that if a Boolean function can be represented by a DNF formula with [MATH] terms, then most of its Fourier weight is conc... |
FEI is also closely related to the fundamental KKL theorem KKL88 stating that for every Boolean function, [MATH] . We define [MATH] , the min-entropy of [MATH] . It is easy to verify that [MATH] . A natural relaxation of FEI is the following weaker Fourier Min-Entropy Influence conjecture: |
Conjecture 2 (FMEI) There exists some [MATH] such that for any [MATH] we have [MATH] KKL can be directly derived from FMEI (and therefore is clearly implied by FEI). In the other direction, one can easily prove FMEI for monotone functions using KKL (see OWZ11 ). We note that FEI for monotone functions is still an open ... |
1.1 Prior Work Despite many years of attention, Conjecture remains open, but some significant steps towards proving it have been made. For example, a weaker folklore version of FEI, where instead of a universal constant [MATH] we settle for a [MATH] factor, is known to be true even for the more general case of real-val... |
Lemma 1 (Weak FEI) Let [MATH] be some function with [MATH] . Then [MATH] This can be proved in several different ways, as done in KMS12 OWZ11 and WWW14 . It should be noted that [MATH] is indeed tight for non-Boolean functions, so proofs of FEI will have to make use of the fact that [MATH] is Boolean-valued. The tightn... |
[EQUATION] and it is easy to verify that [MATH] , and also that [MATH] and [MATH] . For Boolean-valued functions this [MATH] bound has been recently improved by Gopalan et al. in GSTW16 to [MATH] , where [MATH] is max sensitivity of the function: the sensitivity of [MATH] in a function [MATH] , denoted [MATH] , is the ... |
Furthermore, FEI has been verified for several families of Boolean functions. O’Donnel, Wright and Zhou OWZ11 proved it for symmetric functions by using the fact that derivatives of symmetric functions are very noise sensitive. They also prove FEI for the class of read-once decision trees. |
In another paper Das, Pal and Visavaliya DPV11 show that FEI holds with universal constant [MATH] for a random function, as [MATH] is strongly concentrated around its mean [MATH] , and the spectral entropy of a function is always bounded by [MATH] . We give another proof of this fact (with a worse constant), by proving... |
In KMS12 , Keller, Mossel and Schlank generalize FEI to the biased setting. Furthermore, for functions with almost all of their Fourier weight on the lowest [MATH] levels, they upper-bound the spectral entropy by [MATH] |
In the paper OT13 , O’Donnell and Tan study FEI under composition: given functions [MATH] and [MATH] , they ask what properties do [MATH] and [MATH] must satisfy for the FEI conjecture to hold for the disjoint composition [MATH] ? To make progress they present a strengthening of FEI which they call FEI - a generalizati... |
In CKLS16 , Chakraborty et al. prove a relaxation of FEI, bounding the spectral entropy with higher moments of [MATH] , where the original conjecture needs this bound to include only the first moment of [MATH] , namely [MATH] . They also prove FEI for read-once formulas with a more elementary method than the one of O’D... |
Independently, CKLS16 also give upper bounds on the entropy of a Boolean function in terms of several complexity measures - to name a few, they show that [MATH] , and also that [MATH] where [MATH] is the average depth of a decision tree computing [MATH] . This implies FEI for the class [MATH] where [MATH] and [MATH] ar... |
This raises the natural question, whether the [MATH] requirement is actually necessary or merely an artifact of the proof. For the [MATH] complexity measure and other measures strongly related to it, we manage to overcome this condition by making subtle changes to the proof technique of CKLS16 , generalizing the bound ... |
In WWW14 , Wan, Wright and Wu present a new perspective of FEI as a communication (or rather, compression) game: one player randomly samples a set [MATH] according to the distribution [MATH] , and wishes to send it to another player using a short representation. The price of the protocol is the expected number of bits ... |
Lemma Let [MATH] , and let [MATH] be a prefix-free protocol on alphabet [MATH] , except it outputs an empty string on the input [MATH] Then [MATH] |
They use this technique combined with observations regarding the covariance of decision trees to prove a theorem (that is also known due to CKLS16 ) - that FEI holds for the class of functions [MATH] computed by decision trees with constant average depth and [MATH] WWW14 also provide a reduction, showing that removing ... |
Using their protocol technique, WWW14 also achieve [MATH] for read-k decision trees, thus proving FEI for read-k decision trees where [MATH] is constant. They explicitly conjecture that the correct coefficient is actually [MATH] and provide a matching example. We improve their bound to [MATH] , but share their belief t... |
In Hod17 , Hod improves the lower bound on the conjectured universal constant for FEI to [MATH] via lexicographic functions, using composition techniques and biased Fourier analysis. |
1.2 Our Results Intrigued by the implicit and explicit difficulties of FEI for low influence functions, we prove FEI for functions with extremely low influence: |
Theorem 2 Let [MATH] be some constant. Let [MATH] with [MATH] . Then [MATH] This result may seem at first somewhat disappointing, as interesting functions usually don’t have such small total influence. Can we do better than this bound? Apparently not, at least without proving the full conjecture. Using a construction p... |
Theorem 3 Let [MATH] such that [MATH] . Suppose that FEI holds for all [MATH] with [MATH] . Then FEI holds for all Boolean functions. |
For example, proving FEI for the class of functions with [MATH] will be enough to confirm Conjecture This result for functions with extremely low influence raises the question of the opposite extremal case - where the entropy is high, say, [MATH] for some [MATH] . We provide analogous results for this extremal case. |
Theorem 4 Let [MATH] be some constant. For any [MATH] with [MATH] we have [MATH] , where [MATH] is the inverse of the binary entropy function. |
Theorem 5 Let [MATH] such that [MATH] . Suppose that FEI holds for all [MATH] with [MATH] . Then FEI holds for all Boolean functions. |
For example, proving FEI for the class of functions with [MATH] will confirm Conjecture . We also note that the other two extremal cases are easy, namely functions with exponentially low entropy and functions with total influence linear in [MATH] |
Independently from our work on the extremal classes, we also provide some improvements on previously known results. First, we modify the [MATH] bound of CKLS16 to include the influence and variance of the function, thereby showing that FEI holds for the class of functions with constant [MATH] [MATH] |
Theorem 6 Let [MATH] with [MATH] . Then [MATH] . In particular, from the edge isoperimetric inequality, we have [MATH] As a direct corollary, we can deduce FEI for functions with some related complexity measures that are constant. We note that some of these results have been previously known. |
Corollary 7 FEI holds for functions with constant [MATH] , constant sub-cube partition, constant degree, constant decision tree depth, constant decision tree size, constant granularity or constant sparsity. |
We also build and improve on the work of WWW14 . Inspired by their methods, we provide a hopefully promising direction towards proving FEI for read-k DNFs. We give an explicit protocol for the Tribes function which is a read-once DNF, and conjecture its possible generalization to a protocol for read-k DNFs, as a step t... |
Theorem 8 Let [MATH] be a regular read-k DNF, then [MATH] We also improve the result of WWW14 for read-k decision trees. WWW14 define the tree covariance of a decision tree recursively as: [MATH] , where [MATH] represent the functions defined by the left and right children of the root of [MATH] . They come up with a pr... |
Theorem 9 Let [MATH] be computed by a read-k decision tree. Then [MATH] . As a result, FEI holds for read-k decision trees with constant [MATH] |
We believe the tree covariance of a decision tree and its connection to other measures of the function it computes such as its variance and influence, might be of independent interest in the study of decision trees. |
Finally, as an independent result, we refine the known connection between the size of a decision tree, and the spectral norm ( [MATH] ) of the function it computes. It is a well known fact that [MATH] , the size of a decision tree being the number of nodes in it. Our improvement involves the covariance of the nodes in ... |
Proposition 10 For a Boolean function [MATH] that is computed by a decision tree [MATH] [EQUATION] Where [MATH] is the number of nodes that have at least one child that is a leaf. The sum of covariances is over all inner nodes of [MATH] , i.e. nodes that have two non-leaf children. This improved bound is tight in some ... |
Preliminaries 2.1 Fourier Analysis of Boolean Functions It is well known that functions [MATH] can be uniquely expressed as multi-linear polynomials: |
[EQUATION] where [MATH] . This is known as the Fourier expansion of [MATH] , and [MATH] are the Fourier coefficients of the function. For Boolean-valued functions [MATH] Parseval’s identity implies that |
[MATH] , and therefore [MATH] can be viewed as a probability distribution, named the spectral distribution of [MATH] and denoted [MATH] Two of the central complexity measures of a Boolean function can be defined using its spectral distribution: |
Definition The spectral entropy of a function [MATH] is the Shannon-entropy of the squared Fourier coefficients, namely [EQUATION] |
Definition The influence of a function [MATH] (sometimes referred to as its total influence) is [EQUATION] The influence of a Boolean function also has a nice combinatorial interpretation. For [MATH] , the influence of a variable [MATH] in [MATH] is [MATH] , namely the probability that for a uniformly random input flip... |
It is sometimes useful to classify the Fourier coefficients by their level, where the level of [MATH] is [MATH] . The weight of [MATH] at level [MATH] is denoted [MATH] . Note that |
[MATH] . Additionally, we use the following notations: [MATH] , and [MATH] We also use the decision tree model of computation, see O’D14 for a formal definition. Given a tree [MATH] , we call the sub-tree corresponding to the [MATH] edge leaving the root the left sub-tree ( [MATH] ), and call the sub-tree corresponding... |
Given two functions [MATH] define [MATH] . Following the definitions of WWW14 , we define the covariance of a decision tree [MATH] for an internal node [MATH] , let [MATH] be the function computed by [MATH] ’s left sub-tree and [MATH] be the function computed by [MATH] ’s right sub-tree. Then, define: |
[MATH] [MATH] Note that [MATH] can be equivalently defined recursively as [MATH] , with the base case that [MATH] if [MATH] has depth 0. |
A DNF over Boolean variables [MATH] is the logical OR of terms, [MATH] each of which is a logical AND of literals { [MATH] [MATH] }. The number of literals in a term is called its width (sometimes we refer to it as the size of the term). A DNF is read-k if no variable appears in more than [MATH] terms. |
The Tribes function with width [MATH] and [MATH] tribes, is a read-once DNF on [MATH] variables, where all terms are of width exactly [MATH] |
[EQUATION] For [MATH] , we choose [MATH] to be the largest integer such that [MATH] , so [MATH] will be as unbiased as possible. Then we denote by [MATH] , defined only for such pairs of [MATH] , to be the (essentially) unbiased Tribes function on [MATH] variables. Due to Proposition 4.12 in O’D14 [MATH] [MATH] |
Finally, we present the definition of regular DNFs: Definition Let [MATH] . We say [MATH] is a [MATH] -regular DNF (or just “regular”), if there exists some [MATH] s.t. the number of variables in each clause respects [MATH] , and the number of clauses is [MATH] |
Expanding on the mentioned notions of the Shannon entropy [MATH] and the min-entropy [MATH] , the Renyi entropy of a distribution [MATH] (we discuss only [MATH] ) is defines as follows: |
[EQUATION] Where [MATH] are the probabilities of possible instances in [MATH] - in our case, these are the squared Fourier coefficients. |
It can be seen that for [MATH] , the Renyi entropy converges to the Shannon entropy, therefore we denote [MATH] . Furthermore, when [MATH] , the Renyi entropy converges to the min-entropy [MATH] . It is known that for a fixed distribution, the function [MATH] is non-increasing in [MATH] |
2.2 Edge Isoperimetric Inequality The simplest form of the Edge Isoperimetric Inequality states that for any Boolean function [MATH] [MATH] We also rely on the following edge isoperimetric inequality, see e.g. Theorem 2.39 in O’D14 |
Fact 11 Let [MATH] be a Boolean function. Denote [MATH] , then [MATH] Keeping the notation [MATH] , it is easy to see that [MATH] can be “replaced” by the variance, losing only a constant multiplicative factor: |
[EQUATION] Since [MATH] , we have: [EQUATION] Lemma 12 Let [MATH] be a Boolean function, then [MATH] Proof. [EQUATION] Where the second inequality is due to [MATH] |
Lemma 13 Let [MATH] be a Boolean function with [MATH] , then [MATH] Proof. The requirement that [MATH] is necessary, or else the term [MATH] is non-positive. We derive the new inequality from the proof of Lemma 12 |
[EQUATION] 2.3 Tensorization of FEI Let [MATH] [MATH] be two Boolean functions. Define [MATH] to be their tensor product [MATH] , and |
[MATH] In Kal07 it has been noted that FEI tensorizes in the following sense: Fact 14 For Boolean functions [MATH] and [MATH] [MATH] |
[MATH] [MATH] , where [MATH] denotes the number of variables of the function [MATH] We call [MATH] the self-tensorization of [MATH] . As stated in the following lemma, the tensorization technique allows us to deduce FEI for a class closed under self-tensorization (i.e., for all [MATH] we have [MATH] by proving FEI for ... |
Lemma 15 Suppose we have a class of Boolean functions [MATH] that is closed under self-tensorization, a constant [MATH] and some function [MATH] . If [MATH] for all [MATH] , then [MATH] for all [MATH] |
Proof. Let [MATH] be a Boolean function on [MATH] variables. Define [MATH] , and [MATH] [MATH] since [MATH] is closed under self-tensorization, and therefore we have |
[EQUATION] By Fact 14 [EQUATION] Dividing by [MATH] , we get: [EQUATION] Fixing [MATH] and taking [MATH] to infinity, we get [MATH] , and therefore [MATH] |
Additionally, we note that the min-entropy tensorizes as well: for [MATH] and [MATH] as stated above, [MATH] , so a similar proof will suffice for an analogous result. |
Lemma 16 Suppose we have a class of Boolean functions [MATH] that is closed under self-tensorization, a constant [MATH] and some function [MATH] . If [MATH] for all [MATH] , then [MATH] for all [MATH] |
FEI for Low Influence Functions In this section we prove FEI for the class of functions with exponentially low influence (in [MATH] ), and then show that improving this will imply Conjecture . To state this formally, we introduce some notations and consider the following classes of functions: |
The class of Boolean functions on [MATH] variables [MATH] , and the class of all Boolean functions [MATH] The class of functions with exponentially-low influence. For every constant [MATH] , define [MATH] |
The class of functions with “almost” exponentially-low influence. For every function [MATH] such that [MATH] , define [MATH] The class of functions with influence larger than [MATH] [MATH] |
Formally, we show that for any [MATH] FEI holds for the class [MATH] with constant [MATH] . We then show that improving on this result by proving FEI for any [MATH] will actually imply FEI for the class [MATH] . As a simple corollary of Theorem that we will later prove, this will imply FEI for [MATH] , i.e. Conjecture |
3.1 Proving FEI for ELI We restate and prove Theorem as follows: Theorem 17 For all [MATH] [MATH] Proof. Let [MATH] , and denote [MATH] , where [MATH] We use the concentration method presented by CKLS16 to show |
[MATH] . We partition the Fourier coefficients to a family [MATH] , and its complement [MATH] . These families have Fourier weight of [MATH] and [MATH] respectively. By a known formula of entropy partition, we have: |
[EQUATION] where the entropies of the families are of the adequately normalized distributions and [MATH] is the binary entropy function. Note that [MATH] , since [MATH] contains only one element. Also note that [MATH] , as [MATH] . Therefore we have: |
[EQUATION] If [MATH] then by our assumption it follows that [MATH] . From this and a from Weak FEI (Lemma ) we get [MATH] so we are done. Otherwise, we can assume [MATH] . To bound the second term of inequality 3.1 [MATH] , we note that for [MATH] , we have [MATH] , so [MATH] . Acknowledging the fact that [MATH] and ap... |
[EQUATION] We can bound the first term of inequality 3.1 by applying Lemma 13 [EQUATION] Inserting 3.2 3.3 into 3.1 we obtain the wanted result: |
[EQUATION] There is also an alternative proof for theorem 17 using the protocol method of WWW14 : Intuitively, consider the following trivial protocol: if the sampled [MATH] is non-empty, send [MATH] bits, where the [MATH] ’th bit is set to [MATH] if [MATH] and otherwise is [MATH] . If [MATH] is sampled, the protocol s... |
3.2 Proving FEI for AELI Implies FEI Completely Lemma 18 Let [MATH] such that [MATH] . Suppose that FEI holds for some class [MATH] with universal constant [MATH] , Then FEI holds for the class [MATH] with constant [MATH] |
Proof. We follow exactly the same construction appearing in appendix E of WWW14 Let [MATH] . For now we assume [MATH] is balanced (i.e. that [MATH] ), and deal with biased functions later. |
Consider the function [MATH] on [MATH] variables defined as [EQUATION] [MATH] is extremely biased, as it can get the value [MATH] only when [MATH] . By direct calculation, WWW14 show that: |
[EQUATION] and also that: [EQUATION] We would like to argue that [MATH] , so we need to pick a large enough [MATH] accordingly, so that the following inequality will hold: |
[EQUATION] We also want to use self-tensorization on [MATH] , so we need to ensure [MATH] So it would suffice to find [MATH] such that: |
[MATH] [MATH] We can pick [MATH] . For such [MATH] it is clear that [MATH] , and also that [MATH] , where the last inequality uses the fact that [MATH] and that [MATH] is monotone increasing - we can assume w.l.o.g that [MATH] is a monotone function, or otherwise redefine [MATH] with [MATH] |
So by the fact that [MATH] and assuming FEI for [MATH] with universal constant [MATH] [EQUATION] [EQUATION] [EQUATION] The subclass of balanced functions in [MATH] is closed under self-tensorization, so we can use the tensorization technique to get [MATH] hereby completing the proof for balanced functions. |
If [MATH] is biased, we can define [MATH] [MATH] is balanced, [MATH] , so [MATH] , and [MATH] . Therefore we have: [EQUATION] Where the last inequality is the only place where we use the fact that [MATH] (apart from the fact that [MATH] is closed under self-tensorization). |
We would like to extend the lemma from [MATH] to [MATH] . If we examine for a moment the class [MATH] , it is easy to see from Weak FEI that for all [MATH] [MATH] . Therefore [MATH] , so FEI holds for [MATH] . By Lemma 18 and Theorem to be proven in the next section, we can now deduce Theorem as a simple corollary: |
Theorem 19 Let [MATH] such that [MATH] . Suppose that FEI holds for some class [MATH] with universal constant [MATH] , then FEI holds for the class [MATH] with constant [MATH] |
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