Analysis of Boolean functions

In mathematics and theoretical computer science, analysis of Boolean functions is the study of real-valued functions on or (such functions are sometimes known as pseudo-Boolean functions) from a spectral perspective.[1] The functions studied are often, but not always, Boolean-valued, making them Boolean functions. The area has found many applications in combinatorics, social choice theory, random graphs, and theoretical computer science, especially in hardness of approximation, property testing, and PAC learning.

Basic conceptsEdit

We will mostly consider functions defined on the domain  . Sometimes it is more convenient to work with the domain   instead. If   is defined on  , then the corresponding function defined on   is

 

Similarly, for us a Boolean function is a  -valued function, though often it is more convenient to consider  -valued functions instead.

Fourier expansionEdit

Every real-valued function   has a unique expansion as a multilinear polynomial:

 

This is the Hadamard transform of the function  , which is the Fourier transform in the group  . The coefficients   are known as Fourier coefficients, and the entire sum is known as the Fourier expansion of  . The functions   are known as Fourier characters, and they form an orthonormal basis for the space of all functions over  , with respect to the inner product  .

The Fourier coefficients can be calculated using an inner product:

 

In particular, this shows that  , where the expected value is taken with respect to the uniform distribution over  . Parseval's identity states that

 

If we skip  , then we get the variance of  :

 

Fourier degree and Fourier levelsEdit

The degree of a function   is the maximum   such that   for some set   of size  . In other words, the degree of   is its degree as a multilinear polynomial.

It is convenient to decompose the Fourier expansion into levels: the Fourier coefficient   is on level  .

The degree   part of   is

 

It is obtained from   by zeroing out all Fourier coefficients not on level  .

We similarly define  .

InfluenceEdit

The  'th influence of a function   can be defined in two equivalent ways:

 

If   is Boolean then   is the probability that flipping the  'th coordinate flips the value of the function:

 

If   then   doesn't depend on the  'th coordinate.

The total influence of   is the sum of all of its influences:

 

The total influence of a Boolean function is also the average sensitivity of the function. The sensitivity of a Boolean function   at a given point is the number of coordinates   such that if we flip the  'th coordinate, the value of the function changes. The average value of this quantity is exactly the total influence.

The total influence can also be defined using the discrete Laplacian of the Hamming graph, suitably normalized:  .

The Sensitivity conjecture: There exists an absolute constant C > 0, such that for every boolean function f,

          Block Sensitivity  bs(f) ≤ Sensitivity s(f}C. 

was confirmed by Hao Huang [2], [3].

A generalized form of influence is the  -stable influence, defined by:

 

The corresponding total influences is

 

One can prove that a function   has at most “constantly” many “stably-influential” coordinates:  

Noise stabilityEdit

Given  , we say that two random vectors   are  -correlated if the marginal distributions of   are uniform, and  . Concretely, we can generate a pair of  -correlated random variables by first choosing   uniformly at random, and then choosing   according to one of the following two equivalent rules, applied independently to each coordinate:

 

We denote this distribution by  .

The noise stability of a function   at   can be defined in two equivalent ways:

 

For  , the noise sensitivity of   at   is

 

If   is Boolean, then this is the probability that the value of   changes if we flip each coordinate with probability  , independently.

Noise operatorEdit

The noise operator   is an operator taking a function   and returning another function   given by

 

When  , the noise operator can also be defined using a continuous-time Markov chain in which each bit is flipped independently with rate 1. The operator   corresponds to running this Markov chain for   steps starting at  , and taking the average value of   at the final state. This Markov chain is generated by the Laplacian of the Hamming graph, and this relates total influence to the noise operator.

Noise stability can be defined in terms of the noise operator:  .

HypercontractivityEdit

For  , the  -norm of a function   is defined by

 

We also define  

The hypercontractivity theorem states that for any   and  ,

 

Hypercontractivity is closely related to the logarithmic Sobolev inequalities of functional analysis.[4]

A similar result for   is known as reverse hypercontractivity.[5]

p-Biased analysisEdit

In many situations the input to the function is not uniformly distributed over  , but instead has a bias toward   or  . In these situations it is customary to consider functions over the domain  . For  , the p-biased measure   is given by

 

This measure can be generated by choosing each coordinate independently to be 1 with probability   and 0 with probability  .

The classical Fourier characters are no longer orthogonal with respect to this measure. Instead, we use the following characters:

 

The p-biased Fourier expansion of   is the expansion of   as a linear combination of p-biased characters:

 

We can extend the definitions of influence and the noise operator to the p-biased setting by using their spectral definitions.

InfluenceEdit

The  's influence is given by

 

The total influence is the sum of the individual influences:

 

Noise operatorEdit

A pair of  -correlated random variables can be obtained by choosing   independently and  , where   is given by

 

The noise operator is then given by

 

Using this we can define the noise stability and the noise sensitivity, as before.

Russo–Margulis formulaEdit

The Russo–Margulis formula (also called the Margulis–Russo formula[1]) states that for monotone Boolean functions  ,

 

Both the influence and the probabilities are taken with respect to  , and on the right-hand side we have the average sensitivity of  . If we think of   as a property, then the formula states that as   varies, the derivative of the probability that   occurs at   equals the average sensitivity at  .

The Russo–Margulis formula is key for proving sharp threshold theorems such as Friedgut's.

Gaussian spaceEdit

One of the deepest results in the area, the invariance principle, connects the distribution of functions on the Boolean cube   to their distribution on Gaussian space, which is the space   endowed with the standard  -dimensional Gaussian measure.

Many of the basic concepts of Fourier analysis on the Boolean cube have counterparts in Gaussian space:

  • The counterpart of the Fourier expansion in Gaussian space is the Hermite expansion, which is an expansion to an infinite sum (converging in  ) of multivariate Hermite polynomials.
  • The counterpart of total influence or average sensitivity for the indicator function of a set is Gaussian surface area, which is the Minkowski content of the boundary of the set.
  • The counterpart of the noise operator is the Ornstein–Uhlenbeck operator (related to the Mehler transform), given by  , or alternatively by  , where   is a pair of  -correlated standard Gaussians.
  • Hypercontractivity holds (with appropriate parameters) in Gaussian space as well.

Gaussian space is more symmetric than the Boolean cube (for example, it is rotation invariant), and supports continuous arguments which may be harder to get through in the discrete setting of the Boolean cube. The invariance principle links the two settings, and allows deducing results on the Boolean cube from results on Gaussian space.

Basic resultsEdit

Friedgut–Kalai–Naor theoremEdit

If   has degree at most 1, then   is either constant, equal to a coordinate, or equal to the negation of a coordinate. In particular,   is a dictatorship: a function depending on at most one coordinate.

The Friedgut–Kalai–Naor theorem,[6] also known as the FKN theorem, states that if   almost has degree 1 then it is close to a dictatorship. Quantitatively, if   and  , then   is  -close to a dictatorship, that is,   for some Boolean dictatorship  , or equivalently,   for some Boolean dictatorship  .

Similarly, a Boolean function of degree at most   depends on at most   coordinates, making it a junta (a function depending on a constant number of coordinates), where   is an absolute constant equal to at least 1.5, and at most 4.41, as shown by Wellens. The Kindler–Safra theorem[7] generalizes the Friedgut–Kalai–Naor theorem to this setting. It states that if   satisfies   then   is  -close to a Boolean function of degree at most  .

Kahn–Kalai–Linial theoremEdit

The Poincaré inequality for the Boolean cube (which follows from formulas appearing above) states that for a function  ,

 

This implies that  .

The Kahn–Kalai–Linial theorem,[8] also known as the KKL theorem, states that if   is Boolean then  .

The bound given by the Kahn–Kalai–Linial theorem is tight, and is achieved by the Tribes function of Ben-Or and Linial:[9]

 

The Kahn–Kalai–Linial theorem was one of the first results in the area, and was the one introducing hypercontractivity into the context of Boolean functions.

Friedgut's junta theoremEdit

If   is an  -junta (a function depending on at most   coordinates) then   according to the Poincaré inequality.

Friedgut's theorem[10] is a converse to this result. It states that for any  , the function   is  -close to a Boolean junta depending on   coordinates.

Combined with the Russo–Margulis lemma, Friedgut's junta theorem implies that for every  , every monotone function is close to a junta with respect to   for some  .

Invariance principleEdit

The invariance principle[11] generalizes the Berry–Esseen theorem to non-linear functions.

The Berry–Esseen theorem states (among else) that if   and no   is too large compared to the rest, then the distribution of   over   is close to a normal distribution with the same mean and variance.

The invariance principle (in a special case) informally states that if   is a multilinear polynomial of bounded degree over   and all influences of   are small, then the distribution of   under the uniform measure over   is close to its distribution in Gaussian space.

More formally, let   be a univariate Lipschitz function, let  , let  , and let  . Suppose that  . Then

 

By choosing appropriate  , this implies that the distributions of   under both measures are close in CDF distance, which is given by  .

The invariance principle was the key ingredient in the original proof of the Majority is Stablest theorem.

Some applicationsEdit

Linearity testingEdit

A Boolean function   is linear if it satisfies  , where  . It is not hard to show that the Boolean linear functions are exactly the characters  .

In property testing we want to test whether a given function is linear. It is natural to try the following test: choose   uniformly at random, and check that  . If   is linear then it always passes the test. Blum, Luby and Rubinfeld[12] showed that if the test passes with probability   then   is  -close to a Fourier character. Their proof was combinatorial.

Bellare et al.[13] gave an extremely simple Fourier-analytic proof, that also shows that if the test succeeds with probability  , then   is correlated with a Fourier character. Their proof relies on the following formula for the success probability of the test:

 

Arrow's theoremEdit

Arrow's impossibility theorem states that for three and more candidates, the only unanimous voting rule for which there is always a Condorcet winner is a dictatorship.

The usual proof of Arrow's theorem is combinatorial. Kalai[14] gave an alternative proof of this result in the case of three candidates using Fourier analysis. If   is the rule that assigns a winner among two candidates given their relative orders in the votes, then the probability that there is a Condorcet winner given a uniformly random vote is  , from which the theorem easily follows.

The FKN theorem implies that if   is a rule for which there is almost always a Condorcet winner, then   is close to a dictatorship.

Sharp thresholdsEdit

A classical result in the theory of random graphs states that the probability that a   random graph is connected tends to   if  . This is an example of a sharp threshold: the width of the "threshold window", which is  , is asymptotically smaller than the threshold itself, which is roughly  . In contrast, the probability that a   graph contains a triangle tends to   when  . Here both the threshold window and the threshold itself are  , and so this is a coarse threshold.

Friedgut's sharp threshold theorem[15] states, roughly speaking, that a monotone graph property (a graph property is a property which doesn't depend on the names of the vertices) has a sharp threshold unless it is correlated with the appearance of small subgraphs. This theorem has been widely applied to analyze random graphs and percolation.

On a related note, the KKL theorem implies that the width of threshold window is always at most  .[16]

Majority is stablestEdit

Let   denote the majority function on   coordinates. Sheppard's formula gives the asymptotic noise stability of majority:

 

This is related to the probability that if we choose   uniformly at random and form   by flipping each bit of   with probability  , then the majority stays the same:

 .

There are Boolean functions with larger noise stability. For example, a dictatorship   has noise stability  .

The Majority is Stablest theorem states, informally, then the only functions having noise stability larger than majority have influential coordinates. Formally, for every   there exists   such that if   has expectation zero and  , then  .

The first proof of this theorem used the invariance principle in conjunction with an isoperimetric theorem of Borell in Gaussian space; since then more direct proofs were devised.[citation needed]

Majority is Stablest implies that the Goemans–Williamson approximation algorithm for MAX-CUT is optimal, assuming the unique games conjecture. This implication, due to Khot et al.,[17] was the impetus behind proving the theorem.

ReferencesEdit

  1. ^ a b O'Donnell, Ryan (2014). Analysis of Boolean functions. Cambridge University Press. ISBN 978-1-107-03832-5.
  2. ^ Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture, Huang Hao, 19 August 2019, https://arxiv.org/abs/1907.00847
  3. ^ Decades-old computer Science Conjecture Solved in Two Pages,Quanta Magazine, 2019 February, https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725
  4. ^ Diaconis, Persi; Saloff-Coste, Laurent (1996). "Logarithmic Sobolev inequalities for finite Markov chains". Annals of Applied Probability. 6 (3): 695–750. doi:10.1214/aoap/1034968224.
  5. ^ Mossel, Elchanan; Oleszkiewicz, Krzysztof; Sen, Arnab (2013). "On reverse hypercontractivity". GAFA. 23 (3): 1062–1097. arXiv:1108.1210. doi:10.1007/s00039-013-0229-4.
  6. ^ Friedgut, Ehud; Kalai, Gil; Naor, Assaf (2002). "Boolean functions whose Fourier transform is concentrated on the first two levels". Advances in Applied Mathematics. 29 (3): 427–437. doi:10.1016/S0196-8858(02)00024-6.
  7. ^ Kindler, Guy (2002). "16". Property testing, PCP, and juntas (Thesis). Tel Aviv University.
  8. ^ Kahn, Jeff; Kalai, Gil; Linial, Nati (1988). "The influence of variables on Boolean functions.". Proc. 29th Symp. on Foundations of Computer Science. SFCS'88. White Plains: IEEE. pp. 68–80. doi:10.1109/SFCS.1988.21923.
  9. ^ Ben-Or, Michael; Linial, Nathan (1985). "Collective coin flipping, robust voting schemes and minima of Banzhaf values". Proc. 26th Symp. on Foundations of Computer Science. SFCS'85. Portland, Oregon: IEEE. pp. 408–416. doi:10.1109/SFCS.1985.15.
  10. ^ Friedgut, Ehud (1998). "Boolean functions with low average sensitivity depend on few coordinates". Combinatorica. 18 (1): 474–483. CiteSeerX 10.1.1.7.5597. doi:10.1007/PL00009809.
  11. ^ Mossel, Elchanan; O'Donnell, Ryan; Oleszkiewicz, Krzysztof (2010). "Noise stability of functions with low influences: Invariance and optimality". Annals of Mathematics. 171 (1): 295–341. arXiv:math/0503503. doi:10.4007/annals.2010.171.295.
  12. ^ Blum, Manuel; Luby, Michael; Rubinfeld, Ronitt (1993). "Self-testing/correcting with applications to numerical problems". J. Comput. Syst. Sci. 47 (3): 549–595. doi:10.1016/0022-0000(93)90044-W.
  13. ^ Bellare, Mihir; Coppersmith, Don; Håstad, Johan; Kiwi, Marcos; Sudan, Madhu (1995). "Linearity testing in characteristic two". Proc. 36th Symp. on Foundations of Computer Science. FOCS'95.
  14. ^ Kalai, Gil (2002). "A Fourier-theoretic perspective on the Condorcet paradox and Arrow's theorem". Adv. Appl. Math. 29 (3): 412–426. doi:10.1016/S0196-8858(02)00023-4.
  15. ^ Friedgut, Ehud (1999). "Sharp thresholds of graph properties and the k-SAT problem". J. Am. Math. Soc. 12 (4): 1017–1054. doi:10.1090/S0894-0347-99-00305-7.
  16. ^ Friedgut, Ehud; Kalai, Gil (1996). "Every monotone graph property has a sharp threshold". Proc. Am. Math. Soc. 124 (10): 2993–3002. doi:10.1090/S0002-9939-96-03732-X.
  17. ^ Khot, Subhash; Kindler, Guy; Mossel, Elchanan; O'Donnell, Ryan (2007), "Optimal inapproximability results for MAX-CUT and other two-variable CSPs?" (PDF), SIAM Journal on Computing, 37 (1): 319–357, CiteSeerX 10.1.1.130.8886, doi:10.1137/S0097539705447372