Algorithm 1: The Multi-Matrices Factorization Algorithm.
	Input: matrix R; number of the latent features, d; learning rates, η1, η2, η3 and η4; regularization parameters, α and λ; threshold ϵ.
	Output: the estimated matrix, R̂.
	// Initialize U and V.
1	Draw random vectors, U(1),1, U(1),2, …, U(1),n, V1 ∼ N(0, I);
2	for j = 2; j ≤ t; j + + do
3	Let Vj = Vj−1 + Z; here Z ∼ Laplace(0, 1);
4	end
5	for j = 2; j ≤ t; j + + do
6	for i = 1; i ≤ n; i + + do
7	Let U(j),i = U(j-1),i + Z; here Z ∼ Laplace(0, 1);
8	end
9	end
	// Coordinate descent.
10	$W_1={\sum }_{i=1}^n{\sum }_{j=1}^t{(R_{i,j}-U_{(j),i}^T{V}_j)}^2\mathbb{\text{I}}(R_{i,j}{\ne }^{\prime }{\perp }^{\prime })+\alpha {\sum }_{j=1}^t{\Vert U_{(j),1}\Vert }_2^2+\beta {\Vert V_1\Vert }_2^2+\gamma {\sum }_{i=1}^n{\sum }_{j=2}^t{\Vert U_{(j),i}-U_{(j-1),i}\Vert }_1+\lambda {\sum }_{j=2}^t{\Vert V_j-V_{j-1}\Vert }_1$;
11
12	W2 = inf;
13	while |W2 – W1| > ϵ do
14	W2 = W1;
15	for i= 1, 2 …, n do
16	Let $U_{(1),i}^{\mathit{\text{new}}}=U_{(1),i}-{\eta }_1\frac{\partial W_1}{\partial U_{(1),i}}$;
17	end
18	for j > 1 and i = 1, 2 …, n do
19	Let $U_{(j),i}^{\mathit{\text{new}}}=U_{(j),i}-{\eta }_2\frac{\partial W_1}{\partial U_{(j),i}}$
20	end
21	$V_1^{\mathit{\text{new}}}=V_1-{\eta }_3\frac{\partial W_1}{\partial V_1}$;
22	for j = 2 …, t do
23	Let $V_j^{\mathit{\text{new}}}=V_j-{\eta }_4\frac{\partial W_1}{\partial V_j}$;
24	end
25	Replace all U(j),is with $U_{(j),i}^{\mathit{\text{new}}}s$ and Vjs with $V_j^{\mathit{\text{new}}}s$, recompute W1;
26	end
27	return R̂, where ${\widehat{R}}_j=U_{(j)}^T{V}_j$;