  This paper considers approximately sparse signal and low-rank matrix's
recovery via truncated norm minimization $\min_{x}\|x_T\|_q$ and
$\min_{X}\|X_T\|_{S_q}$ from noisy measurements. We first introduce truncated
sparse approximation property, a more general robust null space property, and
establish the stable recovery of signals and matrices under the truncated
sparse approximation property. We also explore the relationship between the
restricted isometry property and truncated sparse approximation property. And
we also prove that if a measurement matrix $A$ or linear map $\mathcal{A}$
satisfies truncated sparse approximation property of order $k$, then the first
inequality in restricted isometry property of order $k$ and of order $2k$ can
hold for certain different constants $\delta_{k}$ and $\delta_{2k}$,
respectively. Last, we show that if $\delta_{t(k+|T^c|)}<\sqrt{(t-1)/t}$ for
some $t\geq 4/3$, then measurement matrix $A$ and linear map $\mathcal{A}$
satisfy truncated sparse approximation property of order $k$. Which should
point out is that when $T^c=\emptyset$, our conclusion implies that sparse
approximation property of order $k$ is weaker than restricted isometry property
of order $tk$.
