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

Amir Shpilka, Avishay Tal, Ben lee Volk
2013 arXiv   pre-print
In this paper we prove results regarding 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.  ... 
arXiv:1304.0371v2 fatcat:qpmqv6v7hnbojoqtj6cm4l2xba

On the structure of boolean functions with small spectral norm

Amir Shpilka, Avishay Tal, Ben lee Volk
2014 Proceedings of the 5th conference on Innovations in theoretical computer science - ITCS '14  
In this paper we prove results regarding 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.  ... 
doi:10.1145/2554797.2554803 dblp:conf/innovations/ShpilkaTV14 fatcat:tbjq4axdaba2nctlno2rrxlnju

On the Structure of Boolean Functions with Small Spectral Norm

Amir Shpilka, Avishay Tal, Ben lee Volk
2015 Computational Complexity  
In this paper we prove results regarding 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.  ... 
doi:10.1007/s00037-015-0110-y fatcat:xfjcqrdpv5dqtd7mf3howb36zi

Relational concepts and the fourier transform: An empirical study [chapter]

Eduardo Pérez, Larry Rendell
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.  ... 
doi:10.1007/3-540-64575-6_67 fatcat:ek3ofgcl6fdllhewxetafap6ua

Complexity Theoretic Aspects of Some Cryptographic Functions [chapter]

Eike Kiltz, Hans Ulrich Simon
2003 Lecture Notes in Computer Science  
It is known that 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.  ... 
doi:10.1007/3-540-45071-8_31 fatcat:6zuo5qwjwfhldl5aj654hk6avm

Boolean functions with small spectral norm, revisited

TOM SANDERS
2018 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
AbstractWe show that if f is a 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  ... 
doi:10.1017/s030500411800035x fatcat:evy7rj6ek5falboy34igpi2hvq

Threshold circuit lower bounds on cryptographic functions

Eike Kiltz, Hans Ulrich Simon
2005 Journal of computer and system sciences (Print)  
It is known that 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.  ... 
doi:10.1016/j.jcss.2005.03.001 fatcat:wgv27njlvfdtbm6twet6x6u3la

Fourier Sparsity, Spectral Norm, and the Log-Rank Conjecture

Hing Yin Tsang, Chung Hoi Wong, Ning Xie, Shengyu Zhang
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
We study 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  ... 
doi:10.1109/focs.2013.76 dblp:conf/focs/TsangWXZ13 fatcat:zrmoukqiwzhf7gl7dybseqzcvy

Fourier sparsity, spectral norm, and the Log-rank conjecture [article]

Hing Yin Tsang, Chung Hoi Wong, Ning Xie, Shengyu Zhang
2013 arXiv   pre-print
Along 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  ... 
arXiv:1304.1245v2 fatcat:t5miqj44oncnrfpp5lurml5534

Testing Linear-Invariant Function Isomorphism [chapter]

Karl Wimmer, Yuichi Yoshida
2013 Lecture Notes in Computer Science  
Exploiting our upper bound, we show that any property is testable if it can be well-approximated by 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] .  ... 
doi:10.1007/978-3-642-39206-1_71 fatcat:aghrolcrtnfrtflrvp6w4z4dau

Harmonic Analysis, Real Approximation, and the Communication Complexity of Boolean Functions

V. Grolmusz
1999 Algorithmica  
, or: for almost all 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  ... 
doi:10.1007/pl00009265 fatcat:zchechp6e5divmgn62t5fzmqmq

$L$-structure in $L$-spaces

F. Cunningham
1960 Transactions of the American Mathematical Society  
Then / is a linear 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  ... 
doi:10.1090/s0002-9947-1960-0115084-7 fatcat:doowbpieczhcxcr3adagy7e77q

Scalar operators and integration

Igor Kluvánek
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  ... 
doi:10.1017/s1446788700031116 fatcat:7lepdlrxtrfmlk2lwiq7lvsq3y

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

Mitali Bafna, Max Hopkins, Tali Kaufman, Shachar Lovett
2021 arXiv   pre-print
Hypercontractivity is 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.  ... 
arXiv:2111.09444v2 fatcat:glm7cfkvgrbn7ck7knabem7fgq

Structure of spectral measures on locally convex spaces

Bertram Walsh
1965 Transactions of the American Mathematical Society  
Though 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).  ... 
doi:10.1090/s0002-9947-1965-0196503-1 fatcat:lvs4dgdkxrabngeyjmkfsl253y
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