$ \min _{\operatorname{rank}\{\boldsymbol{B}\}=\boldsymbol{k}<r=rank(A)}\|\boldsymbol{A}-B\|_{2} ,A,B \in \mathbb{C}^{m \times n}$ and A is given

Question

I would like to solve the following minimization problem.$$ \min _{\operatorname{rank}\{\boldsymbol{B}\}=\boldsymbol{k}<r=rank(A)}\|\boldsymbol{A}-\boldsymbol{B}\|_{2} ,\boldsymbol{A,B} \in \mathbb{C}^{m \times n} $$ where $ A \in \mathbb{C}^{m \times n}, \operatorname{rank}\{A\}=r, 2 \leq r \leq \min \{m, n\} $ is a given matrix.

My idea at first was to go with SVD and try to rewrite the norm. But I didn't reach an interesting result.

Any suggestions??

Answer 1

The minimum does not exist, but the infimum is zero.

For example: for any $\epsilon > 0$, the matrices $$ A = \pmatrix{\epsilon \cdot I_k & 0_{k\times (n-k)}\\ 0_{(m-k)\times k}& 0_{(m-k) \times (m-k)}}, \quad B = \pmatrix{\epsilon \cdot I_r & 0_{r\times (n-r)}\\ 0_{(m-r)\times r}& 0_{(m-r) \times (m-r)}} $$ satisfy the constraints with $\|A - B\|_2 = \epsilon$ (assuming $\|\cdot\|_2$ refers to the spectral norm).