Optimization of Frobenius norm and nuclear norm

Question

How to solve the following optimization problem in $ X \in \mathbb{C}^{N \times M} $?

\begin{equation} \hat{X} = \arg \min_{X} \frac{1}{2} {\left\| X - Y \right\|}_{F}^{2} + \lambda {\left\| X \right\|}_{\ast} \end{equation}

Where $ {\left\| \cdot \right\|}_{F} $ denotes the Frobenius norm and $ {\left\| \cdot \right\|}_{\ast} $ denotes the nuclear norm. $ Y \in \mathbb{C}^{N \times M} $ and $ \lambda $ are known.

are you sure that you need to write it $||x-y||^2_F$ instead of $||x-y||^2_2$. I have seen that people use L2-norm instead of Frobenius norm. — user2806363, Jan 05 '17 at 11:15

score 1 · Answer 1 · edited Mar 19 '20 at 14:42

1

Are you familiar with proximal algorithms? You are asking how to evaluate the prox operator of the nuclear norm. The answer is given in slide 3-41 in DTU 2010 - Algorithms for Large Scale Convex Optimization - Proximal Gradient Method.

edited Mar 19 '20 at 14:42

Royi

10,050

answered Feb 01 '15 at 02:18

littleO

54,048

1

I think the nuance is that question deals with Complex Matrices. Does it hold? As the whole course there is for Real Matrices. – Royi Mar 12 '18 at 06:41

score 0 · Answer 2 · answered Apr 26 '16 at 01:13

0

Theorem 2.1 in the paper A Singular Value Thresholding Algorithm for Matrix Completion

answered Apr 26 '16 at 01:13

Huayu Zhang

583

2

But, it's still true for complex matrices? – Yesid Fonseca V. Mar 11 '18 at 01:47

Optimization of Frobenius norm and nuclear norm

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