<s>
In	O
the	O
mathematical	O
theory	O
of	O
artificial	B-Architecture
neural	I-Architecture
networks	I-Architecture
,	O
universal	B-Algorithm
approximation	I-Algorithm
theorems	I-Algorithm
are	O
results	O
that	O
establish	O
the	O
density	O
of	O
an	O
algorithmically	O
generated	O
class	O
of	O
functions	O
within	O
a	O
given	O
function	O
space	O
of	O
interest	O
.	O
</s>
<s>
Typically	O
,	O
these	O
results	O
concern	O
the	O
approximation	O
capabilities	O
of	O
the	O
feedforward	B-Algorithm
architecture	I-Algorithm
on	O
the	O
space	O
of	O
continuous	O
functions	O
between	O
two	O
Euclidean	O
spaces	O
,	O
and	O
the	O
approximation	O
is	O
with	O
respect	O
to	O
the	O
compact	B-Algorithm
convergence	I-Algorithm
topology	O
.	O
</s>
<s>
However	O
,	O
there	O
are	O
also	O
a	O
variety	O
of	O
results	O
between	O
non-Euclidean	O
spaces	O
and	O
other	O
commonly	O
used	O
architectures	O
and	O
,	O
more	O
generally	O
,	O
algorithmically	O
generated	O
sets	O
of	O
functions	O
,	O
such	O
as	O
the	O
convolutional	B-Architecture
neural	I-Architecture
network	I-Architecture
(	O
CNN	B-Architecture
)	O
architecture	O
,	O
radial	O
basis-functions	O
,	O
or	O
neural	B-Architecture
networks	I-Architecture
with	O
specific	O
properties	O
.	O
</s>
<s>
Most	O
universal	B-Algorithm
approximation	I-Algorithm
theorems	I-Algorithm
can	O
be	O
parsed	O
into	O
two	O
classes	O
.	O
</s>
<s>
The	O
first	O
quantifies	O
the	O
approximation	O
capabilities	O
of	O
neural	B-Architecture
networks	I-Architecture
with	O
an	O
arbitrary	O
number	O
of	O
artificial	O
neurons	O
(	O
"	O
arbitrary	O
width	O
"	O
case	O
)	O
and	O
the	O
second	O
focuses	O
on	O
the	O
case	O
with	O
an	O
arbitrary	O
number	O
of	O
hidden	O
layers	O
,	O
each	O
containing	O
a	O
limited	O
number	O
of	O
artificial	O
neurons	O
(	O
"	O
arbitrary	O
depth	B-Algorithm
"	O
case	O
)	O
.	O
</s>
<s>
In	O
addition	O
to	O
these	O
two	O
classes	O
,	O
there	O
are	O
also	O
universal	B-Algorithm
approximation	I-Algorithm
theorems	I-Algorithm
for	O
neural	B-Architecture
networks	I-Architecture
with	O
bounded	O
number	O
of	O
hidden	O
layers	O
and	O
a	O
limited	O
number	O
of	O
neurons	O
in	O
each	O
layer	O
(	O
"	O
bounded	O
depth	B-Algorithm
and	O
bounded	O
width	O
"	O
case	O
)	O
.	O
</s>
<s>
Universal	B-Algorithm
approximation	I-Algorithm
theorems	I-Algorithm
imply	O
that	O
neural	B-Architecture
networks	I-Architecture
can	O
represent	O
a	O
wide	O
variety	O
of	O
interesting	O
functions	O
when	O
given	O
appropriate	O
weights	O
.	O
</s>
<s>
One	O
of	O
the	O
first	O
versions	O
of	O
the	O
arbitrary	O
width	O
case	O
was	O
proven	O
by	O
George	O
Cybenko	O
in	O
1989	O
for	O
sigmoid	B-Algorithm
activation	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
Kurt	O
Hornik	O
,	O
Maxwell	O
Stinchcombe	O
,	O
and	O
Halbert	O
White	O
showed	O
in	O
1989	O
that	O
multilayer	O
feed-forward	B-Algorithm
networks	I-Algorithm
with	O
as	O
few	O
as	O
one	O
hidden	O
layer	O
are	O
universal	B-Algorithm
approximators	I-Algorithm
.	O
</s>
<s>
Hornik	O
also	O
showed	O
in	O
1991	O
that	O
it	O
is	O
not	O
the	O
specific	O
choice	O
of	O
the	O
activation	B-Algorithm
function	I-Algorithm
but	O
rather	O
the	O
multilayer	O
feed-forward	O
architecture	O
itself	O
that	O
gives	O
neural	B-Architecture
networks	I-Architecture
the	O
potential	O
of	O
being	O
universal	B-Algorithm
approximators	I-Algorithm
.	O
</s>
<s>
Moshe	O
Leshno	O
et	O
al	O
in	O
1993	O
and	O
later	O
Allan	O
Pinkus	O
in	O
1999	O
showed	O
that	O
the	O
universal	O
approximation	O
property	O
is	O
equivalent	O
to	O
having	O
a	O
nonpolynomial	O
activation	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
arbitrary	O
depth	B-Algorithm
case	O
was	O
also	O
studied	O
by	O
a	O
number	O
of	O
authors	O
,	O
such	O
as	O
Gustaf	O
Gripenberg	O
in	O
2003	O
,	O
Dmitry	O
Yarotsky	O
,	O
Zhou	O
Lu	O
et	O
al	O
in	O
2017	O
,	O
Boris	O
Hanin	O
and	O
Mark	O
Sellke	O
in	O
2018	O
,	O
and	O
Patrick	O
Kidger	O
and	O
Terry	O
Lyons	O
in	O
2020	O
.	O
</s>
<s>
In	O
2021	O
,	O
Park	O
et	O
al	O
obtained	O
the	O
minimum	O
width	O
required	O
for	O
the	O
universal	O
approximation	O
of	O
Lp	O
functions	O
using	O
feedforward	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
with	O
ReLU	B-Algorithm
as	O
activation	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
Similar	O
results	O
that	O
can	O
be	O
directly	O
applied	O
to	O
residual	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
were	O
also	O
obtained	O
in	O
the	O
same	O
year	O
by	O
Paulo	O
Tabuada	O
and	O
Bahman	O
Gharesifard	O
using	O
control-theoretic	O
arguments	O
.	O
</s>
<s>
The	O
bounded	O
depth	B-Algorithm
and	O
bounded	O
width	O
case	O
was	O
first	O
studied	O
by	O
Maiorov	O
and	O
Pinkus	O
in	O
1999	O
.	O
</s>
<s>
They	O
showed	O
that	O
there	O
exists	O
an	O
analytic	O
sigmoidal	O
activation	B-Algorithm
function	I-Algorithm
such	O
that	O
two	O
hidden	O
layer	O
neural	B-Architecture
networks	I-Architecture
with	O
bounded	O
number	O
of	O
units	O
in	O
hidden	O
layers	O
are	O
universal	B-Algorithm
approximators	I-Algorithm
.	O
</s>
<s>
Using	O
algorithmic	O
and	O
computer	O
programming	O
techniques	O
,	O
Guliyev	O
and	O
Ismailov	O
constructed	O
a	O
smooth	O
sigmoidal	O
activation	B-Algorithm
function	I-Algorithm
providing	O
universal	O
approximation	O
property	O
for	O
two	O
hidden	O
layer	O
feedforward	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
with	O
less	O
units	O
in	O
hidden	O
layers	O
.	O
</s>
<s>
It	O
was	O
constructively	O
proved	O
in	O
2018	O
paper	O
that	O
single	O
hidden	O
layer	O
networks	O
with	O
bounded	O
width	O
are	O
still	O
universal	B-Algorithm
approximators	I-Algorithm
for	O
univariate	O
functions	O
,	O
but	O
this	O
property	O
is	O
no	O
longer	O
true	O
for	O
multivariable	O
functions	O
.	O
</s>
<s>
Several	O
extensions	O
of	O
the	O
theorem	O
exist	O
,	O
such	O
as	O
to	O
discontinuous	O
activation	B-Algorithm
functions	I-Algorithm
,	O
noncompact	O
domains	O
,	O
certifiable	O
networks	O
,	O
</s>
<s>
random	O
neural	B-Architecture
networks	I-Architecture
,	O
and	O
alternative	O
network	O
architectures	O
and	O
topologies	O
.	O
</s>
<s>
A	O
spate	O
of	O
papers	O
in	O
the	O
1980s	O
--	O
1990s	O
,	O
from	O
George	O
Cybenko	O
and	O
Kurt	O
Hornik	O
etc	O
,	O
established	O
several	O
universal	B-Algorithm
approximation	I-Algorithm
theorems	I-Algorithm
for	O
arbitrary	O
width	O
and	O
bounded	O
depth	B-Algorithm
.	O
</s>
<s>
Such	O
an	O
can	O
also	O
be	O
approximated	O
by	O
a	O
network	O
of	O
greater	O
depth	B-Algorithm
by	O
using	O
the	O
same	O
construction	O
for	O
the	O
first	O
layer	O
and	O
approximating	O
the	O
identity	O
function	O
with	O
later	O
layers	O
.	O
</s>
<s>
The	O
'	O
dual	O
 '	O
versions	O
of	O
the	O
theorem	O
consider	O
networks	O
of	O
bounded	O
width	O
and	O
arbitrary	O
depth	B-Algorithm
.	O
</s>
<s>
A	O
variant	O
of	O
the	O
universal	B-Algorithm
approximation	I-Algorithm
theorem	I-Algorithm
was	O
proved	O
for	O
the	O
arbitrary	O
depth	B-Algorithm
case	O
by	O
Zhou	O
Lu	O
et	O
al	O
.	O
</s>
<s>
They	O
showed	O
that	O
networks	O
of	O
width	O
n+4	O
with	O
ReLU	B-Algorithm
activation	B-Algorithm
functions	I-Algorithm
can	O
approximate	O
any	O
Lebesgue	O
integrable	O
function	O
on	O
n-dimensional	O
input	O
space	O
with	O
respect	O
to	O
distance	O
if	O
network	O
depth	B-Algorithm
is	O
allowed	O
to	O
grow	O
.	O
</s>
<s>
In	O
the	O
same	O
paper	O
it	O
was	O
shown	O
that	O
ReLU	B-Algorithm
networks	O
with	O
width	O
n+1	O
were	O
sufficient	O
to	O
approximate	O
any	O
continuous	O
function	O
of	O
n-dimensional	O
input	O
variables	O
.	O
</s>
<s>
Universal	B-Algorithm
approximation	I-Algorithm
theorem	I-Algorithm
(	O
L1	O
distance	O
,	O
ReLU	B-Algorithm
activation	O
,	O
arbitrary	O
depth	B-Algorithm
,	O
minimal	O
width	O
)	O
.	O
</s>
<s>
Moreover	O
,	O
there	O
exists	O
a	O
function	O
and	O
some	O
,	O
for	O
which	O
there	O
is	O
no	O
fully-connected	O
ReLU	B-Algorithm
network	O
of	O
width	O
less	O
than	O
satisfying	O
the	O
above	O
approximation	O
bound	O
.	O
</s>
<s>
Remark	O
:	O
If	O
the	O
activation	O
is	O
replaced	O
by	O
leaky-ReLU	O
,	O
and	O
the	O
input	O
is	O
restricted	O
in	O
a	O
compact	O
domain	O
,	O
then	O
the	O
exact	O
minimum	O
width	O
is	O
.	O
</s>
<s>
Quantitative	O
Refinement	O
:	O
In	O
the	O
case	O
where	O
,	O
when	O
and	O
and	O
where	O
is	O
the	O
ReLU	B-Algorithm
activation	I-Algorithm
function	I-Algorithm
then	O
,	O
the	O
exact	O
depth	B-Algorithm
and	O
width	O
for	O
a	O
ReLU	B-Algorithm
network	O
to	O
achive	O
error	O
is	O
also	O
known	O
.	O
</s>
<s>
Together	O
,	O
the	O
central	O
result	O
of	O
yields	O
the	O
following	O
universal	B-Algorithm
approximation	I-Algorithm
theorem	I-Algorithm
for	O
networks	O
with	O
bounded	O
width	O
(	O
cf	O
.	O
</s>
<s>
Universal	B-Algorithm
approximation	I-Algorithm
theorem	I-Algorithm
(	O
Uniform	O
non-affine	O
activation	O
,	O
arbitrary	O
depth	B-Algorithm
,	O
constrained	O
width	O
)	O
.	O
</s>
<s>
Let	O
be	O
any	O
non-affine	O
continuous	O
function	O
which	O
is	O
continuously	O
differentiable	O
at	O
at	O
least	O
one	O
point	O
,	O
with	O
nonzero	O
derivative	B-Algorithm
at	O
that	O
point	O
.	O
</s>
<s>
Let	O
denote	O
the	O
space	O
of	O
feed-forward	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
with	O
input	O
neurons	O
,	O
output	O
neurons	O
,	O
and	O
an	O
arbitrary	O
number	O
of	O
hidden	O
layers	O
each	O
with	O
neurons	O
,	O
such	O
that	O
every	O
hidden	O
neuron	O
has	O
activation	B-Algorithm
function	I-Algorithm
and	O
every	O
output	O
neuron	O
has	O
the	O
identity	O
as	O
its	O
activation	B-Algorithm
function	I-Algorithm
,	O
with	O
input	O
layer	O
,	O
and	O
output	O
layer	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
is	O
dense	O
in	O
with	O
respect	O
to	O
the	O
topology	O
of	O
uniform	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
Quantitative	O
Refinement	O
:	O
The	O
number	O
of	O
layers	O
and	O
the	O
width	O
of	O
each	O
layer	O
required	O
to	O
approximate	O
f	O
to	O
precision	O
known	O
;	O
moreover	O
,	O
the	O
result	O
hold	O
true	O
when	O
and	O
are	O
replaced	O
with	O
any	O
non-positively	O
curved	O
Riemannian	B-Architecture
manifold	I-Architecture
.	O
</s>
<s>
Certain	O
necessary	O
conditions	O
for	O
the	O
bounded	O
width	O
,	O
arbitrary	O
depth	B-Algorithm
case	O
have	O
been	O
established	O
,	O
but	O
there	O
is	O
still	O
a	O
gap	O
between	O
the	O
known	O
sufficient	O
and	O
necessary	O
conditions	O
.	O
</s>
<s>
The	O
first	O
result	O
on	O
approximation	O
capabilities	O
of	O
neural	B-Architecture
networks	I-Architecture
with	O
bounded	O
number	O
of	O
layers	O
,	O
each	O
containing	O
a	O
limited	O
number	O
of	O
artificial	O
neurons	O
was	O
obtained	O
by	O
Maiorov	O
and	O
Pinkus	O
.	O
</s>
<s>
Their	O
remarkable	O
result	O
revealed	O
that	O
such	O
networks	O
can	O
be	O
universal	B-Algorithm
approximators	I-Algorithm
and	O
for	O
achieving	O
this	O
property	O
two	O
hidden	O
layers	O
are	O
enough	O
.	O
</s>
<s>
It	O
says	O
that	O
activation	B-Algorithm
functions	I-Algorithm
providing	O
universal	O
approximation	O
property	O
for	O
bounded	O
depth	B-Algorithm
bounded	O
width	O
networks	O
exist	O
.	O
</s>
<s>
Using	O
certain	O
algorithmic	O
and	O
computer	O
programming	O
techniques	O
,	O
Guliyev	O
and	O
Ismailov	O
efficiently	O
constructed	O
such	O
activation	B-Algorithm
functions	I-Algorithm
depending	O
on	O
a	O
numerical	O
parameter	O
.	O
</s>
<s>
The	O
developed	O
algorithm	O
allows	O
one	O
to	O
compute	O
the	O
activation	B-Algorithm
functions	I-Algorithm
at	O
any	O
point	O
of	O
the	O
real	O
axis	O
instantly	O
.	O
</s>
<s>
Universal	B-Algorithm
approximation	I-Algorithm
theorem	I-Algorithm
:	O
Let	O
be	O
a	O
finite	O
segment	O
of	O
the	O
real	O
line	O
,	O
and	O
be	O
any	O
positive	O
number	O
.	O
</s>
<s>
Then	O
one	O
can	O
algorithmically	O
construct	O
a	O
computable	O
sigmoidal	O
activation	B-Algorithm
function	I-Algorithm
,	O
which	O
is	O
infinitely	O
differentiable	O
,	O
strictly	O
increasing	O
on	O
,	O
-strictly	O
increasing	O
on	O
,	O
and	O
satisfies	O
the	O
following	O
properties	O
:	O
</s>
<s>
In	O
the	O
"	O
depth-width	O
"	O
terminology	O
,	O
the	O
above	O
theorem	O
says	O
that	O
for	O
certain	O
activation	B-Algorithm
functions	I-Algorithm
depth	B-Algorithm
-	O
width	O
-	O
networks	O
are	O
universal	B-Algorithm
approximators	I-Algorithm
for	O
univariate	O
functions	O
and	O
depth	B-Algorithm
-	O
width	O
-	O
networks	O
are	O
universal	B-Algorithm
approximators	I-Algorithm
for	O
-variable	O
functions	O
(	O
)	O
.	O
</s>
<s>
The	O
popular	O
graph	O
convolutional	B-Architecture
neural	I-Architecture
networks	I-Architecture
(	O
GCNs	O
or	O
GNNs	O
)	O
can	O
be	O
made	O
as	O
discriminative	O
as	O
the	O
Weisfeiler	O
–	O
Leman	O
graph	O
isomorphism	O
test	O
.	O
</s>
<s>
In	O
2020	O
,	O
a	O
universal	B-Algorithm
approximation	I-Algorithm
theorem	I-Algorithm
result	O
was	O
established	O
by	O
Brüel-Gabrielsson	O
,	O
showing	O
that	O
graph	O
representation	O
with	O
certain	O
injective	O
properties	O
is	O
sufficient	O
for	O
universal	O
function	O
approximation	O
on	O
bounded	O
graphs	O
and	O
restricted	O
universal	O
function	O
approximation	O
on	O
unbounded	O
graphs	O
,	O
with	O
an	O
accompanying	O
#edges	O
#nodes	O
-runtime	O
method	O
that	O
performed	O
at	O
state	O
of	O
the	O
art	O
on	O
a	O
collection	O
of	O
benchmarks	O
.	O
</s>
