AbstractFor a parameter dimension $$d\in {\mathbb {N}}$$
d
∈
N
, we consider the approximation of many-parametric maps $$u: [-\,1,1]^d\rightarrow {\mathbb R}$$
u
:
[
-
1
,
1
]
d
→
R
by deep ReLU neural networks. The input dimension d may possibly be large, and we assume quantitative control of the domain of holomorphy of u: i.e., u admits a holomorphic extension to a Bernstein polyellipse $${{\mathcal {E}}}_{\rho _1}\times \cdots \times {{\mathcal {E}}}_{\rho _d} \subset {\mathbb {C}}^d$$
E
ρ
1
×
⋯
×
E
ρ
d
⊂
C
d
of semiaxis sums $$\rho _i>1$$
ρ
i
>
1
containing $$[-\,1,1]^{d}$$
[
-
1
,
1
]
d
. We establish the exponential rate $$O(\exp (-\,bN^{1/(d+1)}))$$
O
(
exp
(
-
b
N
1
/
(
d
+
1
)
)
)
of expressive power in terms of the total NN size N and of the input dimension d of the ReLU NN in $$W^{1,\infty }([-\,1,1]^d)$$
W
1
,
∞
(
[
-
1
,
1
]
d
)
. The constant $$b>0$$
b
>
0
depends on $$(\rho _j)_{j=1}^d$$
(
ρ
j
)
j
=
1
d
which characterizes the coordinate-wise sizes of the Bernstein-ellipses for u. We also prove exponential convergence in stronger norms for the approximation by DNNs with more regular, so-called “rectified power unit” activations. Finally, we extend DNN expression rate bounds also to two classes of non-holomorphic functions, in particular to d-variate, Gevrey-regular functions, and, by composition, to certain multivariate probability distribution functions with Lipschitz marginals.
Wick rotations in deformation quantization
