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  • 整体思路以及计算方式
  • 代码
  • 简评
  1. LongConv

Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks

PreviousLongConvNextParallelizing Legendre Memory Unit Training

Last updated 2 years ago

论文地址：

  • https://papers.nips.cc/paper/2019/hash/952285b9b7e7a1be5aa7849f32ffff05-Abstract.html

整体思路以及计算方式

一种RNN结构，模型计算的是输入信号在某个窗口内关于勒让德多项式的系数，达到某种程度的最优，系数的微分方程为：

θm˙(t)=Am(t)+Bu(t)\theta \dot{\mathbf{m}}(t)=\mathbf{A} \mathbf{m}(t)+\mathbf{B} u(t)θm˙(t)=Am(t)+Bu(t)

其中：

A=[a]ij∈Rd×d,aij=(2i+1){−1i<j(−1)i−j+1i≥jB=[b]i∈Rd×1,bi=(2i+1)(−1)i,i,j∈[0,d−1]\begin{aligned} \mathbf{A}&=[a]_{i j} \in \mathbb{R}^{d \times d}, \quad a_{i j}=(2 i+1) \begin{cases}-1 & i<j \\ (-1)^{i-j+1} & i \geq j\end{cases} \\ \mathbf{B}&=[b]_{i} \in \mathbb{R}^{d \times 1}, \quad b_{i}=(2 i+1)(-1)^{i}, \quad i, j \in[0, d-1] \end{aligned}AB​=[a]ij​∈Rd×d,aij​=(2i+1){−1(−1)i−j+1​i<ji≥j​=[b]i​∈Rd×1,bi​=(2i+1)(−1)i,i,j∈[0,d−1]​

离散化可得：

mt=A‾mt−1+B‾ut\mathbf{m}_{t}=\mathbf{\overline A} \mathbf{m}_{t-1}+\mathbf{\overline B} u_{t}mt​=Amt−1​+But​

其中：

最后的模型结构为：

代码

  • https://github.com/hrshtv/pytorch-lmu

  • https://github.com/nengo/keras-lmu

简评

感觉S4应该或多或少从这篇文章受到某些启发。

A‾=(Δt/θ)A+I,B‾=(Δt/θ)B{\mathbf{\overline A}}=(\Delta t / \theta) \mathbf{A}+\mathbf{I}, \quad {\mathbf{\overline B}}=(\Delta t / \theta) \mathbf{B}A=(Δt/θ)A+I,B=(Δt/θ)B
ht=f(Wxxt+Whht−1+Wmmt)ut=ex⊤xt+eh⊤ht−1+em⊤mt−1\mathbf{h}_{t}=f\left(\mathbf{W}_{\mathbf{x}} \mathbf{x}_{t}+\mathbf{W}_{\mathbf{h}} \mathbf{h}_{t-1}+\mathbf{W}_{\mathbf{m}} \mathbf{m}_{t}\right)\\ u_{t}=\mathbf{e}_{\mathbf{x}}^{\top} \mathbf{x}_{t}+\mathbf{e}_{\mathbf{h}}{ }^{\top} \mathbf{h}_{t-1}+\mathbf{e}_{\mathbf{m}}^{\top} \mathbf{m}_{t-1}ht​=f(Wx​xt​+Wh​ht−1​+Wm​mt​)ut​=ex⊤​xt​+eh​⊤ht−1​+em⊤​mt−1​