How to solve total variation problem when the feature of points is a vector?

I’m trying to implement a total variation problem described in the paper “l1-sparse Reconstruction of Sharp Point Set Surfaces”. It’s different from the case of total variation in image, the feature of a point is not a scalar value but a vector.
Let N be the features matrix of the whole point set, n_i be the feature of point p_i, N^in be the measured feature matrix, E denote the adjacency set, the problem can be formulated as:

\min_N \sum_{pi,pj\in E}||n_i-n_j||_2 s.t. ||n_i-n_i^in||_2 \leq \alpha

The paper said this problem can be recast to a SOCP problem and solved by CVX efficiently. How to do the translation? Thanks in advance!