Exercise 3.4.6

Follow example 3, we need minimize $\mathit{NMF}=\parallel A-X{\text{∥}}_F^2+I_{\text{+}}(C)+I_{\text{+}}(R)$ with $X=\mathit{CR}$, so we can compute

• $(X_1,C_1) = argmin$\left[\parallel A-X{\text{∥}}_F^2+\frac{1}{2}\rho \parallel X-C{R}_0+U_0{\text{∥}}_F^2\right]=\mathit{\$with}$ X\; \ge \; 0,C\; \ge \; 0$$

Let $C=\left[\begin{array}{cc}\hfill c_{11}\hfill & \hfill c_{12}\hfill \\ \hfill c_{21}\hfill & \hfill c_{22}\hfill \end{array}\right]$,$X=\left[\begin{array}{cc}\hfill x_{11}\hfill & \hfill x_{12}\hfill \\ \hfill x_{21}\hfill & \hfill x_{22}\hfill \end{array}\right]$, we have

Now take derivatives w.r.t. to $X,C$, we have

Note, we also need satisfy $X\ge 0$ and $C\ge 0$, Solve the equations we have $X_1=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$ and $C_1=\left[\begin{array}{cc}\hfill 1.5\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 3\hfill \end{array}\right]$

• Now solve for $R_1=\text{argmin}\parallel X_1-C_{1}R+U_0{\text{∥}}_F^2$ with $R\ge 0$, that is

Take derivatives w.r.t. $R$, we have

Solve for the equations we have $R=\left[\begin{array}{cc}\hfill \frac{4}{3}\hfill & \hfill \frac{2}{9}\hfill \\ \hfill 0\hfill & \hfill \frac{2}{3}\hfill \end{array}\right]\ge 0$

• Lastly, we have $U_1=U_0+X_1-C_1{R}_1=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 5\hfill \end{array}\right]$

Note $X_1=C_1{R}_1=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$ which is what we have solved in problem 5.