I am trying to solve below minimization problem (rank $k$ approximation of a matrix) which is somewhat sparse SVD
\begin{equation*}
\begin{aligned}
& \underset{B,\Lambda}{\text{minimize}}
& & ||A-B\Lambda B^T||_F^2 \\
& \text{subject to}
& & |b_k|_1 \leq \lambda, \; k = 1, \ldots, K.
\end{aligned}
\end{equation*}
where $A \in R^{n \times n}$ is a symmteric matrix, $B \in R^{n \times K}$, $\Lambda \in R^{K \times K}$ is a diagonal matrix and $b_k$ is $k^{th}$ column of $B$ matrix. I am using alternate minimization to solve the above problem and proximal operator for incorporating $L1$ constraint. Whenever I try to decrease the value of $\lambda$, the algorithm always return some $b_k$ 0, where $k$ is close to $K$. Below is an example of $B$ matrix 
I want columns of the matrix sparse, I am not sure how to do this.