TY - GEN

T1 - On the structure of boolean functions with small spectral norm

AU - Shpilka, Amir

AU - Tal, Avishay

AU - Volk, Ben Lee

PY - 2014

Y1 - 2014

N2 - In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is ||f̂||1 = Σ α |f̂(α)|). Specifically, we prove the following results for functions f: {0, 1}n → {0,1} with ||f̂||1 = A. 1. There is a subspace V of co-dimension at most A2 such that f\V is constant. 2. f can be computed by a parity decision tree of size 2 A2n2A. (a parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.) 3. f can be computed by a De Morgan formula of size O(2A2n2A+2) and by a De Morgan formula of depth O(A2 + log(n) · A). 4. If in addition f has at most s nonzero Fourier coefficients, then f can be computed by a parity decision tree of depth A2 log s. 5. For every ε > 0 there is a parity decision tree of depth O(A2 + log(1/ε)) and size 2 O(A2) · min{1/ε2, O(log(1/ε)) 2A} that e-approximates f. Furthermore, this tree can be learned, with probability 1 - δ, using poly(n, exp(A2), 1/ε, log(1/δ)) membership queries. All the results above (except 3) also hold (with a slight change in parameters) for functions f: ℤp n → {0,1}.

AB - In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is ||f̂||1 = Σ α |f̂(α)|). Specifically, we prove the following results for functions f: {0, 1}n → {0,1} with ||f̂||1 = A. 1. There is a subspace V of co-dimension at most A2 such that f\V is constant. 2. f can be computed by a parity decision tree of size 2 A2n2A. (a parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.) 3. f can be computed by a De Morgan formula of size O(2A2n2A+2) and by a De Morgan formula of depth O(A2 + log(n) · A). 4. If in addition f has at most s nonzero Fourier coefficients, then f can be computed by a parity decision tree of depth A2 log s. 5. For every ε > 0 there is a parity decision tree of depth O(A2 + log(1/ε)) and size 2 O(A2) · min{1/ε2, O(log(1/ε)) 2A} that e-approximates f. Furthermore, this tree can be learned, with probability 1 - δ, using poly(n, exp(A2), 1/ε, log(1/δ)) membership queries. All the results above (except 3) also hold (with a slight change in parameters) for functions f: ℤp n → {0,1}.

KW - Analysis of Boolean Functions

KW - Decision Trees

KW - Spectral Norm

UR - http://www.scopus.com/inward/record.url?scp=84893322608&partnerID=8YFLogxK

U2 - 10.1145/2554797.2554803

DO - 10.1145/2554797.2554803

M3 - פרסום בספר כנס

AN - SCOPUS:84893322608

SN - 9781450322430

T3 - ITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science

SP - 37

EP - 47

BT - ITCS 2014 - Proceedings of the 2014 Conference on Innovations in Theoretical Computer Science

PB - Association for Computing Machinery

Y2 - 12 January 2014 through 14 January 2014

ER -