What is meant by "regularized rank-1 approximation"? Suppose we have the following formula:
$$\forall i: \{d_i, x_{[i]}^T\}={\mathrm{argmin}_{d,z}} 1/2 ||E_i-dz^T||_F^2+ \lambda|z||_1  $$
I really don't get how the following sentence is concluded and why:

Note that the above problem is indeed a regularized rank-1 approximation of $E_i$.

Could you explain the meaning of the above sentence? 
For more context, please see this paper, page 887, formula #7.
 A: To find the best rank-one approximation of a given matrix $A$. If the SVD of $A=U\Sigma V^T$ is given, then $A_1=\sigma_1u_1v_1^T $, where $u_1$ and $v_1$ correspond to the left and right singular vectors corresponding to the largest singular value $\sigma_1$, is the best rank-one approximation. 
A: It is not just about finding the best rank-1 approximation to the matrix in the Frobenius norm sense. The term $\lambda\left|z\right|_1$ regularizes one of the factors of the outer product - i.e. that factor is not allowed to have a too large 1-norm. 
Regularization is done to impose constraints that the solution needs to be "nice" in one way or another. It can be shown that low-rank approximations of tensors can get "arbitrarily close" to a high rank tensor or matrix in a norm sense, but then usually the norm of the individual factors skyrocket. The regularization is to avoid that.
A: In the context of principal component analysis I suggest the following reference for low-rank matrix approximation: 
H. Shen, Jianhua Z. Huang (2008). Sparse Principal Component Analysis via    Regularized Low Rank Matrix Approximation, Journal of Multivariate Analysis, 99, 1015-1034" which can be downloaded from https://www.unc.edu/~haipeng/publication/sparsePCA.pdf
Please also check the following two references for matrix decomposition with low reconstruction error:
Eckart, C., Young, G., 1936. The approximation of one matrix by another of lower rank. Psychometrika 1, 211–218.
Puntanen, S., Styan, G., Isotalo, J., 2011. Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty. Springer.
