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On the Representational Power of Restricted Boltzmann Machines for Symmetric Functions and Boolean Functions

机译:关于对称函数和布尔函数限制Boltzmann机器的代表性

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Restricted Boltzmann machines (RBMs) are used to build deep-belief networks that are widely thought to be one of the first effective deep learning neural networks. This paper studies the ability of RBMs to represent distributions over {0, 1}(n) via softplus/hardplus RBM networks. It is shown that any distribution whose density depends on the number of 1's in their input can be approximated with arbitrarily high accuracy by an RBM of size 2n + 1, which improves the result of a previous study by reducing the size from n(2) to 2n + 1. A theorem for representing partially symmetric Boolean functions by softplus RBM networks is established. Accordingly, the representational power of RBMs for distributions whose mass represents the Boolean functions is investigated in comparison with that of threshold circuits and polynomial threshold functions. It is shown that a distribution over {0, 1}(n) whose mass represents a Boolean function can be computed with a given margin delta by an RBM of size and parameters bounded by polynomials in n, if and only if it can be computed by a depth-2 threshold circuit with size and parameters bounded by polynomials in n.
机译:限制的Boltzmann机器(RBMS)用于构建深信网络,广泛认为是第一个有效的深度学习神经网络之一。本文研究了RBMS通过SoftPlus / HardPlus RBM网络表示{0,1}(n)的分布的能力。结果表明,其密度取决于其输入中的1的数量的任何分布可以通过大小的尺寸2n + 1的RBM任意高精度来近似,这通过减少N(2)的尺寸来提高先前研究的结果到2N + 1。建立了由SoftPlus RBM网络表示部分对称布尔函数的定理。因此,与阈值电路和多项式阈值功能相比,研究了质量代表了质量代表布尔函数的分布的RBMS的代表性。结果表明,其质量代表布尔函数的分布可以通过rbm的大小和参数在n,if any项中使用多项式的rbm来计算{0,1}(n)的分布。通过深度-2阈值电路,具有由n的多项式界定的大小和参数。

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