Obtain B to minimize $\|\boldsymbol{X}-\boldsymbol{B}\|_{\boldsymbol{F}}^{2}$ with constraint on the spectral norm of B to be ≤ 1

Question

I stumbled upon this problem where I have to

I want to obtain B such that we Minimize $\|\boldsymbol{X}-\boldsymbol{B}\|_{\boldsymbol{F}}^{2}$ subject to the constraint that the spectral norm of is less than or equal to one.

I tried to proceed using the SVD decomposition and trying to involve some inequalities of norms. but with no result. Any suggestions??

Answer 1

According to Wikipedia, the Frobenius norm is equivalent to $\displaystyle \Vert X-B\Vert_F^2=\sum_{i=1}^{min(m,n)}\sigma_i^2(X-B)$

Write X in its svd form $X=\sum_i\lambda_iu_iv_i^T$

Write B as $\sum_ib_iu_iv_i^T$

Then $X-B=\sum_i(\lambda_i-b_i)u_iv_i^T$

Take $b_i=min\{\lambda_i, 1\}$

Then this achieves the minimum frobenius norm.

Disclaimer: not sure if this works.