We present a method for enumerating linear threshold functions of n-dimensional binary inputs, for neural nets. Our starting point is the geometric lattice L/sub n/ of hyperplane intersections in the dual (weight) space. We show how the hyperoctahedral group O/sub n+1/, the symmetry group of the (n+1)-dimensional hypercube, can be used to construct a symmetry-adapted poset of hyperplane intersections /spl Delta//sub n/ which is much more compact and tractable than L/sub n/. A generalized Zeta function and its inverse, the generalized Mobius function, are defined on /spl Delta//sub n/. Symmetry-adapted posets of hyperplane intersections for three-, four-, and five-dimensional inputs are constructed and the number of linear threshold functions is computed from the generalized Mobius function. Finally, we show how equivalence classes of linear threshold functions are enumerated by unfolding the symmetry-adapted poset of hyperplane intersections into a symmetry-adapted face poset. It is hoped that our construction will lead to ways of placing asymptotic bounds on the number of equivalence classes of linear threshold functions.
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机译：我们提出了一种用于神经网络的n维二进制输入的线性阈值函数的枚举方法。我们的起点是双（权重）空间中超平面相交的几何格L / sub n /。我们展示了如何使用（n + 1）维超立方体的对称群O / sub n + 1 /的八面体组来构造超平面相交点/ spl Delta // sub n /的对称适应性球体比L / sub n /更紧凑和易于处理。广义Zeta函数及其反函数（广义Mobius函数）在/ spl Delta // sub n /上定义。构造了用于三维，四维和五维输入的超平面相交的对称适应性姿态，并根据广义Mobius函数计算了线性阈值函数的数量。最后，我们展示了如何通过将超平面相交的对称适应的坐姿展开成对称适应的面部坐姿来枚举线性阈值函数的等价类。希望我们的构造将导致在线性阈值函数的等价类数上放置渐近边界的方法。
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