<s>
The	O
information	B-General_Concept
bottleneck	I-General_Concept
method	I-General_Concept
is	O
a	O
technique	O
in	O
information	O
theory	O
introduced	O
by	O
Naftali	O
Tishby	O
,	O
Fernando	O
C	O
.	O
Pereira	O
,	O
and	O
William	O
Bialek	O
.	O
</s>
<s>
It	O
is	O
designed	O
for	O
finding	O
the	O
best	O
tradeoff	O
between	O
accuracy	O
and	O
complexity	O
(	O
compression	B-General_Concept
)	O
when	O
summarizing	O
(	O
e.g.	O
</s>
<s>
clustering	B-Algorithm
)	O
a	O
random	O
variable	O
X	O
,	O
given	O
a	O
joint	O
probability	O
distribution	O
p(X,Y )	O
between	O
X	O
and	O
an	O
observed	O
relevant	O
variable	O
Y	O
-	O
and	O
self-described	O
as	O
providing	O
"	O
a	O
surprisingly	O
rich	O
framework	O
for	O
discussing	O
a	O
variety	O
of	O
problems	O
in	O
signal	O
processing	O
and	O
learning	O
"	O
.	O
</s>
<s>
Applications	O
include	O
distributional	O
clustering	B-Algorithm
and	O
dimension	B-Algorithm
reduction	I-Algorithm
,	O
and	O
more	O
recently	O
it	O
has	O
been	O
suggested	O
as	O
a	O
theoretical	O
foundation	O
for	O
deep	B-Algorithm
learning	I-Algorithm
.	O
</s>
<s>
It	O
generalized	O
the	O
classical	O
notion	O
of	O
minimal	O
sufficient	O
statistics	O
from	O
parametric	B-General_Concept
statistics	I-General_Concept
to	O
arbitrary	O
distributions	O
,	O
not	O
necessarily	O
of	O
exponential	O
form	O
.	O
</s>
<s>
The	O
information	O
bottleneck	O
can	O
also	O
be	O
viewed	O
as	O
a	O
rate	B-General_Concept
distortion	I-General_Concept
problem	O
,	O
with	O
a	O
distortion	O
function	O
that	O
measures	O
how	O
well	O
Y	O
is	O
predicted	O
from	O
a	O
compressed	O
representation	O
T	O
compared	O
to	O
its	O
direct	O
prediction	O
from	O
X	O
.	O
</s>
<s>
They	O
conjectured	O
that	O
the	O
training	O
process	O
of	O
a	O
DNN	O
consists	O
of	O
two	O
separate	O
phases	O
;	O
1	O
)	O
an	O
initial	O
fitting	O
phase	O
in	O
which	O
increases	O
,	O
and	O
2	O
)	O
a	O
subsequent	O
compression	B-General_Concept
phase	O
in	O
which	O
decreases	O
.	O
</s>
<s>
in	O
countered	O
the	O
claim	O
of	O
Shwartz-Ziv	O
and	O
Tishby	O
,	O
stating	O
that	O
this	O
compression	B-General_Concept
phenomenon	O
in	O
DNNs	O
is	O
not	O
comprehensive	O
,	O
and	O
it	O
depends	O
on	O
the	O
particular	O
activation	O
function	O
.	O
</s>
<s>
In	O
particular	O
,	O
they	O
claimed	O
that	O
the	O
compression	B-General_Concept
does	O
not	O
happen	O
with	O
ReLu	O
activation	O
functions	O
.	O
</s>
<s>
Shwartz-Ziv	O
and	O
Tishby	O
disputed	O
these	O
claims	O
,	O
arguing	O
that	O
Saxe	O
et	O
al	O
had	O
not	O
observed	O
compression	B-General_Concept
due	O
to	O
weak	O
estimates	O
of	O
the	O
mutual	O
information	O
.	O
</s>
<s>
used	O
a	O
rate-optimal	O
estimator	O
of	O
mutual	O
information	O
to	O
explore	O
this	O
controversy	O
,	O
observing	O
that	O
the	O
optimal	O
hash-based	O
estimator	O
reveals	O
the	O
compression	B-General_Concept
phenomenon	O
in	O
a	O
wider	O
range	O
of	O
networks	O
with	O
ReLu	O
and	O
maxpooling	O
activations	O
.	O
</s>
<s>
have	O
argued	O
that	O
the	O
observed	O
compression	B-General_Concept
is	O
a	O
result	O
of	O
geometric	O
,	O
and	O
not	O
of	O
information-theoretic	O
phenomena	O
,	O
a	O
view	O
that	O
has	O
been	O
shared	O
also	O
in	O
.	O
</s>
<s>
In	O
the	O
present	O
method	O
,	O
joint	O
sample	O
probabilities	O
are	O
found	O
by	O
use	O
of	O
a	O
Markov	B-Algorithm
transition	I-Algorithm
matrix	I-Algorithm
method	O
and	O
this	O
has	O
some	O
mathematical	O
synergy	O
with	O
the	O
bottleneck	O
method	O
itself	O
.	O
</s>
<s>
Treating	O
samples	O
as	O
states	O
,	O
and	O
a	O
normalised	O
version	O
of	O
as	O
a	O
Markov	O
state	O
transition	B-Algorithm
probability	I-Algorithm
matrix	I-Algorithm
,	O
the	O
vector	O
of	O
probabilities	O
of	O
the	O
'	O
states	O
 '	O
after	O
steps	O
,	O
conditioned	O
on	O
the	O
initial	O
state	O
,	O
is	O
.	O
</s>
<s>
In	O
the	O
following	O
soft	O
clustering	B-Algorithm
example	O
,	O
the	O
reference	O
vector	O
contains	O
sample	O
categories	O
and	O
the	O
joint	O
probability	O
is	O
assumed	O
known	O
.	O
</s>
<s>
presented	O
the	O
following	O
iterative	O
set	O
of	O
equations	O
to	O
determine	O
the	O
clusters	O
which	O
are	O
ultimately	O
a	O
generalization	O
of	O
the	O
Blahut-Arimoto	O
algorithm	O
,	O
developed	O
in	O
rate	B-General_Concept
distortion	I-General_Concept
theory	I-General_Concept
.	O
</s>
<s>
The	O
following	O
case	O
examines	O
clustering	B-Algorithm
in	O
a	O
four	O
quadrant	O
multiplier	O
with	O
random	O
inputs	O
and	O
two	O
categories	O
of	O
output	O
,	O
,	O
generated	O
by	O
.	O
</s>
<s>
The	O
statistical	O
soft	O
clustering	B-Algorithm
definition	O
has	O
some	O
overlap	O
with	O
the	O
verbal	O
fuzzy	O
membership	O
concept	O
of	O
fuzzy	O
logic	O
.	O
</s>
