$$\label{eq:LRF11}
{p(\mathcal{U}|\mathcal{V},\mathbf{\theta} )=\prod\limits_{i=1}^{N}{(\frac{\gamma}{{{(2\pi {{\sigma }^{2}})}^{D/2}}}{{e}^{-\frac{{{\| {\mathbf{v}_{i}}-\mathbf{f}_{vfc}({\mathbf{u}_{i}}) \|}_{2}}}{2{{\sigma }^{2}}}}}+\frac{1-\gamma }{a})}},$$ where $\mathcal{\theta}=\left\{{\mathbf{f}_{vfc},\sigma^{2},\gamma}\right\}$ is a set of unknown parameters, $\mathbf{f}_{vfc}$ is the vector field expected to be recovered, $\gamma$ is the mixing coefficient of the mixture probability model, i.e, $p(z_{i}=1)=\gamma$, $\mathcal{U}$ and $\mathcal{V}$ respectively are sets of $\mathbf{u}$ and $\mathbf{v}$, $\sigma$ is the uniform standard deviation of Gaussian distribution, $\frac{1}{a}$ is the probability density of the uniform distribution and $D$ is the dimension of the output space. VFC employs the EM [@Dempster1977Maximum] algorithm to deal with the maximum likelihood estimation with latent variables. At E-step, the diagonal element of a diagonal matrix $\mathbf{P}$, i.e., $p_{i}=p(z_{i}=1|\mathbf{u}_{i},\mathbf{v}_{i},\theta)$, can be computed by the Bayes rule $$\label{eq:LRF12}
{{p_i} = \frac{{\gamma {e^{ - \frac{{{{\| {\mathbf{v}_{i} - \mathbf{f}_{vfc}(\mathbf{u}_{i})} \|}_2}}}{{2{\sigma ^2}}}}}}}{{\gamma {e^{ - \frac{{{{\| {\mathbf{v}_{i} - \mathbf{f}_{vfc}(\mathbf{u}_{i})} \|}_2}}}{{2{\sigma ^2}}}}} + (1 - \gamma )\frac{{{{(2\pi {\sigma ^2})}^{D/2}}}}{a}}}}.$$ At M-step, a coefficient matrix $\mathbf{C}$ is created first by $$\label{eq:LRF13}
{(\mathbf{K}_{Gauss}+\lambda_{vfc}{\sigma^{2}}\mathbf{P}^{-1})\mathbf{C}=\mathcal{V}},$$ where $\mathbf{K}_{Gauss}$ is a matrix consisting of the Gaussian kernel $k(\mathbf{u}_{i},\mathbf{u}_{j}) = {e^{ - \beta {{\| {\mathbf{u}_{i} - \mathbf{u}_{j}} \|}_2}}}$ and $\lambda_{vfc}$ is a regularization constant. Second, the vector field $\mathbf{f}_{vfc}$ is estimated by $$\label{eq:LRF14}
{\mathbf{f}_{vfc}(\mathbf{u}) = \sum\limits_{i = 1}^{N} {k}(\mathbf{u},\mathbf{u}_{i})\mathbf{c}_{i}},$$ where $\mathbf{c}_i\in{\mathbf{C}}$. Third, values of $\sigma^{2}$ and $\gamma$ are updated 