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.
On
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.
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.