### Abstract

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 w_{1}, w_{2}, ..., w_{n} of each linear threshold function satisfy ∑_{i}|w_{i}| ≤ 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 2^{Ω(√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.

Original language | English |
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Title of host publication | Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings |

Pages | 568-579 |

Number of pages | 12 |

DOIs | |

Publication status | Published - Sep 1 2011 |

Event | 36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011 - Warsaw, Poland Duration: Aug 22 2011 → Aug 26 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6907 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011 |
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Country | Poland |

City | Warsaw |

Period | 8/22/11 → 8/26/11 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings*(pp. 568-579). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6907 LNCS). https://doi.org/10.1007/978-3-642-22993-0_51

**Lower bounds for linear decision trees via an energy complexity argument.** / Uchizawa, Kei; Takimoto, Eiji.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6907 LNCS, pp. 568-579, 36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011, Warsaw, Poland, 8/22/11. https://doi.org/10.1007/978-3-642-22993-0_51

}

TY - GEN

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

AU - Uchizawa, Kei

AU - Takimoto, Eiji

PY - 2011/9/1

Y1 - 2011/9/1

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.

UR - http://www.scopus.com/inward/record.url?scp=80052133590&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052133590&partnerID=8YFLogxK

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

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

M3 - Conference contribution

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

ER -