Boolean circuits of McCulloch-Pitts threshold gates are a classic model of neural computation studied heavily in the late 20th century as a model of general computation. Recent advances in large-scale neural computing hardware has made their practical implementation a near-term possibility. We describe a theoretical approach for multiplying two $N$ by $N$ matrices that integrates threshold gate logic with conventional fast matrix multiplication algorithms, that perform $O(N^\omega)$ arithmetic operations for a positive constant $\omega < 3$. Our approach converts such a fast matrix multiplication algorithm into a constant-depth threshold circuit with approximately $O(N^\omega)$ gates. Prior to our work, it was not known whether the $\Theta(N^3)$-gate barrier for matrix multiplication was surmountable by constant-depth threshold circuits. Dense matrix multiplication is a core operation in convolutional neural network training. Performing this work on a neural architecture instead of off-loading it to a GPU may be an appealing option.

中文翻译：

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用于矩阵乘法的恒定深度和亚立方尺寸阈值电路

McCulloch-Pitts 阈值门的布尔电路是 20 世纪后期作为一般计算模型进行大量研究的经典神经计算模型。大规模神经计算硬件的最新进展使它们的实际实现成为近期的可能性。我们描述了一种将两个 $N$ 乘以 $N$ 矩阵的理论方法，该方法将阈值门逻辑与传统的快速矩阵乘法算法相结合，对正常数 $\omega < 3 执行 $O(N^\omega)$ 算术运算$. 我们的方法将这种快速矩阵乘法算法转换为具有大约 $O(N^\omega)$ 门的恒定深度阈值电路。在我们的工作之前，不知道矩阵乘法的 $\Theta(N^3)$-门屏障是否可以通过恒定深度阈值电路克服。密集矩阵乘法是卷积神经网络训练中的核心操作。在神经架构上执行这项工作而不是将其卸载到 GPU 可能是一个有吸引力的选择。