<s>
Quadratic	B-Algorithm
unconstrained	I-Algorithm
binary	I-Algorithm
optimization	I-Algorithm
(	O
QUBO	O
)	O
,	O
also	O
known	O
as	O
unconstrained	B-Algorithm
binary	I-Algorithm
quadratic	I-Algorithm
programming	I-Algorithm
(	O
UBQP	B-Algorithm
)	O
,	O
is	O
a	O
combinatorial	O
optimization	O
problem	O
with	O
a	O
wide	O
range	O
of	O
applications	O
from	O
finance	O
and	O
economics	O
to	O
machine	O
learning	O
.	O
</s>
<s>
QUBO	O
is	O
an	O
NP	O
hard	O
problem	O
,	O
and	O
for	O
many	O
classical	O
problems	O
from	O
theoretical	O
computer	O
science	O
,	O
like	O
maximum	O
cut	O
,	O
graph	O
coloring	O
and	O
the	O
partition	B-Algorithm
problem	I-Algorithm
,	O
embeddings	O
into	O
QUBO	O
have	O
been	O
formulated	O
.	O
</s>
<s>
Embeddings	O
for	O
machine	O
learning	O
models	O
include	O
support-vector	B-Algorithm
machines	I-Algorithm
,	O
clustering	B-Algorithm
and	O
probabilistic	O
graphical	O
models	O
.	O
</s>
<s>
Moreover	O
,	O
due	O
to	O
its	O
close	O
connection	O
to	O
Ising	O
models	O
,	O
QUBO	O
constitutes	O
a	O
central	O
problem	O
class	O
for	O
adiabatic	O
quantum	O
computation	O
,	O
where	O
it	O
is	O
solved	O
through	O
a	O
physical	O
process	O
called	O
quantum	B-Algorithm
annealing	I-Algorithm
.	O
</s>
<s>
We	O
are	O
given	O
a	O
real-valued	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
,	O
whose	O
entries	O
define	O
a	O
weight	O
for	O
each	O
pair	O
of	O
indices	O
within	O
the	O
binary	O
vector	O
.	O
</s>
<s>
As	O
an	O
illustrative	O
example	O
of	O
how	O
QUBO	O
can	O
be	O
used	O
to	O
encode	O
an	O
optimization	O
problem	O
,	O
we	O
consider	O
the	O
problem	O
of	O
cluster	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
One	O
way	O
to	O
derive	O
a	O
clustering	B-Algorithm
is	O
to	O
consider	O
the	O
pairwise	O
distances	O
between	O
points	O
.	O
</s>
