TY - GEN

T1 - Lower bounds for linear decision trees via an energy complexity argument

AU - Uchizawa, Kei

AU - Takimoto, Eiji

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2011

Y1 - 2011

N2 - A linear decision tree is a binary decision tree in which a classification rule at each internal node is defined by a linear threshold function. In this paper, we consider a linear decision tree T where the weights w1, w2, ..., wn of each linear threshold function satisfy ∑i|wi| ≤ w for an integer w, and prove that if T computes an n-variable Boolean function of large unbounded-error communication complexity (such as the Inner-Product function modulo two), then T must have 2Ω(√n)/w leaves. To obtain the lower bound, we utilize a close relationship between the size of linear decision trees and the energy complexity of threshold circuits; the energy of a threshold circuit C is defined to be the maximum number of gates outputting "1," where the maximum is taken over all inputs to C. In addition, we consider threshold circuits of depth ω(1) and bounded energy, and provide two exponential lower bounds on the size (i.e., the number of gates) of such circuits.

AB - A linear decision tree is a binary decision tree in which a classification rule at each internal node is defined by a linear threshold function. In this paper, we consider a linear decision tree T where the weights w1, w2, ..., wn of each linear threshold function satisfy ∑i|wi| ≤ w for an integer w, and prove that if T computes an n-variable Boolean function of large unbounded-error communication complexity (such as the Inner-Product function modulo two), then T must have 2Ω(√n)/w leaves. To obtain the lower bound, we utilize a close relationship between the size of linear decision trees and the energy complexity of threshold circuits; the energy of a threshold circuit C is defined to be the maximum number of gates outputting "1," where the maximum is taken over all inputs to C. In addition, we consider threshold circuits of depth ω(1) and bounded energy, and provide two exponential lower bounds on the size (i.e., the number of gates) of such circuits.

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U2 - 10.1007/978-3-642-22993-0_51

DO - 10.1007/978-3-642-22993-0_51

M3 - Conference contribution

AN - SCOPUS:80052133590

SN - 9783642229923

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 568

EP - 579

BT - Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings

T2 - 36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011

Y2 - 22 August 2011 through 26 August 2011

ER -