<s>
A	O
CUR	B-Algorithm
matrix	I-Algorithm
approximation	I-Algorithm
is	O
a	O
set	O
of	O
three	O
matrices	B-Architecture
that	O
,	O
when	O
multiplied	O
together	O
,	O
closely	O
approximate	O
a	O
given	O
matrix	O
.	O
</s>
<s>
The	O
matrices	B-Architecture
are	O
more	O
interpretable	O
;	O
The	O
meanings	O
of	O
rows	O
and	O
columns	O
in	O
the	O
decomposed	O
matrix	O
are	O
essentially	O
the	O
same	O
as	O
their	O
meanings	O
in	O
the	O
original	O
matrix	O
.	O
</s>
<s>
Formally	O
,	O
a	O
CUR	B-Algorithm
matrix	I-Algorithm
approximation	I-Algorithm
of	O
a	O
matrix	O
A	O
is	O
three	O
matrices	B-Architecture
C	O
,	O
U	O
,	O
and	O
R	O
such	O
that	O
C	O
is	O
made	O
from	O
columns	O
of	O
A	O
,	O
R	O
is	O
made	O
from	O
rows	O
of	O
A	O
,	O
and	O
that	O
the	O
product	O
CUR	O
closely	O
approximates	O
A	O
.	O
</s>
<s>
There	O
are	O
many	O
possible	O
CUR	B-Algorithm
matrix	I-Algorithm
approximations	I-Algorithm
,	O
and	O
many	O
CUR	B-Algorithm
matrix	I-Algorithm
approximations	I-Algorithm
for	O
a	O
given	O
rank	O
.	O
</s>
<s>
The	O
CUR	B-Algorithm
matrix	I-Algorithm
approximation	I-Algorithm
is	O
often	O
used	O
in	O
place	O
of	O
the	O
low-rank	O
approximation	O
of	O
the	O
SVD	O
in	O
principal	B-Application
component	I-Application
analysis	I-Application
.	O
</s>
<s>
The	O
CUR	B-Algorithm
matrix	I-Algorithm
approximation	I-Algorithm
is	O
not	O
unique	O
and	O
there	O
are	O
multiple	O
algorithms	O
for	O
computing	O
one	O
.	O
</s>
<s>
Formally	O
,	O
a	O
CURT	O
tensor	O
approximation	O
of	O
a	O
tensor	O
A	O
is	O
three	O
matrices	B-Architecture
and	O
a	O
(	O
core	O
-	O
)	O
tensor	O
C	O
,	O
R	O
,	O
T	O
and	O
U	O
such	O
that	O
C	O
is	O
made	O
from	O
columns	O
of	O
A	O
,	O
R	O
is	O
made	O
from	O
rows	O
of	O
A	O
,	O
T	O
is	O
made	O
from	O
tubes	O
of	O
A	O
and	O
that	O
the	O
product	O
U(C,R,T )	O
(	O
where	O
the	O
-th	O
entry	O
of	O
it	O
is	O
)	O
closely	O
approximates	O
A	O
.	O
</s>
