<s>
Algebraic	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
(	O
ASP	O
)	O
is	O
an	O
emerging	O
area	O
of	O
theoretical	O
signal	O
processing	O
(	O
SP	O
)	O
.	O
</s>
<s>
In	O
the	O
algebraic	O
theory	O
of	O
signal	O
processing	O
,	O
a	O
set	O
of	O
filters	O
is	O
treated	O
as	O
an	O
(	O
abstract	O
)	O
algebra	O
,	O
a	O
set	O
of	O
signals	O
is	O
treated	O
as	O
a	O
module	O
or	O
vector	O
space	O
,	O
and	O
convolution	B-Language
is	O
treated	O
as	O
an	O
algebra	O
representation	O
.	O
</s>
<s>
The	O
advantage	O
of	O
algebraic	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
is	O
its	O
generality	O
and	O
portability	O
.	O
</s>
<s>
In	O
the	O
original	O
formulation	O
of	O
algebraic	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
by	O
Puschel	O
and	O
Moura	O
,	O
the	O
signals	O
are	O
collected	O
in	O
an	O
-module	O
for	O
some	O
algebra	O
of	O
filters	O
,	O
and	O
filtering	O
is	O
given	O
by	O
the	O
action	O
of	O
on	O
the	O
-module	O
.	O
</s>
<s>
To	O
be	O
an	O
algebra	O
homomorphism	O
,	O
must	O
not	O
only	O
be	O
a	O
linear	O
transformation	O
,	O
but	O
also	O
satisfy	O
the	O
propertyGiven	O
a	O
signal	O
,	O
convolution	B-Language
of	O
the	O
signal	O
by	O
a	O
filter	O
yields	O
a	O
new	O
signal	O
.	O
</s>
<s>
The	O
image	O
of	O
a	O
generator	O
is	O
called	O
a	O
shift	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
In	O
all	O
practically	O
all	O
examples	O
,	O
convolutions	B-Language
are	O
formed	O
as	O
polynomials	O
in	O
generated	O
by	O
shift	B-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
In	O
discrete	B-General_Concept
signal	O
processing	O
(	O
DSP	O
)	O
,	O
the	O
signal	O
space	O
is	O
the	O
set	O
of	O
complex-valued	O
functions	O
with	O
bounded	O
energy	O
(	O
i.e.	O
</s>
<s>
square-integrable	B-Algorithm
functions	I-Algorithm
)	O
.	O
</s>
<s>
The	O
shift	B-Algorithm
operator	I-Algorithm
is	O
given	O
by	O
the	O
linear	O
endomorphism	O
.	O
</s>
<s>
The	O
filter	O
space	O
is	O
the	O
algebra	O
of	O
polynomials	O
with	O
complex	O
coefficients	O
and	O
convolution	B-Language
is	O
given	O
by	O
where	O
is	O
an	O
element	O
of	O
the	O
algebra	O
.	O
</s>
<s>
There	O
a	O
few	O
possible	O
choices	O
for	O
a	O
graph	O
shift	B-Algorithm
operator	I-Algorithm
(	O
GSO	O
)	O
.	O
</s>
<s>
The	O
(	O
un	O
)	O
normalized	O
weighted	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
is	O
a	O
popular	O
choice	O
,	O
as	O
well	O
as	O
the	O
(	O
un	O
)	O
normalized	O
graph	B-Algorithm
Laplacian	I-Algorithm
.	O
</s>
<s>
If	O
is	O
the	O
GSO	O
,	O
then	O
a	O
graph	O
convolution	B-Language
is	O
the	O
linear	O
transformation	O
for	O
some	O
,	O
and	O
convolution	B-Language
of	O
a	O
graph	O
signal	O
by	O
a	O
filter	O
yields	O
a	O
new	O
graph	O
signal	O
.	O
</s>
<s>
These	O
objects	O
include	O
including	O
quivers	O
,	O
graphons	B-Device
,	O
semilattices	O
,	O
finite	O
groups	O
,	O
and	O
Lie	O
groups	O
,	O
and	O
others	O
.	O
</s>
<s>
The	O
z	B-Algorithm
transform	I-Algorithm
is	O
a	O
prototypical	O
example	O
of	O
an	O
intertwining	O
map	O
.	O
</s>
<s>
Inspired	O
by	O
a	O
recent	O
perspective	O
that	O
popular	O
graph	O
neural	O
networks	O
(	O
GNNs	O
)	O
architectures	O
are	O
in	O
fact	O
convolutional	B-Architecture
neural	I-Architecture
networks	I-Architecture
(	O
CNNs	B-Architecture
)	O
,	O
recent	O
work	O
has	O
been	O
focused	O
on	O
developing	O
novel	O
neural	O
network	O
architectures	O
from	O
the	O
algebraic	O
point-of-view	O
.	O
</s>
<s>
An	O
algebraic	O
neural	O
network	O
is	O
a	O
composition	O
of	O
algebraic	O
convolutions	B-Language
,	O
possibly	O
with	O
multiple	O
features	O
and	O
feature	O
aggregations	O
,	O
and	O
nonlinearities	O
.	O
</s>
