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Generalization of hinging hyperplanes

机译：铰接超平面的推广

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摘要

The model of hinging hyperplanes (HH) can approximate a large class of nonlinear functions to arbitrary precision, but represent only a small part of continuous piecewise-linear (CPWL) functions in two or more dimensions. In this correspondence, the influence of this drawback for black-box modeling is first illustrated by a simple example. Then it is shown that the above shortcoming can be amended by adding a sufficient number of linear functions to current hinges. It is proven that any CPWL function of n variables can be represented by a sum of hinges containing at most n+1 linear functions. Hence the model of a sum of such expanded hinges is a general representation for all CPWL functions. The structure of the novel general representation is much simpler than the existing generalized canonical representation that consists of nested absolute-value functions. This characteristic is very useful for black-box modeling. Based on the new general representation, an upper bound on the number of nestings of nested absolute-value functions of a generalized canonical representation is established, which is much smaller than the known result.
机译：铰接超平面（HH）模型可以将一类大型非线性函数近似为任意精度，但在二维或更多维中仅代表连续分段线性（CPWL）函数的一小部分。在此对应关系中，首先通过一个简单的示例说明此缺陷对黑盒建模的影响。然后表明，可以通过向当前的铰链添加足够数量的线性函数来弥补上述缺点。事实证明，n个变量的任何CPWL函数都可以由最多包含n + 1个线性函数的铰链之和表示。因此，这种扩展铰链之和的模型是所有CPWL功能的一般表示。新颖的一般表示的结构比现有的由嵌套绝对值函数组成的一般规范表示要简单得多。此特性对于黑盒建模非常有用。基于新的一般表示，建立了广义规范表示的嵌套绝对值函数的嵌套数量的上限，该上限比已知结果小得多。

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