# 自编码器

## 基本结构

${\displaystyle \phi :{\mathcal {X}}\rightarrow {\mathcal {F}}}$
${\displaystyle \psi :{\mathcal {F}}\rightarrow {\mathcal {X}}}$
${\displaystyle \phi ,\psi ={\underset {\phi ,\psi }{\operatorname {arg\,min} }}\,\|{\mathcal {X}}-(\psi \circ \phi ){\mathcal {X}}\|^{2}}$

${\displaystyle \mathbf {h} =\sigma (\mathbf {Wx} +\mathbf {b} )}$

${\displaystyle \mathbf {h} }$通常表示编码、潜变量或潜在表示。${\displaystyle \sigma }$是一个逐元素的激活函数（例如sigmoid函数线性整流函数）。${\displaystyle \mathbf {W} }$是权重矩阵，${\displaystyle \mathbf {b} }$是偏置向量。权重和偏置通常随机初始化，并在训练期间通过反向传播迭代更新。自编码器的解码阶段映射${\displaystyle \mathbf {h} }$到重构${\displaystyle \mathbf {x'} }$（与${\displaystyle \mathbf {x} }$形状一致）：

${\displaystyle \mathbf {x'} =\sigma '(\mathbf {W'h} +\mathbf {b'} )}$

${\displaystyle {\mathcal {L}}(\mathbf {x} ,\mathbf {x'} )=\|\mathbf {x} -\mathbf {x'} \|^{2}=\|\mathbf {x} -\sigma '(\mathbf {W'} (\sigma (\mathbf {Wx} +\mathbf {b} ))+\mathbf {b'} )\|^{2}}$

## 參考

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