# Projection of a Symmetric Matrix onto the Positive Semi Definite (PSD) Cone Under the Nuclear Norm

Question: Given a symmetric matrix, $$S$$, what is the solution to the optimization problem:

$$\arg\min_{P \in \mathcal{S}_{\ge 0}} || S - P ||_N$$

where $$|| \cdot ||_N$$ denotes the nuclear norm, i.e. the sum of the singular values, or equivalently in the case of a symmetric matrix, the sum of the absolute value of the eigenvalues?

In other words, given a symmetric matrix $$S$$, what is the closest matrix $$P \succeq 0$$ in the positive semidefinite cone (denoted $$\mathcal{S}_{\ge 0}$$) with respect to the nuclear norm?

Guess: I would guess that the answer is the same as for the spectral norm and the Frobenius norm, provided that the nuclear norm is unitarily invariant (is it? I don't know, and I can't find a reference). Namely the answer in those cases is apparently:

$$V \Lambda^+ V^\top$$

where $$V \Lambda V^\top$$ is the eigenvalue decomposition of $$S$$, and $$V^+$$ is the diagonal matrix that is the same as $$\Lambda$$ except that any negative entries have been replaced with zero.

The following argument is obviously wrong, but imagine we would have that for any symmetric matrix $$R = U K U^\top$$, after applying a rotation to get $$\tilde{R} = V K V^\top$$, we would still have that (in fantasy land) $$||S - R||_N = ||S - \tilde{R}||_N$$, then we could assume without loss of generality that $$U = V = I$$, and then so calculating the nuclear norm of the difference would reduce to finding the nuclear norm $$|| \Lambda - K ||_N$$. But this is easy, since it's just the sum of the absolute values of the diagonals, $$\sum_i| \lambda_i - \kappa_i|$$, and then the only way to minimize this (is the following claim actually true?), while ensuring that all of the $$\kappa_i \ge 0$$, is to take $$\kappa_i = \max\{ \lambda_i , 0 \}$$, i.e. $$K = \Lambda^+$$.

Of course the claim about taking $$V=U=I$$ without loss of generality is obviously wrong, since then there would be no unique minimizer/no unique solution to the optimization problem, but the PSD cone is convex and the nuclear norm is a convex function.

However, it would seem to suffice to be able to show that $$|| V \Lambda V^\top - V K V^\top||_N < || V \Lambda V^\top - U K U^\top||_N$$ for any $$U$$ orthogonal and $$U \not=V$$. So I guess two claims of questionable merit:

Claim 1: The minimizer of $$\min_{i, \kappa_i \ge 0} \sum_{i} | \lambda_i - \kappa_i |$$ is $$\kappa_i = \max \{0, \lambda_i\}$$ for all $$i$$.

Claim 2: $$|| V \Lambda V^\top - V K V^\top||_N < || V \Lambda V^\top - U K U^\top||_N$$ for any $$U$$ orthogonal and $$U \not=V$$. (And any diagonal matrix $$K$$.)

The following questions are all related but don't answer my question, because they either involve projection onto a set different than the PSD cone, or projection with respect to a distance different from that induced by the nuclear norm (or w.r.t. an unspecified distance). So this question does not seem to be a duplicate of any of these.