Supplement: Matrix Completion with Quantified Uncertainty through Low Rank Gaussian Copula

For the first equality, we use Eq. In practice, the result is more useful for small d, such as d = 0. Let us first state a generalization of our Theorem 2. Theorem 4. Suppose x LRGC(W, σ The proof applies to each missing dimension j M. Let us further define s For a detailed treatment of sub-Gaussian random distributions, see [10]. K p for all p 1 with some K > 0. The sub-Gaussian norm of x is defined as ||x|| Our Lemma 2 is Lemma 17 in [1], which is also a simplified version of Theorem 1 in [4]. To compute (2) and (3), we use the law of total expectation similar as in Section 1.1 by first treating z R. The computation for all cases are similar. We take the first case as an example.