Proximal Operator of Spectral Norm (Schatten Norm) of a Matrix

Solution 1:

Basically, for any Schatten Norm the algorithm is pretty simple.

If we use Capital Letter $ A $ for Matrix and Small Letter for Vector then:

$$ {\operatorname*{Prox}}_{\lambda \left\| \cdot \right\|_{p}} \left( A \right) = \arg \min_{X} \frac{1}{2} \left\| X - A \right\|_{F}^{2} + \lambda \left\| X \right\|_{p} $$

Where $ \left\| X \right\|_{p} $ is the $ p $ Schatten Norm of $ X $.

Defining $ \boldsymbol{\sigma} \left( X \right) $ as a vector of the Singular Values of $ X $ (See the singular value decomposition).

Then the Proximal Operator Calculation is as following:

1. Apply the SVD on $ A $: $ A \rightarrow U \operatorname*{diag} \left( \boldsymbol{\sigma} \left( A \right) \right) {V}^{T} $.
2. Extract the vector of Singular Values $ \boldsymbol{\sigma} \left( A \right) $.
3. Calculate the Proximal Operator of the extracted vector using Vector Norm $ p $: $ \hat{\boldsymbol{\sigma}} \left( A \right) = {\operatorname*{Prox}}_{\lambda \left\| \cdot \right\|_{p}} \left( \boldsymbol{\sigma} \left( A \right) \right) = \arg \min_{x} \frac{1}{2} \left\| x - \boldsymbol{\sigma} \left( A \right) \right\|_{2}^{2} + \lambda \left\| x \right\|_{p} $.
4. Return the Proximal of the Matrix Norm: $ \hat{A} = {\operatorname*{Prox}}_{\lambda \left\| \cdot \right\|_{p}} \left( A \right) = U \operatorname*{diag} \left( \hat{\boldsymbol{\sigma}} \left( A \right) \right) {V}^{T} $.

The mapping of Matrix Norm into Schatten Norm:

• Frobenius Norm - Given by $ p = 2 $ in Schatten Norm.
• Nuclear Norm - Given by $ p = 1 $ in Schatten Norm.
• Spectral Norm (The $ {L}_{2} $ Induced Norm of a Matrix) - Given by $ p = \infty $ in Schatten Norm.