I assume that you have defined \norm more or less by \newcommand{\norm}[1]{\left\Vert#1\right\Vert}.

The solution proposed by localghost works if you replace \begin{split} by \begin{aligned}[t], and \end{split} by \end{aligned}. Anyway, I would remark the following facts:

- The equation number usually is vertically centered, which is not the case in your code. This can be simply achieved by replacing align by a combination of equation and split.
- You break lines which contain brackets in such a way that the opening bracket falls in one line and the closing one, in the second line. If you use \left - \right to fix the size of brackets, it may happen that the opening and closing brackets have different sizes, depending on the mathematical surrounding expressions. I find better to manually fix sizes.
- Since breaks happens inside a bracketed expression, the alignment point should on the left of the opening bracket.

Taking these points into account, I would propose the following code:

\begin{equation}

\begin{split}

E_{ic}&=\!

\begin{aligned}[t]

\frac{1}{N}\sum_{z_k\in V} \Bigl[&\norm{m_f(z_k;\mu_f)

+ m_b(z_k+m_f(z_k;\mu_f);\mu_b)}^2 \Bigr. \\

\Bigl. & +\norm{m_b(z_k;\mu_b) + m_f(z_k+m_b(z_k;\mu_b); \mu_f)}^2 \Bigr]

\end{aligned}\\

&=\!

\begin{aligned}[t]

\frac{1}{N}\sum_{z_k\in V} \Biggl[

&\norm{\begin{bmatrix}

m_{fx}(z_k;\mu_{fx}) + m_{bx}(z_k+m_f(z_k;\mu_{f});\mu_{bx})\\

m_{fy}(z_k;\mu_{fy}) + m_{by}(z_k+m_f(z_k;\mu_{f});\mu_{by})

\end{bmatrix}}^2 \Biggr. \\

\Biggl. & +\norm{\begin{bmatrix}

m_{bx}(z_k;\mu_{bx}) + m_{fx}(z_k+m_b(z_k;\mu_{b}); \mu_{fx})\\

m_{by}(z_k;\mu_{by}) + m_{fy}(z_k+m_b(z_k;\mu_{b}); \mu_{fy})

\end{bmatrix}}^2 \Biggr].

\end{aligned}

\end{split}

\end{equation}

I have also added a point at the end. Mathematical expressions should be punctuated with commas and points, as done with ordinary text.