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On the Structure of Boolean Functions with Small Spectral Norm
[article]

2013
*
arXiv
*
pre-print

In this paper we prove results regarding

arXiv:1304.0371v2
fatcat:qpmqv6v7hnbojoqtj6cm4l2xba
*Boolean**functions**with**small**spectral**norm*(*the**spectral**norm**of*f is f̂_1=∑_α|f̂(α)|). ... Specifically, we prove*the*following results for*functions*f:{0,1}^n →{0,1}*with*f̂_1=A. 1. ... There are still many intriguing open problems related to*the**structure**of**Boolean**functions**with**small**spectral**norm*. ...##
###
On the structure of boolean functions with small spectral norm

2014
*
Proceedings of the 5th conference on Innovations in theoretical computer science - ITCS '14
*

In this paper we prove results regarding

doi:10.1145/2554797.2554803
dblp:conf/innovations/ShpilkaTV14
fatcat:tbjq4axdaba2nctlno2rrxlnju
*Boolean**functions**with**small**spectral**norm*(*the**spectral**norm**of*f is f 1 = α |f (α)|). ... Specifically, we prove*the*following results for*functions*f : {0, 1} n → {0, 1}*with*f 1 = A. ... There are still many intriguing open problems related to*the**structure**of**Boolean**functions**with**small**spectral**norm*. ...##
###
On the Structure of Boolean Functions with Small Spectral Norm

2015
*
Computational Complexity
*

In this paper we prove results regarding

doi:10.1007/s00037-015-0110-y
fatcat:xfjcqrdpv5dqtd7mf3howb36zi
*Boolean**functions**with**small**spectral**norm*(*the**spectral**norm**of*f is f 1 = α |f (α)|). ... Specifically, we prove*the*following results for*functions*f : {0, 1} n → {0, 1}*with*f 1 = A. ... There are still many intriguing open problems related to*the**structure**of**Boolean**functions**with**small**spectral**norm*. ...##
###
Relational concepts and the fourier transform: An empirical study
[chapter]

1998
*
Lecture Notes in Computer Science
*

*One*

*of*such approaches, based

*on*

*the*Fourier transform

*of*

*Boolean*

*functions*, is studied and compared empirically to others, based

*on*constructing new features or extracting relations from propositional ... This characterization, which implicates parameters

*of*Fourier complexity, other measures

*of*concept di culty, and

*the*relational

*structure*

*of*

*the*target concepts, is also discussed

*with*respect to di cult ... We thank Lenny Pitt for bringing to our attention

*the*literature

*on*learning through

*the*Fourier transform, and for his suggestions. ...

##
###
Complexity Theoretic Aspects of Some Cryptographic Functions
[chapter]

2003
*
Lecture Notes in Computer Science
*

It is known that

doi:10.1007/3-540-45071-8_31
fatcat:6zuo5qwjwfhldl5aj654hk6avm
*the*distributed*Boolean**function*represented by M is hard to compute in various restricted models*of*computation if*the**spectral**norm*is bounded from above by N 1−ε , where ε > 0 denotes ... In this work, we are interested in non-trivial upper bounds*on**the**spectral**norm**of*binary matrices M from {−1, 1} N×N . ... proving lower bounds*on**the*circuit size for concrete families*of**Boolean**functions*. ...##
###
Boolean functions with small spectral norm, revisited

2018
*
Mathematical proceedings of the Cambridge Philosophical Society (Print)
*

AbstractWe show that if f is a

doi:10.1017/s030500411800035x
fatcat:evy7rj6ek5falboy34igpi2hvq
*Boolean**function**on*F2n*with**spectral**norm*at most M then there is some L ≤ exp(M3+o(1)) and subspaces V1,. . .,VL such that f = Σi ± 1Vi. ... Acknowledgment*The*author should like to thank Ben Green for suggesting that*the*writing*of*this paper might make*the*work*of*[San16] more accessible. ...*The*above result includes a*structure*theorem for*Boolean**functions*(*with**small**spectral**norm*) as they are*functions*taking particular integer values, but does not address*the*question*of*which linear ...##
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Threshold circuit lower bounds on cryptographic functions

2005
*
Journal of computer and system sciences (Print)
*

It is known that

doi:10.1016/j.jcss.2005.03.001
fatcat:wgv27njlvfdtbm6twet6x6u3la
*the*distributed*Boolean**function*represented by M is hard to compute in various restricted models*of*computation if*the**spectral**norm*is bounded from above by N 1−ε , where ε > 0 denotes ... In this work, we are interested in non-trivial upper bounds*on**the**spectral**norm**of*binary matrices M from {−1, 1} N×N . ...*small**spectral**norms*. ...##
###
Fourier Sparsity, Spectral Norm, and the Log-Rank Conjecture

2013
*
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
*

We study

doi:10.1109/focs.2013.76
dblp:conf/focs/TsangWXZ13
fatcat:zrmoukqiwzhf7gl7dybseqzcvy
*Boolean**functions**with*sparse Fourier spectrum or*small**spectral**norm*, and show their applications to*the*Log-rank Conjecture for XOR*functions*f (x ⊕ y)a fairly large class*of**functions*including ...*The*rank*of**the*communication matrix M f for such*functions*is exactly*the*Fourier sparsity*of*f . ... ACKNOWLEDGMENTS We are indebted to Andrej Bogdanov for an enlightening discussion at*the*early stage*of**the*work, and we would also like to thank Irit Dinur, Elena Grigorescu, Ronitt Rubinfeld, Rocco Servedio ...##
###
Fourier sparsity, spectral norm, and the Log-rank conjecture
[article]

2013
*
arXiv
*
pre-print

Along

arXiv:1304.1245v2
fatcat:t5miqj44oncnrfpp5lurml5534
*the*way we also show several*structural*results about*Boolean**functions**with**small*F2-degree or*small**spectral**norm*, which could be*of*independent interest. ... We study*Boolean**functions**with*sparse Fourier coefficients or*small**spectral**norm*, and show their applications to*the*Log-rank Conjecture for XOR*functions*f(x⊕ y) --- a fairly large class*of**functions*... In a related setting, Green and Sanders [GS08] showed that*Boolean**functions**with*a*small**spectral**norm*(i.e.*the*ℓ 1 -*norm**of**the*Fourier spectrum) can be decomposed into a*small*number*of*signed indicator ...##
###
Testing Linear-Invariant Function Isomorphism
[chapter]

2013
*
Lecture Notes in Computer Science
*

Exploiting our upper bound, we show that any property is testable if it can be well-approximated by

doi:10.1007/978-3-642-39206-1_71
fatcat:aghrolcrtnfrtflrvp6w4z4dau
*functions**with**small**spectral**norm*. ... We show that*the*query complexity to test g-isomorphism is essentially determined by*the**spectral**norm**of*g. ... We note*the*strong connection*of*learning*functions**with**small**spectral**norm*due to Kushilevitz and Mansour [16] . ...##
###
Harmonic Analysis, Real Approximation, and the Communication Complexity of Boolean Functions

1999
*
Algorithmica
*

, or: for almost all

doi:10.1007/pl00009265
fatcat:zchechp6e5divmgn62t5fzmqmq
*Boolean**function**the*distribution*of**the*Fourier-eoefficients is "even": they eannot be divided into two classes:*one**with**small*LI,*the*other*with**small*L 2*norms*. ... However, if*the*Fourier-coefficienu*of*a*Boolean**function*f are unevenly distributed, more exactly, Ü they ean be divided into two groups:*one**with**small*LI*norm*(say, L), and*the*other*with**small*enough ...##
###
$L$-structure in $L$-spaces

1960
*
Transactions of the American Mathematical Society
*

Then / is a linear

doi:10.1090/s0002-9947-1960-0115084-7
fatcat:doowbpieczhcxcr3adagy7e77q
*functional**of**norm*1. For/ in S*the*composition //*of*/*with*/is a*functional**on*8. Instead*of*ffu we shall write fu, when /" is a characteristic*function*. Lemma 7.5. ... In §2 we show that*the*i-*structure**of*any Banach space consists*of*a complete*Boolean*algebra*of*projections. Preliminary algebra having to do*with**Boolean*rings is covered in §3. ... Then 2 is compact and Hausdorff,*the**functions**on*2 obtained by transferring to 2*functions*in 9Jc ...##
###
Scalar operators and integration

1988
*
Journal of the Australian Mathematical Society
*

*The*notion

*of*a scalar operator

*on*a Banach space, in

*the*sense

*of*N. ... Dunford, extends to arbitrary Banach spaces

*the*idea

*of*an operator

*with*diagonalizable matrix

*on*a finite-dimensional space. It proved ... Accordingly, a set

*of*operators W C B(E) is called a

*Boolean*quasialgebra [5] Recall that, if A is a commutative Banach algebra

*with*unit, then

*the*

*structure*space, A,

*of*A is

*the*set

*of*all homomorphisms ...

##
###
Hypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments
[article]

2021
*
arXiv
*
pre-print

Hypercontractivity is

arXiv:2111.09444v2
fatcat:glm7cfkvgrbn7ck7knabem7fgq
*one**of**the*most powerful tools in*Boolean**function*analysis. ... Our results lead to a new understanding*of**the**structure**of**Boolean**functions**on*HDX, including a tight analog*of**the*KKL Theorem and a new characterization*of*non-expanding sets. ...*One**of**the*most well-studied problems in*the*analysis*of**boolean**functions*is understanding*the**structure**of**functions**with*low influence. ...##
###
Structure of spectral measures on locally convex spaces

1965
*
Transactions of the American Mathematical Society
*

Though

doi:10.1090/s0002-9947-1965-0196503-1
fatcat:lvs4dgdkxrabngeyjmkfsl253y
*spectral*measures -i.e. countably additive idempotent-valued set*functions*defined*on*a cr-algebra*of*sets*with*values in =S? ... are intimately connected*with**spectral*measures -*the*operators*with*their "resolutions*of**the*identity,"*the**Boolean*algebras (if "complete" in a suitable sense)*with**the*measures, defined*on**the*Stone ...*The**norm*q*on*Eq~ is compatible*with*(X,Q,pq). ...
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