Let $Y \in \mathbb R^{n \times c}$ and $r$ be an integer with $1 \le r \le \min(n,c)$. Consider the problem

$$\text{minimize} \|Y-X\|_{\text{Fro}}^2\text{ over }X \in \mathbb R^{n \times c}\text{ subject to rank}(X) \le r.$$

By the Eckart-Young-Mirsky theorem shows that the choice $X = \hat{Y}(r) := Y\Pi(r)$ solves this problem, where $\Pi(r) := V(r)V(r)^T$ the projection onto the subspace spanned by the first $r$ principal vectors $V(r) : = [V_1,\ldots,V_r] \in \mathbb R^{c \times r}$ of $Y$.

**Question:** What is $\hat{df}(r) := \text{tr}\left(\frac{\partial vec(\hat{Y}(r))}{\partial vec(Y)}\right) = \sum_{i,j,k,l} \frac{\partial \hat{Y}_{ij}(r)}{\partial Y_{kl}}$ ?

**Motivation:** Such computations are necessary in applying SURE (Stein's Unbiased Risk Estimator) theory for tuning penalized least squares models, as a powerful alternative to cross-validation. Here "df" stands for "degrees of freedom" and is a measure of the complexity of the approximator model.

**Observation:** It's not hard to show that
$$\hat{df}(r) \ge \hat{df}_{\text{naive}}(r),$$

 where $\hat{df}_{\text{naive}}(r):= nc - (n-r)(c-r)$ is number of free parameters needed to specify an $n$-by-$c$ matrix of rank $r$. Theorem 5.2 of [this paper][1] computes the correction $\hat{df}(r) - \hat{df}_{\text{naive}}(r)$ exactly to be

$$\hat{df}(r) - \hat{df}_{\text{naive}}(r) = 2\sum_{k=1}^{\text{rank}(Y)}\sum_{l=r+1}^{\min(n,c)}\frac{\lambda_l}{\lambda_k-\lambda_l},$$
where $\lambda_1 \ge \lambda_2 > \ldots > \lambda_r > \lambda_{r+1} \ge \ldots \lambda_{\text{rank}(Y)}$ are the eigenvalues of $Y^TY$.
However, the method used by the referred paper is direct "brute-force" and involves some very long formulae. I was wondering whether there could be a cleaner (i.e synthetic) way to obtain this.



  [1]: https://arxiv.org/pdf/1210.2464.pdf