Why the trace norm of a tensor is a good approximation to its rank?

Question

In many literature, they use the trace norm of a tensor to approximate its rank, which is defined as follows: $$\|\mathcal{X}\|_{\ast}:=\sum_{i=1}^{n}\alpha_i\|X_{(i)}\|_{\ast}$$ where ${\alpha_i}'s$ are constants satisfying $\alpha_i\geq0$ and $\sum_{i=1}^n\alpha_i=1$, $X_{(i)}$ is the mode-$i$ unfolding of $\mathcal{X}$, and $\|X_{(i)}\|_{\ast}$ is the trace norm of the 2D matrix $X_{(i)}$ which is defined as follows: $$\|X_{(i)}\|_{\ast}=\sum_i\sigma_i(X_{(i)})$$ where $\sigma_i(X_{(i)})$ denotes the $i$th largest sigular value of $X_{(i)}$.