►
Description
Subutai Ahmad goes through the framework and results of a study he conducted on dimensionality and sparsity in a deep learning network. Using the GSC dataset, he explores the correlation between dimensionality and the size and accuracy of sparse networks. He also assesses whether there are scaling laws for sparsity in deep learning, similar to the mathematical algorithms for sparse distributed representations in the brain.
“How Can We Be So Dense? The Benefits of Using Highly Sparse Representations” paper: https://arxiv.org/abs/1903.11257
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A
These
are
just
some
experiments
that
I
ran
over
the
holiday
break.
There
are
a
bunch
of
questions
that
keep
coming
up
about
sparsity
and
dimensionality,
so
I
wanted
to
just
run
a
few
experiments
just
to
get
some
results
and
have
some
initial
results.
I
don't
think
I
have
anything
definitive
yet,
but
but
the
basic
idea
is,
you
know,
as
the
title
says,
how
low
can
we
go
with
sparsity?
A
And
so
specifically,
you
know
it's
a
study
of
dimensionality
and
sparsity
in
the
context
of
deep
network,
specifically
the
gsc
network,
and
so
the
agenda
is
I'll.
Talk
about
some
of
the
questions.
I'd
like
to
answer
that
I
keep
getting
asked
all
the
time
I'll
talk
about
the
network,
experiment,
design
and
then
I'll
show
some
results.
First,
with
networks
that
only
have
sparse
weights
and
then
networks
that
have
both
sparse
weights
and
sparse
activations.
B
A
Why
does
it
keep
doing
that?
Okay,
so
questions?
We
should
answer
I'd
like
to
answer
so
the
high
level
thing
is
so
we
have
a
whole
bunch
of
intuitions
and
math
around
sdrs
from
art
neuroscience
work,
and
the
question
is
how
well
do
these
translate
to
deep
learning
networks,
and
I
tried
to
address
this
in
the
house
of
dance
paper.
A
You
know
some
of
the
stuff
around
robustness
and
things
like
that,
and
I
showed
some
of
the
scaling
laws
for
scalar
vectors
as
well,
but
we
didn't
really
look
at
the
question
of
dimensionality
and
scaling
in
the
context
of
sort
of
full,
deep
learning
networks.
So
the
question
is,
you
know:
do
we
have
natural
scaling
laws
for
sparsity
in
in
deep
learning,
I
mean
the
same
way
that
we
have
intuitions
for
sdrs.
A
A
If
you
have
a
if
you're,
given
a
budget
off
a
given,
you
know
fixed
number
of
non-zero
weights
or
non-zero
parameters.
What's
the
best
network
size?
Okay,
is
it
better
to
have
a
small,
dense
network,
or
is
it
actually
better
to
have
a
really
large,
really
sparse
network?
Okay?
So
that's
so.
These
are
all
sort
of
variations
of
the
same
theme
of
looking
at.
How
does
dimensionality
affect
performance
in
in
deep
networks?
A
It's
difficult
to
have
really
sparse
small
networks
for
obvious
reasons
that
work
well,
because
if
it's
just
small,
then
you
make
it
really
sparse.
You
have
hard,
you
know
you
don't
really
end
up
having
any
weights
or
synapses
from
a
performance
side
from
the
architecture
side,
if
you
reduce
the
number
of
non-zero
weights,
that's
going
to
increase
performance.
A
So
to
really
do
that,
does
this
require
increasing
the
dimensionality?
Is
that
what
we
actually
have
to
do
to
to
get
to
the
optimal
performance
from
a
continued
learning
side?
We
know
that
increasing
sparsity
reduces
interference,
and
so
that
should
help
improve
that
should
improve
continual
learning.
Well
to
do
that.
Do
we
really
need
to
increase
dimensionality
of
networks
in
there
and
again,
you
know
we
haven't
really
hammered
on
this
dimensionality
question.
Typically,
what
we've
done
like
in
the
house
of
dance
paper?
A
We
trained
a
dense
network
that
worked
pretty
well,
and
then
we
tried
to
sparsify
that
dense
network.
We
did
a
little
bit
with
dimensionality
and
I'll
show
a
couple
of
things,
but
we
didn't
really
hammer
on
this
dimensionality
question.
I
think
there's
a
really
key
question
and
it
comes
up
in
a
lot
of
conversations
that
I
have
with
with
other
people
and
so
some
background
results.
This
is
kind
of
the
typical
chart
we
had
with
sdr
math.
A
This
is
actually
the
context
of
scalar
vectors,
I
think,
but
we
can
see
that
there
are
many
different
ways
to
do
this,
but
when
you
do
a
vector
dot
product,
we
know
that
the
error
decreases
exponentially
with
the
dimensionality
and
sort
of
intuitively.
What
I've
been
telling
people
is
you
can
the
information
content
of
a
vector
if
you're?
A
If
you
have
a
fixed
budget
of
let's
say
exactly
a
non-zeros
in
a
vector,
the
information
content
of
that
number
of
non-zero
numbers
increases
as
the
dimensionality
of
the
vector
increases,
because
the
exact
locations
of
where
they
are
are
become
more
informative,
as
the
dimensionality
increases
here,
we're
just
looking
at
the
error
when
matching
two
vectors,
we're
not
really
looking
at
the
context
of
a
full
deep
network.
So
it's
it's
unclear
how
this
would
translate
to
a
full
deep
network.
Would
we
get
curves
anything
like
this
in
a
deep
network
with
the
temple
memory?
A
We
actually
did
this.
This
was
a
from
a
poster
at
cosine
2018.
We
varied
the
dimensionality
and
we
looked
at
the
sequence
prediction
error
and
we
got
nice
curves.
That
looked
a
lot
like
you
know
what
you'd
expect
from
the
map
and
in
the
hdm
side.
This
is
completely
not
surprising,
because
the
sdr
math
was
designed
specifically
to
match
what
we
did
with
the
temporal
memory
and
the
spatial
program.
A
Okay,
so,
oh
and
the
other
thing
that
the
only
kind
of
hint
that
a
result
I
had
before
was
given
in
this
table
that
I've
shown
a
few
times
and
this
these
are
results
from
the
google
speech
commands.
If
you
look
at
the
top
two
rows,
we
have
accuracies
for
dense
and
sparse
networks,
so
these
they
both
have
roughly
the
same
test
score
accuracy.
A
But
if
you,
if
you
look
at
the
number
of
parameters
or
non-zero
weights
here,
the
dense
network
has
many
more
weights
than
the
sparse
network,
and
I
I
did
a
little
bit
of
trying
to
create
smaller
versions
of
dense
networks
to
see
if
we
can
get
to
answer
this
question.
This
are
small,
dense
networks
sufficient
and
what
I
found-
and
I
didn't
really
hammer
on
it,
but
I
found
back
then,
is
if
you
try
to
make
it
much
smaller,
you
really
suffered
in
terms
of
accuracy,
okay.
A
So
this
was
very
suggestive
that
for
some
reason
that
that
dimensionality
may
be
important
here,
because
the
sparse
and
dense
networks
both
have
the
same
kind
of
roughly
dimensionality
and
if
you
try
to
reduce
the
dimensionality
to
get
smaller,
dense
networks
you
suffered
in
accuracy,
okay,
so
this
was
kind
of
an
intriguing
thing,
but
again
back
then
I
didn't
really
focus
on
it.
Isn't.
A
A
So
what
I
did
is
you
know
I
should
say
this:
I'm
not
claiming
this
there's
a
huge
results
or
anything
like
that.
This
is
just
something
fun.
I
wanted
to
do
over
the
holidays
and
just
try
to
get
some
answers
because
I
get
getting
asked
this
all
the
time
and
it
would
be
really
nice
to
have
something
a
little
more
concrete
to
say.
A
I'll
get
to
that
yeah
yeah,
I
I
did
do
that.
I
didn't
do
it
as
exhaustively
as
I
would
like,
but
I
did
I'll
show
you
some
results
on
that
and
that's
explained
here.
Actually,
so
I
guess
it's
more
than
a
five:
it's
a
six-dimensional
or
seven-dimensional
search
space.
So
I
have
a
dense
network
baseline,
where
I
created
dense
networks
of
different
sizes.
By
varying
these
three
things,
there
was
no.
The
weight.
Sparsity
was
always
zero
percent
for
the
dense
network.
A
Obviously,
so
there
were
25
total
configurations
and
I
did
four
trials
for
each
configuration
just
to
take
care
of
care
of
randomness.
So
there
were
about
a
hundred
total
runs
for
the
dense
networks.
For
this
sparse
networks.
There's
a
lot
more
because
not
only
do
I
vary
the
the
number
of
channels
and
sizes,
but
I
also
vary
the
weight
sparse
cities
like
I
said
so
there
were
actually
940
configurations,
so
I
did
about
3700.
Total
runs
with
with
sparse
weights
only
and
then
per
michelangelo's
question.
A
I
did
some
stuff
with
sparse
activations
as
well.
So
here
I
just
kept
the
k
winners
to
be
10
on
for
cnn
2
and
the
linear
layer,
and
I
I
ran
smaller
numbers
of
configuration,
420
configurations
here
in
1680..
So
obviously
this
is
not
a
complete
exploration
of
the
space
like
I
can
do
better.
This
was
just
kind
of
what
I
ended
up.
A
B
C
I
I
I'm
not
exactly
sure
anybody
did
a
great
search,
but
like
you're
you're
wearing
the
sparsity
of
these
two
cnn
layers
right.
Are
you
varying
them
together,
like
they,
both
are
going
to
be
going
from
sparse
to
less
sparse
or
maybe
one's
going
to
be
sparse,
someone's
going
to
be
dense
in
one
of
these
trials,
I
mean.
C
B
C
A
B
A
A
No,
this
is
a
mistake
I
made
actually
so
I
should
have
cnn
k
winners
after
cnn
one,
and
I
was
just
trying
to
control
the
number
of
variations,
and
I
didn't
think
too
hard
about
it,
as
I
was
doing
the
graphs
and
stuff
last
couple
of
days.
I
realized
that
this
is
actually
an
issue.
It's
a
flaw
in
this
design
and
okay.
I
think
it
will
actually
affect
the
results.
A
A
Okay,
so
here's
some
results
with
sparse
weights
only
and
so
no
sparse,
activations
or
k
winners
here.
So
these
are
kind
of
results
for
all
of
the
trials,
so
the
blue
is
the
sparse
weights
only
and
as
a
baseline,
I
have
the
the
oranges,
the
dense
networks,
so
obviously-
and
this
is
plotting
the
number
of
non-zero
weights
or
non-zero
parameters
versus
accuracy
and
obviously
the
dense
networks
have
many
more
non-zero
parameters.
A
A
Because
I
didn't
have
that
before
in
the
in
the
gsc
stuff,
but
I
hadn't
really
focused
on
dimensionality
as
much
back
then
so
this
was
kind
of
this
was
the
first
surprise,
and
so,
if
you
kind
of
zoom
in
so
I'm
just
zooming
in
on
here
now,
this
is
kind
of
all
the
sparse.
The
blue
dots
are
all
of
the
sparse
networks
with
sparse
weights.
A
So
then
I
wanted
to
see
how
dimensionality
kind
of
played
into
it,
and
I
had
a
really
hard
time
trying
to
there's
so
many
variables
here.
I
was
trying
to
figure
out
how
to
plot
this
stuff.
So
what
I
came
up
with
for
now-
and
I
want
to
find
better
metrics-
is
I
defined,
because
I
wanted
to
have
one
axis,
which
is
dimensionality
and
another
axis,
which
is
accuracy
so,
but
there
are
many
ways
that
dimensionality
is
changing,
and
so
what
I
did
is
I
defined
dimensionality
as
being
the
size
of
cnn.
A
This
is
a
cnn2,
the
cnn2
layer,
multiplied
by
the
number
of
units
in
the
in
this
l3,
the
linear
layer,
and
take
the
square
root
of
that.
So
that's
what
I
I
defined.
The
dimensionality
as
and
this
plot
shows
accuracy
as
a
function
of
dimensionality
by
itself,
and
you
can
see
that
there's
a
clear
trend
here
that,
as
you
get
higher
dimensionality
the
accuracy
the
improves.
So
this
is
each
dot.
A
You
have
to
be
kind
of
in
this
range
and,
and
they
kind
of-
and
you
can
see
the
accuracy
kind
of
obviously
has
to
tail
off
at
some
point.
But
so
this
was
kind
of
a
I.
I
expected
something
like
this,
but
I
did
not
expect
these
numbers
to
be
better
than
the
best
dense
network
I
had,
but
you
do
see
a
clear
trend.
Improvement
with
with
dimensionality.
E
A
A
A
There's
a
there
is
a
trend
down
as
dimensionality
increases,
but
it
seemed
to
kind
of
it
seems
to
kind
of
bottom
out
here:
okay
and
so
it's
kind
of
suggestive,
and
we
we
definitely
saw
this
in
htms
and
the
the
math
kind
of
is
consistent
with
this-
that
there's
a
natural
floor
of
like
a
the
number
of
non-zero
weights.
You
want
to
have
in
a
neuron
below
which
it's
really
hard
to
get
below
that.
A
If
you
look
at
the
linear
layer
that
might
be
something
like
you
know,
20
to
30
non-zero
weights
per
neuron,
it's
it's
hard
to
go
below
that
and
still
get
good
results,
and
so
so
that
was
that's
interesting.
That's
consistent
with
my
intuitions
and-
and
that
was
good
to
see
and
again
I'm
not
saying
this
is
definitive.
It's
just
suggestive.
A
The
interesting
thing
per
your
question
kevin
is
that
what
this
means
is
that
if
you
can't
go
below
a
certain
floor
at
beyond
here,
if
you
start
increasing
dimensionality
you're,
just
going
to
increase
the
total
number
of
weights
right,
because
if
you
need
to
have
a
minimum
number
of
weights
per
neuron
and
you
increase
the
number
of
neurons
you're
not
going
to
reduce
the
number
of
weights
you're
going
to
increase
the
total
number
of
weights.
So
there's.
D
A
D
E
A
C
A
C
E
Yeah,
basically,
I
I
mean,
given
the
problems
trying
to
solve
the
dimensionality
there's
some.
You
know,
there's
fundamentally
some
information
beyond
which
it
doesn't
make
any
sense
to
add
anything
more
to
the
to
the
problem.
It's
the
rest
of
it
is
is
going
to
be
in
the
noise.
So
when
you
change
the
problem,
you
know
more
categories
or
continuous
learning,
or
something
like
that.
You
know
everything
shifts
around,
but
it's
just
interesting
to
kind
of
compartmentalize
the
notion
that
okay,
there
is
an
ultimate
limit
here
for
this
particular
problem.
C
A
C
A
A
You
know
kind
of
in
the
20
to
30
range
per
or
per
neuron,
which
is
similar
to
what
you
get
in
active
dendrites,
so
that
was
kind
of
interesting
side
thing,
but
it
does
mean
that
you
can't
just
keep
increasing
the
dimensionality
and
forever
expect
to
get
fewer
and
fewer
non-zero
things.
There's
going
to
be
some
maximum
point
and
beyond
that,
it's
going
to
start
increasing
again.
A
A
A
A
E
A
A
Yeah,
at
least
it
didn't
get
much
worse,
that's
good
some
people.
So
that's
an
interesting
point
actually
so
in
in
deep
learning.
Very
few
cases
places
have
sparse,
activations
and
sparse
weights,
and
when
I
was
talking
about
this
with
machine
learning
people,
they
were
surprised
that
it
would
even
work
at
all.
If
you
had
sparse
activations
and
sparse
weights
that
seemed
crazy
to
them.
B
Yeah,
I
think
we
discussed
in
the
past
that
having
sparse
activations
create
a
very
strange
loss
surface
with
it's
not
a
smooth
one
with
very
peaks
and
valleys
yeah.
I
think
that's
the
intuition
behind
you
know.
People
don't
believe
sparse.
Activations
are
actually
gonna
work.
Well
in
this
sort
of
optimization
problems.
We
have.
C
Yeah,
you
know
if
one
thing
we've
learned
over
the
years
is
that
our
visualizations
of
these
you
know
these
peaks
and
valleys
these
these
high
dimensional
spaces
are
just
totally
intuitively
wrong.
You
know
we
imagine
these
things
in
two
or
three
dimensions,
but
anyway,
I
think
that's
a
weak
argument.
Anyway.
I'm
glad
to
see,
I
think,
it's
clear
that
this
is
it.
A
Yeah
and
I'll
show
some
more
results
along
these
lines.
Here,
some
of
these
same
charts.
Here
you
can
see
the
accuracy
versus
dimensionality
either
the
red
dots
are
the
sparse
weights
and
you
can
see
again
that
for
for
these
load
this,
this
range
that,
for
the
red
dots,
seem
to
be
higher
than
the
blue
dots.
A
The
absolute
accuracy
doesn't
change
if
you
look
at
the
best
overall,
sparse
weights
network
and
the
best
overall,
sparse
weights
and
activations
they're
the
same,
but
this
curve
seems
to
you
know
top
out
sooner
than
the
blue
boot
again,
it's
it's.
These
are
noisy
things
and
a
lot
of
problems
with
this
experiment
design.
But
you
can
kind
of
roughly
see
this
at.
B
A
Here's
the
kind
of
number
of
weights
per
neuron
again
to
me
if
I
squint
a
little
bit,
it
seems
to
me
like
the
the
sparse
activation
stuff,
is
a
little
bit
better.
It's
more
clear,
I
think,
in
the
cnn2
here
this,
the
orange
dots
are
lower
than
the
blue
dots,
the
number
of
weights
per
neuron
here
in
the
linear
layer,
it's
a
little
less
clear,
but
you
can
see
this
curve.
A
A
A
Yeah
no,
I
did
not
expect
that
at
all
and
yeah.
I
didn't
expect
that
when
I
came
into
this
and
then
if
you
look
at
the
similar
table
with
sparse
weights
and
k
winners
here,
it's
much
clearer
all
of
the
the
parameters
here
are
much
smaller.
Now
with
once
you
add,
k
winners.
So
if
you
look
at
the
best
network
that
about
had
about
97
accuracy,
the
the
smallest
here
is
now
64
000.
So
that's
a
new
record
for
in.
D
A
A
The
I
didn't
really
spend
too
much
time,
exploring
the
cnn
one
size
much
which
could
have
an
impact.
The
thing
that
kevin
mentioned,
I
think,
is
actually
a
pretty
big
deal.
I
didn't
have
k
winners
at
cnn,
one,
which
means
actually
the
weights
at
cnn2,
were
getting
dense
output
activation.
So
this
could
actually
significantly
impact
the
the
non-zeros
and
the
cnn2
weights.
So
I
should.
A
A
This
is
just
exploring
you
know
a
larger
search
space
here,
but
again
that
could
make
things
even
better.
There's
noise.
Here
I
could,
I
should
have
more
trials
per
configuration
on
the
f4,
I
think
better
charts.
Sometimes
it
was
really
hard
to
decide
what
dimensions
to
plot
in
here.
So
I
tried
to
do
the
best
I
could,
but
I'm
open
to
suggestions.
A
A
You
know,
can
sparse
high
dimensional
networks
reach
higher
accuracy
than
dense
networks?
It
seems
like
the
answer.
Is
yes,
but
more
work
to
be
done
there.
There
may
be
a
natural
floor
to
the
number
of
weights.
You
need
per
neuron,
you
know,
regardless
of
dimensionality,
and
so
once
you
reach
this,
then
increasing
dimensionality
just
increases
the
number
of
weights.
This
floor
may
be
surprisingly
so
close
to
biological
numbers,
which
would
be
kind
of
bizarre
to
me,
but
that
would
be
kind
of
fun
that
happened.
A
Sparse
weights
and
sparse
activations
lead
to
sparser
overall
networks.
I
think
we
have
a
new
record
for
gsc
and
accuracy
and
size
some
one
takeaway
is
it's
very
tempting
for
us
to
do
experiments
with
small,
dense
networks
or
even
small,
sparse
networks,
and
it's
okay
for
debugging,
but
we
really
shouldn't
test
our
key
properties
that
we're
interested
in
small
networks.
We
really
need
to
have
the
discipline
to
go
with
larger,
sparse
networks.
Is
that.
A
I
think
that's
an
issue,
it's
just
easier
to
look
at
stuff
with
small
networks,
and
then
you
can
get
into
this
trap
of
over
thinking.
Oh
something's,
not
working
or
something's
working
one
way
or
another,
whereas
you
really
need
to
make
sure
you
test
with
large,
sparse
networks,
and
we
saw
this
with
htms
as
well
when
we
were
initially
doing
it.
There
was
a
lot
of
pressure.
A
A
Yeah,
it's
just,
I
think.
Initially,
the
tendency
is
okay,
you're,
just
trying
to
debug
your
experiment
to
find
out
there's
a
code
problem.
Well,
it's
easier
to
look
at
these
things.
If
it's
small
and
then
okay,
it's
running
faster
and
I'm
doing
this.
Okay,
now
I'm
getting
some
charts
and
I'm
looking
at
things
and
then
you
sort
of
forget
that
okay,
we
should
just
we
have
to
move
to
large
network,
there's
also
the
issue
of
comparing
against
existing
results
and
so
there's
a
pressure
to
do
a
b
comparisons
which.
B
B
A
B
D
A
A
Lot
more
stuff,
I
can
fix
all
the
problems
I
talked
about
in
the
previous
slide.
You
know.
Clearly
we
need
to
look
at
more
complex
networks
like
resnet50
and
other
things.
Maybe
I
didn't
look
at
how
fast
the
network
learns
that
that
would
be
kind
of.
I
have
all
the
data
that
would
be
kind
of
interesting
to
look
at
this.
A
It
would
be
interesting
to
look
at
michelangelo,
has
been
doing
this
visualizations
of
sparsity,
sparse
activations
would
be
kind
of
interesting
to
look
at
all
of
this
anyway.
There's
a
lot
more
that
could
be
done
here,
but
I
thought
I'd
just
stop
it
here
and
just
share
it
with
you
guys
and
decide
what
to
do
next.
E
I
I
have
a
slightly
extemporaneous
question
when
you
were
talking
about
the
biological
numbers
that
you
know
the
20
to
40.
You
know
kind
of
interesting
range
that
we
we've
been
looking
at.
Is
that
conserved
across
mammals
or
I'm
just
kind
of
curious?
If,
if,
if
that
number
holds
true
for
other
than
primates,.
A
C
C
Yeah
so
you're
you're
phrasing,
the
opposite
kevin
like
oh,
does
it
apply?
I
I
I
don't
think
there's
a
lot
of
data
on
this.
If
you
correct
me,
if
anyone
else
knows
otherwise,
it's
not
like
this.
First
of
all,
it's
hard
to
actually
determine
these
things
in
biological
networks,
and-
and
these
are
done
with
you-
know,
slice
preparations
or
whatever,
and
so
it's
not
like
there's
a
huge
amount
of
data
about
this.
So
but
my
assumption
would
be
that
it
would
be
preserved.
It's
more
of
a
mathematical
thing.
I
don't.
C
A
E
A
C
Floor
here
so
there's
been
some
modeling.
Excuse
me,
you
know
you
can
model
these
stuff
in,
like
the
neuron
models,
which
are
very
detailed,
biological
membrane
models
and
those
come
out
about
the
same
thing
too.
So
anyways,
I
don't
think
it's
going
to
answer
the
question,
but
I'd
be
surprised
about
substantiation.
A
Yeah
yeah,
I
agree
the
ish.
The
one
wild
card
is
that
here
we're
using
32-bit
floating
point
numbers.
So
in
theory
there
could
be
a
lot
more
information
per
number.
So
maybe
you
could
argue
that.
Oh,
the
number
should
be
much
smaller
here
because
we're
dealing
with
floating
point,
whereas
on
synapses
we
assume
binary
or
at
least
very
coarse
resolution.
C
D
B
A
A
A
On
is
at
least
easily
run
on
multiple
have
a
head,
node
and
multiple
slave
nodes.
If
I
remember
yeah,
you
could
just
have
to
it's
just
the
idea
to
work
yeah
that
yeah,
it
was
easier
for
me
to
just
you
know,
wait
for
you
to
fix
vernon
and
I'll
just
do
it
in
vernon.
That's
rather
than
mucking
too
much
with
the
old
code.
D
B
B
D
A
A
Yeah
you
said
it,
but
but
the
point
is
that
if
you
go
below
that
you're
not
going
to
get
good
results,
obviously
right,
so
if
you,
if
you
say
okay,
I
want,
I
want
good
results,
I'm
not
going
to
accept
bad
results,
I'm
going
to
get
good
results.
At
that
point.
There
is
a
natural
floor
to
the
number
of
weights.