Data_Sheet_1_Signal Perceptron: On the Identifiability of Boolean Function Spaces and Beyond.zip

In a seminal book, Minsky and Papert define the perceptron as a limited implementation of what they called “parallel machines.” They showed that some binary Boolean functions including XOR are not definable in a single layer perceptron due to its limited capacity to learn only linearly separable functions. In this work, we propose a new more powerful implementation of such parallel machines. This new mathematical tool is defined using analytic sinusoids—instead of linear combinations—to form an analytic signal representation of the function that we want to learn. We show that this re-formulated parallel mechanism can learn, with a single layer, any non-linear k-ary Boolean function. Finally, to provide an example of its practical applications, we show that it outperforms the single hidden layer multilayer perceptron in both Boolean function learning and image classification tasks, while also being faster and requiring fewer parameters.

在一部开创性专著中，明斯基（Minsky）与佩珀特（Papert）将感知机（Perceptron）定义为他们所称的“并行机器”的一种受限实现方式。他们证明，包括异或（XOR）在内的部分二元布尔函数无法在单层感知机中被表征，原因在于单层感知机的学习能力受限，仅能学习线性可分函数。本研究提出了一种此类并行机器的新型、更具表达能力的实现方案。该新型数学工具摒弃传统的线性组合方式，转而采用解析正弦函数来构建待学习函数的解析信号表征。我们证明，这种重构的并行机制仅需单层结构，即可学习任意非线性k元布尔函数。最后，为展示其实际应用价值，我们验证了该方法在布尔函数学习与图像分类任务中均优于单隐层多层感知机，同时具备更快的运算速度与更少的参数需求。