TY - GEN

T1 - Lower bounds for linear decision trees with bounded weights

AU - Uchizawa, Kei

AU - Takimoto, Eiji

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2015.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - In this paper, we consider a linear decision tree such that a linear threshold function at each internal node has a bounded weight: the sum of the absolute values of its integer weights is at most w. We prove that if a Boolean function f is computable by such a linear decision tree of size (i.e., the number of leaves) s and rank r, then f is also computable by a depth-2 threshold circuit containing at most s(2w+1)r threshold gates with weight at most (2w+1)r+1 in the bottom level. Combining a known lower bound on the size of depth-2 threshold circuits, we obtain a 2Ω(n/ logw) lower bound on the size of linear decision trees computing the Inner-Product function modulo 2, which improves on the previous bound 2√n if w = 2o(√n).

AB - In this paper, we consider a linear decision tree such that a linear threshold function at each internal node has a bounded weight: the sum of the absolute values of its integer weights is at most w. We prove that if a Boolean function f is computable by such a linear decision tree of size (i.e., the number of leaves) s and rank r, then f is also computable by a depth-2 threshold circuit containing at most s(2w+1)r threshold gates with weight at most (2w+1)r+1 in the bottom level. Combining a known lower bound on the size of depth-2 threshold circuits, we obtain a 2Ω(n/ logw) lower bound on the size of linear decision trees computing the Inner-Product function modulo 2, which improves on the previous bound 2√n if w = 2o(√n).

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U2 - 10.1007/978-3-662-46078-8_34

DO - 10.1007/978-3-662-46078-8_34

M3 - Conference contribution

AN - SCOPUS:84922032283

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

SP - 412

EP - 422

BT - SOFSEM 2015

A2 - Italiano, Giuseppe F.

A2 - Margaria-Steffen, Tiziana

A2 - Pokorný, Jaroslav

A2 - Quisquater, Jean-Jacques

A2 - Wattenhofer, Roger

PB - Springer Verlag

T2 - 41st International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2015

Y2 - 24 January 2015 through 29 January 2015

ER -