TITLE: Why is minimizing the nuclear norm of a matrix a good surrogate for minimizing the rank?
QUESTION [34 upvotes]: A method called "Robust PCA" solves the matrix decomposition problem
$$L^*, S^* = \arg \min_{L, S} \|L\|_* + \|S\|_1 \quad \text{s.t. } L + S = X$$
as a surrogate for the actual problem
$$L^*, S^* = \arg \min_{L, S} rank(L) + \|S\|_0 \quad \text{s.t. } L + S = X,$$ i.e. the actual goal is to decompose the data matrix $X$ into a low-rank signal matrix $L$ and a sparse noise matrix $S$. In this context: why is the nuclear norm a good approximation for the rank of a matrix? I can think of matrices with low nuclear norm but high rank and vice-versa. Is there any intuition one can appeal to?

REPLY [10 votes]: To be accurate, it has been shown that the $\ell_1$ norm is the convex envelope of the $\| \cdot \|_0$ pseudo-norm while the nuclear norm is the convex envelope of the rank.
As a reminder, the convex envelope is the tightest convex surrogate of a function. An important property is that a function and its convex envelope have the same global minimizer.