Algorithm 1 Block Updating Algorithm

Input: data $({\mathit{x}}_{i1},{\mathit{x}}_{i2},\dots ,{\mathit{x}}_{id},y_i),i=1,\cdots ,n$; where ${\mathit{x}}_{ir}={(x_{ir1},\dots ,x_{ir{p}_r})}^{\mathrm{T}}$ is the r-th modality for product i with $p_r$ number of features

Initialization:$\theta ={\mathbf{1}}_d$; ${\lambda }_0$: solving the smoothing spline problem as [43], and tuning ${\lambda }_0$ according to cross-validation; ${\beta }_r$: initialized via ridge regression, $r=1,\cdots ,d$.

Repeat

Select the tuning parameter ${\lambda }_2$ based on cross-validation

Repeat until $\alpha$, $\mathit{c}$, and $\mathit{\theta }$ coverage:

Step 1: $\underset{\alpha ,\mathit{c}}{argmin}{\left(\mathit{y}-\alpha {\mathbf{1}}_n-R_{\theta }^{*}\mathit{c}\right)}^{\mathrm{T}}\left(\mathit{y}-\alpha {\mathbf{1}}_n-R_{\theta }^{*}\mathit{c}\right)+n{\lambda }_0{\mathit{c}}^{\mathrm{T}}{R}_{\theta }^{*}\mathit{c}$

Step 2: $\underset{\theta }{argmin}{\left(\mathit{z}-G\mathit{\theta }\right)}^{\mathrm{T}}\left(\mathit{z}-G\mathit{\theta }\right)+n{\lambda }_2{\sum }_{r=1}^d{\theta }_r,\mathrm{subject}\mathrm{to}{\theta }_r\ge 0,r=1,\dots ,d.$

Select the tuning parameter ${\lambda }_1$ based on cross-validation

Repeat until ${\mathit{\beta }}_r$ coverage:

Step 1: $\underset{{\mathit{\beta }}_r}{argmin}{\sum }_i{\left(y_i-\alpha -{\sum }_{r=1}^d{f}_r\left({\mathit{x}}_{ir}^{\mathrm{T}}{\mathit{\beta }}_r\right)\right)}^2+{\lambda }_1{\sum }_r{∥{\mathit{\beta }}_r∥}_1.$