<s>
Quantum	B-Algorithm
Clustering	I-Algorithm
(	O
QC	O
)	O
is	O
a	O
class	O
of	O
data-clustering	B-Algorithm
algorithms	O
that	O
use	O
conceptual	O
and	O
mathematical	O
tools	O
from	O
quantum	O
mechanics	O
.	O
</s>
<s>
QC	O
belongs	O
to	O
the	O
family	O
of	O
density-based	O
clustering	B-Algorithm
algorithms	I-Algorithm
,	O
where	O
clusters	O
are	O
defined	O
by	O
regions	O
of	O
higher	O
density	O
of	O
data	O
points	O
.	O
</s>
<s>
(	O
This	O
step	O
is	O
a	O
particular	O
example	O
of	O
kernel	B-General_Concept
density	I-General_Concept
estimation	I-General_Concept
,	O
often	O
referred	O
to	O
as	O
a	O
Parzen-Rosenblatt	O
window	O
estimator	O
.	O
)	O
</s>
<s>
QC	O
then	O
uses	O
gradient	B-Algorithm
descent	I-Algorithm
to	O
move	O
each	O
data	O
point	O
‘	O
downhill’	O
in	O
the	O
landscape	O
,	O
causing	O
points	O
to	O
gather	O
together	O
in	O
nearby	O
minima	O
,	O
thus	O
revealing	O
clusters	O
within	O
the	O
data	O
set	O
.	O
</s>
<s>
QC	O
has	O
a	O
single	O
main	O
hyperparameter	B-General_Concept
,	O
which	O
is	O
the	O
width	O
sigma	O
of	O
the	O
Gaussian	O
distribution	O
around	O
each	O
data	O
point	O
.	O
</s>
<s>
Developed	O
by	O
Marvin	O
Weinstein	O
and	O
David	O
Horn	O
in	O
2009	O
,	O
Dynamic	O
Quantum	B-Algorithm
Clustering	I-Algorithm
(	O
DQC	O
)	O
extends	O
the	O
basic	O
QC	O
algorithm	O
in	O
several	O
ways	O
.	O
</s>
<s>
DQC	O
uses	O
the	O
same	O
potential	O
landscape	O
as	O
QC	O
,	O
but	O
it	O
replaces	O
classical	O
gradient	B-Algorithm
descent	I-Algorithm
with	O
quantum	O
evolution	O
.	O
</s>
<s>
Thus	O
,	O
quantum	O
evolution	O
for	O
each	O
point	O
acts	O
as	O
a	O
form	O
of	O
non-local	O
gradient	B-Algorithm
descent	I-Algorithm
in	O
the	O
potential	O
.	O
</s>
<s>
The	O
biggest	O
problem	O
in	O
non-convex	O
gradient	B-Algorithm
descent	I-Algorithm
is	O
often	O
the	O
existence	O
of	O
many	O
small	O
and	O
uninteresting	O
local	O
minima	O
where	O
points	O
can	O
get	O
stuck	O
as	O
they	O
descend	O
.	O
</s>
<s>
(	O
This	O
problem	O
tends	O
to	O
get	O
worse	O
as	O
the	O
number	O
of	O
dimensions	O
increases	O
,	O
which	O
is	O
part	O
of	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
.	O
)	O
</s>
<s>
DQC	O
’s	O
use	O
of	O
non-local	O
gradient	B-Algorithm
descent	I-Algorithm
and	O
tunneling	O
presents	O
a	O
solution	O
to	O
this	O
problem	O
.	O
</s>
<s>
DQC	O
introduces	O
2	O
new	O
hyperparameters	B-General_Concept
:	O
the	O
time	O
step	O
,	O
and	O
the	O
mass	O
of	O
each	O
data	O
point	O
(	O
which	O
controls	O
the	O
degree	O
of	O
tunneling	O
behavior	O
)	O
.	O
</s>
<s>
For	O
a	O
data	O
set	O
of	O
n	O
points	O
,	O
DQC	O
creates	O
a	O
set	O
of	O
n	O
quantum	O
eigenstates	O
for	O
use	O
in	O
its	O
underlying	O
calculations	O
;	O
the	O
eigenstates	O
are	O
orthonormal	B-Algorithm
,	O
and	O
each	O
one	O
is	O
a	O
linear	O
combination	O
of	O
the	O
Gaussian	O
distributions	O
representing	O
each	O
data	O
point	O
.	O
</s>
<s>
Use	O
of	O
a	O
PCA	B-Application
coordinate	O
system	O
is	O
helpful	O
for	O
these	O
visualizations	O
;	O
viewing	O
the	O
trajectories	O
in	O
the	O
first	O
3	O
PCA	B-Application
dimensions	O
packs	O
as	O
much	O
information	O
as	O
possible	O
into	O
a	O
single	O
visualization	O
.	O
</s>
