用非平衡响应理论理解循环神经网络
Understanding Recurrent Neural Networks Using Nonequilibrium Response Theory
摘要：
循环神经网络（RNN）是在机器学习中广泛用于分析序列数据的大脑启发模型。目前的工作是对RNN如何使用非平衡统计力学的响应理论来处理输入信号的更深入理解的贡献。对于一类由输入信号驱动的连续时间随机RNN（SRNN），我们推导出其输出的Volterra型级数表示。这种表示是可解释的，并且将输入信号从SRNN架构中分离出来。该系列的内核是某些递归定义的相关函数相对于未扰动的动态，完全确定的输出。利用这种表示及其对粗糙路径理论的影响，我们确定了一个通用的功能-响应功能，这原来是输入信号的张量积和自然支持基础的签名。特别是，我们表明，SRNN，只有在读出层中的权重优化和隐藏层中的权重保持固定，而不是优化，可以被视为内核机器上运行的再生核希尔伯特空间与响应功能。
英文摘要：
Recurrent neural networks (RNNs) are brain-inspired models widely used in machine learning for analyzing sequential data. The present work is a contribution towards a deeper understanding of how RNNs process input signals using the response theory from nonequilibrium statistical mechanics. For a class of continuous-time stochastic RNNs (SRNNs) driven by an input signal, we derive a Volterra type series representation for their output. This representation is interpretable and disentangles the input signal from the SRNN architecture. The kernels of the series are certain recursively defined correlation functions with respect to the unperturbed dynamics that completely determine the output. Exploiting connections of this representation and its implications to rough paths theory, we identify a universal feature -- the response feature, which turns out to be the signature of tensor product of the input signal and a natural support basis. In particular, we show that SRNNs, with only the weights in the readout layer optimized and the weights in the hidden layer kept fixed and not optimized, can be viewed as kernel machines operating on a reproducing kernel Hilbert space associated with the response feature.
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