Algorithm 1. MOG algorithm for LRMF.

input: $X=(x_1,x_2,\dots x_n)\in R^{d\times n}$

Randomly initialize $\mathrm{\Pi },\mathrm{\Sigma },V$

repeat

$\left(EStep\right):Evaluate{\gamma }_{ijk}fori=1,\dots .,n,j=1,\dots ,d,k=1,\dots ,Ma{x}_{k}by$ Eq. (6)

$\left(MStepfor\mathrm{\Pi },\mathrm{\Sigma }\right):Evaluate{\pi }_k,{\sigma }_k^{2}fork=1,\dots ,Ma{x}_{k}by$ Eq. (10)

$\left(MStepfor\mathrm{U},\mathrm{V}\right):EvaluateU,Vbysolving\underset{U,V}{min}\parallel W\odot \left(X-U{V}^T\right){\parallel }_F^2$

through WALS where $W=\sqrt{{\sum }_{k=1}^{Ma{x}_k}{\displaystyle \frac{{\gamma }_{ijk}}{2\pi {\sigma }_k^2}}}fori=1,\dots .,d,j=1,\dots ,n$

$\left(AutomaticMa{x}_{k}tuning\right):IfKL\left(N\left({\mu }_i,{\sigma }_i^2\right),N\left({\mu }_j,{\sigma }_j^2\right)\right)<\epsilon forsome$

$i,j,thencombine{i}^{th},j^{th}gaussiancomponentintoaunique$

$gaussianbyletting{\pi }_i={\pi }_i+{\pi }_j,{\sigma }_i^2=\left({\pi }_i{\sigma }_i^2+{\pi }_j{\sigma }_j^2\right)/\left({\pi }_i+{\pi }_j\right)$

${\mu }_i={\displaystyle \frac{{\pi }_i{\mu }_i+{\pi }_j{\mu }_j}{{\pi }_i+{\pi }_j}}$

$removingthe{j}^{th}gaussianfrom\mathrm{\Pi },\mathrm{\Sigma }.letMa{x}_k=Ma{x}_k-1$

until meet the stop criteria

return U, V