Algorithm 1.

Multimodal OT Autoencoder (two-modality case)

Data matrices $X^{(t,s)}$, training iterations $M$, batch size $B$, autoencoder $f_1$, $g_1$, $f_2$, $g_2$ with weights $\theta$, learning rate $\alpha$, loss weight tuning parameters ${\lambda }_{\text{recon}}$, ${\lambda }_{\mathrm{o}\mathrm{t}}$, OT update frequency $K$.

Initialize all ${\gamma }_s^{(t,t+1)}$, $1\le t\le T-1$ matrices with zero matrices.

for $iteration=1,2,\dots ,M$ do

Sample cells $ℬ={\{{ℬ}^{\left(t\right)}\}}_{t=1}^T$ where each subset ${ℬ}^{\left(t\right)}$ has $B$ cells.

Compute $L_{\text{recon}}$, $L_{\mathrm{o}\mathrm{t}}$, $L_{\text{modality}}$

Compute $L={\lambda }_{\text{recon}}{L}_{recon}+{\lambda }_{ot}{L}_{ot}+L_{modality}$

Perform gradient descent step on autoencoder weights $\theta \leftarrow \theta -\alpha {\nabla }_{\theta }L$

if $M\mathrm{\%}K==0$ then

Update ${\gamma }_s^{(t,t+1)}$, $1\le t\le T-1$, $s=1,2$ using current embeddings.

end if

end for