Algorithm 1. Sparsity penalized stacked denoising autoencoders for single cell RNA-seq data (scSDAE).

Input: normalized expression vector $\mathit{x}=\left(x_1,\cdots x_n\right)$, network layer width vector $\mathit{d}\mathit{i}\mathit{m}=\left(n,d_1,\cdots d_k\right)$ with $d_k$ denoted the bottleneck layer width

Output: imputed expression vector ${\mathit{x}}_{imputed}$

for$i$ in$\left\{1,\cdots k\right\}$ do

if $i==1$ do

corrupt $\mathit{x}$ into $\tilde{\mathit{x}}$ with noise, build neural network $N{N}_1\left(n\to d_1\to n\right)$: $\mathit{x}\prime ={\mathit{g}}_{{\mathit{\omega }}_1^{\prime }}({\mathit{f}}_{{\mathit{\omega }}_1}\left(\tilde{\mathit{x}}\right))$ and train $N{N}_1$ to minimize $L_{scSDAE}\left(\mathit{x},\mathit{x}\prime \right)$; predict ${\mathit{h}}_1={\mathit{f}}_{{\mathit{\omega }}_1}\left(\mathit{x}\right)$

else do

corrupt ${\mathit{h}}_{i-1}$ into ${\tilde{\mathit{h}}}_{i-1}$ with noise, build neural network $N{N}_i\left(d_{i-1}\to d_i\to d_{i-1}\right)$: ${\mathit{h}}_{i-1}^{\prime }={\mathit{g}}_{{\mathit{\omega }}_i^{\prime }}({\mathit{f}}_{{\mathit{\omega }}_i}\left({\tilde{\mathit{h}}}_{i-1}\right))$ and train $N{N}_i$ to minimize formula $||{\mathit{h}}_{i-1}-{\mathit{h}}_{i-1}^{\prime }||{}^2/d_{i-1}$

if$i<k$do

predict ${\mathit{h}}_i={\mathit{f}}_{{\mathit{\omega }}_i}\left({\mathit{h}}_{i-1}\right)$

end for

build neural network $NN$: ${\mathit{x}}^{\prime }={\mathit{g}}_{{\mathit{\omega }}_1^{\prime }}\circ \cdots \circ {\mathit{g}}_{{\mathit{\omega }}_k^{\prime }}\left({\mathit{f}}_{{\mathit{\omega }}_k}\circ \cdots \circ {\mathit{f}}_{{\mathit{\omega }}_1}\left(\mathit{x}\right)\right)$ and train $NN$ to minimize $L_{scSDAE}\left(\mathit{x},\mathit{x}\prime \right)$

predict ${\mathit{x}}_{out}={\mathit{g}}_{{\mathit{\omega }}_1^{\prime }}\circ \cdots \circ {\mathit{g}}_{{\mathit{\omega }}_k^{\prime }}\left({\mathit{f}}_{{\mathit{\omega }}_k}\circ \cdots \circ {\mathit{f}}_{{\mathit{\omega }}_1}\left(\mathit{x}\right)\right)$

calculate the mask vector $\mathit{m}$: $m_j=\mathrm{I}\left(x_j>0\right),j=1\cdots n$ ($\mathrm{I}(·)$ is the indicator function.)

calculate the output ${\mathit{x}}_{imputed}={\mathit{x}}_{out}\ast \left(\mathit{1}-\mathit{m}\right)+\mathit{x}\ast \mathit{m}$ ($\ast$ denotes element-wise multiplication.)