  Given a function dictionary $\cal D$ and an approximation budget
$N\in\mathbb{N}^+$, nonlinear approximation seeks the linear combination of the
best $N$ terms $\{T_n\}_{1\le n\le N}\subseteq{\cal D}$ to approximate a given
function $f$ with the minimum approximation
error\[\varepsilon_{L,f}:=\min_{\{g_n\}\subseteq{\mathbb{R}},\{T_n\}\subseteq{\cal
D}}\|f(x)-\sum_{n=1}^N g_n T_n(x)\|.\]Motivated by recent success of deep
learning, we propose dictionaries with functions in a form of compositions,
i.e.,\[T(x)=T^{(L)}\circ T^{(L-1)}\circ\cdots\circ T^{(1)}(x)\]for all
$T\in\cal D$, and implement $T$ using ReLU feed-forward neural networks (FNNs)
with $L$ hidden layers. We further quantify the improvement of the best
$N$-term approximation rate in terms of $N$ when $L$ is increased from $1$ to
$2$ or $3$ to show the power of compositions. In the case when $L>3$, our
analysis shows that increasing $L$ cannot improve the approximation rate in
terms of $N$.
  In particular, for any function $f$ on $[0,1]$, regardless of its smoothness
and even the continuity, if $f$ can be approximated using a dictionary when
$L=1$ with the best $N$-term approximation rate $\varepsilon_{L,f}={\cal
O}(N^{-\eta})$, we show that dictionaries with $L=2$ can improve the best
$N$-term approximation rate to $\varepsilon_{L,f}={\cal O}(N^{-2\eta})$. We
also show that for H\"older continuous functions of order $\alpha$ on
$[0,1]^d$, the application of a dictionary with $L=3$ in nonlinear
approximation can achieve an essentially tight best $N$-term approximation rate
$\varepsilon_{L,f}={\cal O}(N^{-2\alpha/d})$. Finally, we show that
dictionaries consisting of wide FNNs with a few hidden layers are more
attractive in terms of computational efficiency than dictionaries with narrow
and very deep FNNs for approximating H\"older continuous functions if the
number of computer cores is larger than $N$ in parallel computing.
