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I want to solve the following optimization problem:

Let $S \in \mathbb{R}^{p \times p}$ be a symmetric matrix, and fix $n \in \mathbb{R}$ such that $n$ is significantly less than $p$ and the rank of $S$ is at most $n$. Now, I want to find a positive-semidefinite real matrix $R \in \mathbb{R}^{p \times p}$ that minimizes

$\lVert S-R \rVert_{F} + \lambda \lVert R \rVert_{*},$

where $\lVert \cdot \rVert_{F}$ and $\lVert \cdot \rVert_{*}$ are Frobenius norm and nuclear norm, respectively, and $\lambda > 0$ is a regularization parameter to be tuned via cross-validation. What is the best way to solve this optimization problem?

Eco-nometrician
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1 Answers1

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You may use the solution in The Proximal Operator of the Nuclear Norm / Schatten Norm.

It will give you a closed form solution for any $\lambda$.

Royi
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