<s>
In	O
statistics	O
,	O
machine	O
learning	O
and	O
algorithms	O
,	O
a	O
tensor	B-Algorithm
sketch	I-Algorithm
is	O
a	O
type	O
of	O
dimensionality	B-Algorithm
reduction	I-Algorithm
that	O
is	O
particularly	O
efficient	O
when	O
applied	O
to	O
vectors	O
that	O
have	O
tensor	B-Device
structure	O
.	O
</s>
<s>
Such	O
a	O
sketch	O
can	O
be	O
used	O
to	O
speed	O
up	O
explicit	O
kernel	B-Algorithm
methods	I-Algorithm
,	O
bilinear	O
pooling	B-General_Concept
in	O
neural	B-Architecture
networks	I-Architecture
and	O
is	O
a	O
cornerstone	O
in	O
many	O
numerical	O
linear	B-Architecture
algebra	O
algorithms	O
.	O
</s>
<s>
A	O
tensor	B-Algorithm
sketch	I-Algorithm
has	O
the	O
extra	O
property	O
that	O
if	O
for	O
some	O
vectors	O
such	O
that	O
,	O
the	O
transformation	O
can	O
be	O
computed	O
more	O
efficiently	O
.	O
</s>
<s>
For	O
higher-order	O
tensors	B-Device
,	O
such	O
as	O
,	O
the	O
savings	O
are	O
even	O
more	O
impressive	O
.	O
</s>
<s>
The	O
term	O
tensor	B-Algorithm
sketch	I-Algorithm
was	O
coined	O
in	O
2013	O
describing	O
a	O
technique	O
by	O
Rasmus	O
Pagh	O
from	O
the	O
same	O
year	O
.	O
</s>
<s>
Originally	O
it	O
was	O
understood	O
using	O
the	O
fast	O
Fourier	O
transform	O
to	O
do	O
fast	O
convolution	B-Language
of	O
count	B-Algorithm
sketches	I-Algorithm
.	O
</s>
<s>
Later	O
research	O
works	O
generalized	O
it	O
to	O
a	O
much	O
larger	O
class	O
of	O
dimensionality	B-Algorithm
reductions	I-Algorithm
via	O
Tensor	B-Device
random	O
embeddings	O
.	O
</s>
<s>
Tensor	B-Device
random	O
embeddings	O
were	O
introduced	O
in	O
2010	O
in	O
a	O
paper	O
on	O
differential	O
privacy	O
and	O
were	O
first	O
analyzed	O
by	O
Rudelson	O
et	O
al	O
.	O
</s>
<s>
were	O
the	O
first	O
to	O
study	O
the	O
subspace	O
embedding	O
properties	O
of	O
tensor	B-Algorithm
sketches	I-Algorithm
,	O
particularly	O
focused	O
on	O
applications	O
to	O
polynomial	B-Algorithm
kernels	I-Algorithm
.	O
</s>
<s>
In	O
this	O
context	O
,	O
the	O
sketch	O
is	O
required	O
not	O
only	O
to	O
preserve	O
the	O
norm	O
of	O
each	O
individual	O
vector	O
with	O
a	O
certain	O
probability	O
but	O
to	O
preserve	O
the	O
norm	O
of	O
all	O
vectors	O
in	O
each	O
individual	O
linear	B-Architecture
subspace	O
.	O
</s>
<s>
This	O
is	O
a	O
much	O
stronger	O
property	O
,	O
and	O
it	O
requires	O
larger	O
sketch	O
sizes	O
,	O
but	O
it	O
allows	O
the	O
kernel	B-Algorithm
methods	I-Algorithm
to	O
be	O
used	O
very	O
broadly	O
as	O
explored	O
in	O
the	O
book	O
by	O
David	O
Woodruff	O
.	O
</s>
<s>
The	O
face-splitting	O
product	O
is	O
defined	O
as	O
the	O
tensor	B-Device
products	O
of	O
the	O
rows	O
(	O
was	O
proposed	O
by	O
V	O
.	O
Slyusar	O
in	O
1996	O
for	O
radar	B-Application
and	O
digital	B-Algorithm
antenna	I-Algorithm
array	I-Algorithm
applications	O
)	O
.	O
</s>
<s>
Since	O
this	O
operation	O
can	O
be	O
computed	O
in	O
linear	B-Architecture
time	O
,	O
can	O
be	O
multiplied	O
on	O
vectors	O
with	O
tensor	B-Device
structure	O
much	O
faster	O
than	O
normal	O
matrices	O
.	O
</s>
<s>
,	O
where	O
and	O
are	O
independent	O
count	B-Algorithm
sketch	I-Algorithm
matrices	O
and	O
is	O
vector	O
convolution	B-Language
.	O
</s>
<s>
They	O
show	O
that	O
,	O
amazingly	O
,	O
this	O
equals	O
–	O
a	O
count	B-Algorithm
sketch	I-Algorithm
of	O
the	O
tensor	B-Device
product	O
!	O
</s>
<s>
Since	O
is	O
an	O
orthonormal	B-Algorithm
matrix	O
,	O
does	O
n't	O
impact	O
the	O
norm	O
of	O
and	O
may	O
be	O
ignored	O
.	O
</s>
<s>
The	O
problem	O
with	O
the	O
original	O
tensor	B-Algorithm
sketch	I-Algorithm
algorithm	O
was	O
that	O
it	O
used	O
count	B-Algorithm
sketch	I-Algorithm
matrices	O
,	O
which	O
are	O
n't	O
always	O
very	O
good	O
dimensionality	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
In	O
2020	O
it	O
was	O
shown	O
that	O
any	O
matrices	O
with	O
random	O
enough	O
independent	O
rows	O
suffice	O
to	O
create	O
a	O
tensor	B-Algorithm
sketch	I-Algorithm
.	O
</s>
<s>
The	O
paper	O
also	O
shows	O
that	O
the	O
dependency	O
on	O
is	O
necessary	O
for	O
constructions	O
using	O
tensor	B-Device
randomized	O
projections	O
with	O
Gaussian	O
entries	O
.	O
</s>
<s>
With	O
this	O
method	O
,	O
we	O
only	O
apply	O
the	O
general	O
tensor	B-Algorithm
sketch	I-Algorithm
method	O
to	O
order	O
2	O
tensors	B-Device
,	O
which	O
avoids	O
the	O
exponential	O
dependency	O
in	O
the	O
number	O
of	O
rows	O
.	O
</s>
<s>
It	O
can	O
be	O
proved	O
that	O
combining	O
dimensionality	B-Algorithm
reductions	I-Algorithm
like	O
this	O
only	O
increases	O
by	O
a	O
factor	O
.	O
</s>
<s>
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
where	O
each	O
diagonal	O
entry	O
is	O
independently	O
.	O
</s>
<s>
If	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
is	O
replaced	O
by	O
one	O
which	O
has	O
a	O
tensor	B-Device
product	O
of	O
values	O
on	O
the	O
diagonal	O
,	O
instead	O
of	O
being	O
fully	O
independent	O
,	O
it	O
is	O
possible	O
to	O
compute	O
fast	O
.	O
</s>
<s>
For	O
an	O
example	O
of	O
this	O
,	O
let	O
be	O
two	O
independent	O
vectors	O
and	O
let	O
be	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
shows	O
that	O
if	O
has	O
rows	O
,	O
then	O
for	O
any	O
vector	O
with	O
probability	O
,	O
while	O
allowing	O
fast	O
multiplication	O
with	O
degree	O
tensors	B-Device
.	O
</s>
<s>
Jin	O
et	O
al.	O
,	O
the	O
same	O
year	O
,	O
showed	O
a	O
similar	O
result	O
for	O
the	O
more	O
general	O
class	O
of	O
matrices	O
call	O
RIP	B-Algorithm
,	O
which	O
includes	O
the	O
subsampled	O
Hadamard	O
matrices	O
.	O
</s>
<s>
They	O
showed	O
that	O
these	O
matrices	O
allow	O
splitting	O
into	O
tensors	B-Device
provided	O
the	O
number	O
of	O
rows	O
is	O
.	O
</s>
<s>
These	O
fast	O
constructions	O
can	O
again	O
be	O
combined	O
with	O
the	O
recursion	O
approach	O
mentioned	O
above	O
,	O
giving	O
the	O
fastest	O
overall	O
tensor	B-Algorithm
sketch	I-Algorithm
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
do	O
so-called	O
"	O
data	O
aware	O
"	O
tensor	B-Device
sketching	O
.	O
</s>
<s>
Kernel	B-Algorithm
methods	I-Algorithm
are	O
popular	O
in	O
machine	O
learning	O
as	O
they	O
give	O
the	O
algorithm	O
designed	O
the	O
freedom	O
to	O
design	O
a	O
"	O
feature	O
space	O
"	O
in	O
which	O
to	O
measure	O
the	O
similarity	O
of	O
their	O
data	O
points	O
.	O
</s>
<s>
Typical	O
examples	O
are	O
the	O
radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
kernel	I-Algorithm
,	O
,	O
and	O
polynomial	B-Algorithm
kernels	I-Algorithm
such	O
as	O
.	O
</s>
<s>
When	O
used	O
this	O
way	O
,	O
the	O
kernel	B-Algorithm
method	I-Algorithm
is	O
called	O
"	O
implicit	O
"	O
.	O
</s>
<s>
Sometimes	O
it	O
is	O
faster	O
to	O
do	O
an	O
"	O
explicit	O
"	O
kernel	B-Algorithm
method	I-Algorithm
,	O
in	O
which	O
a	O
pair	O
of	O
functions	O
are	O
found	O
,	O
such	O
that	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
the	O
polynomial	B-Algorithm
kernel	I-Algorithm
we	O
get	O
and	O
,	O
where	O
is	O
the	O
tensor	B-Device
product	O
and	O
where	O
.	O
</s>
<s>
The	O
idea	O
of	O
tensor	B-Algorithm
sketch	I-Algorithm
is	O
that	O
we	O
can	O
compute	O
approximate	O
functions	O
where	O
can	O
even	O
be	O
smaller	O
than	O
,	O
and	O
which	O
still	O
have	O
the	O
property	O
that	O
.	O
</s>
<s>
This	O
method	O
was	O
shown	O
in	O
2020	O
to	O
work	O
even	O
for	O
high	O
degree	O
polynomials	O
and	O
radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
kernels	I-Algorithm
.	O
</s>
<s>
where	O
is	O
the	O
tensor	B-Device
product	O
.	O
</s>
<s>
Since	O
we	O
can	O
compute	O
a	O
(	O
linear	B-Architecture
)	O
approximation	O
to	O
efficiently	O
,	O
we	O
can	O
sum	O
those	O
up	O
to	O
get	O
an	O
approximation	O
for	O
the	O
complete	O
product	O
.	O
</s>
<s>
Bilinear	O
pooling	B-General_Concept
is	O
the	O
technique	O
of	O
taking	O
two	O
input	O
vectors	O
,	O
from	O
different	O
sources	O
,	O
and	O
using	O
the	O
tensor	B-Device
product	O
as	O
the	O
input	O
layer	O
to	O
a	O
neural	B-Architecture
network	I-Architecture
.	O
</s>
<s>
In	O
the	O
authors	O
considered	O
using	O
tensor	B-Algorithm
sketch	I-Algorithm
to	O
reduce	O
the	O
number	O
of	O
variables	O
needed	O
.	O
</s>
<s>
This	O
again	O
corresponds	O
to	O
the	O
original	O
tensor	B-Algorithm
sketch	I-Algorithm
.	O
</s>
