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The unpacking
algorithm is simply the other way around.

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So if you pack the tensor

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as follows, you would like to unpack them

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to extract the real values
of the two bit weights.

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And most importantly,
you want to unpack these tensors

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so that you can also perform
some operations on top of these tensors.

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Yeah.

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So if you have for example packages tensor
then you would unpack it like this.

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So 010000 0 or 1 one and then one zero.

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So let's also try to do it step by step.

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So unpacked weights
the signature would be the same as well.

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You take in packed uint8 tensor
number of bits.

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So to get the expected number of values
you can just use this formula.

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Yeah.

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It's simply
the number of packed tensors multiplied

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by eight divided by the number of bits.

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NUM steps is also going to be the same
for each packed tensor.

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We're expecting to extract
eight divided by number of bits

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tensors.

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Let's initialize
our unpack tensor.

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So yeah by just initializing
some torch zeros.

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The values for you the eight.

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And let's keep track

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of the index of the unpacked tensor.

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So yeah we're going to go for a simple
and naive approach.

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Yeah.

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We're just going to loop over the packed
weights one by one

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and extract

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for each pack tensor number of steps.

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So here for two bits four.

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So again if we consider this example okay.

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So this is our packed weight.

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We want to extract
step by step this value in two bit.

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And this one and so on.

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So let's consider the first iteration.

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So for the first packed weight.

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So this is in our simple case
this is one value right.

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So in the first iteration
you uint8 tensor of I is this value.

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So if we want to extract

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only these first two bits,

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in an iterative manner,
we're going to shift it

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by bits times j.

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So j starts from zero.

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And then it goes until three.

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So for the first iteration
this is going to be zero.

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We're going to do nothing.

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But it's fine because we just want to take
the first two bits.

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We're going to use the same bitwise
or operation

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on the unpacked tensor.

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So let's also write down.

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Here.

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The stages of the unpacked tensor.

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So we have num values here

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is four.

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So we have four values in four bit.

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For the first index
we're going to make bitwise or operation

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between these full zeros
and this tensor shifted by zero.

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So basically just this tensor.

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And in the first iteration

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it's going to be
an exact replication of the packed tensor.

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So you may be wondering
"what do we do to remove

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all these bits that are before zero one?"

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But don't worry,

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you're going to see that in a few moments.

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So this is on the first iteration.

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Of course.

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Don't forget to increment unpacked index.

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And let's try to see
what happens on the next iteration.

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So in the next iteration
we want to consider these two bits

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and put them here, right.

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So now unpacked index is going to be one
J is going to be one.

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And you uint
tensor is still going to be zero.

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Because we only have
a single packed tensor.

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So this time instead of shifting by zero
we're shifting by

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two bits
bits times j because j is equal to one.

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So in the second iteration.

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So this is first iteration.

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Second iteration.

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In the second iteration we're going
to shift that by two bits on the right.

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So, the shifted

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tensor would look like this.

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So if you shift something on the right

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the shifted values would be equal to zero.

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And then you're going
to perform a bitwise or.

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So since this is full zeros

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the unpack tensor would look like this.

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And then third iteration.

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So j is going to be equal to two.

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So we're going to shift this by four.

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All right.
So it's going to look like this.

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And we're going to do again
bitwise or between

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this value and our shifted value.

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Perfect. And then last iteration.

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All right. We're almost there.

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There is just one thing.

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So you may be wondering,
"we're not having exactly the same results

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as our target tensor",
which, again, would look like this, right?

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But now we have

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these bits that we want to remove
from our, unpacked tensors.

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So to do that, we're just going to perform
a simple trick.

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So, basically the idea is
that we're just going to create a mask

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and use that mask to perform
a bitwise operation.

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That will always give us here

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zero.

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And always make sure that these values
stay the same.

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So if we consider this mask
and let's say this value,

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if we perform a bitwise and operation

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between these two values,

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so one and 000100

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and so on zero everywhere.

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Except for the last two bits

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where you have 010 and 10111.

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So all the bits before these two bits

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that we want to keep
were correctly masked out a zero.

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And the two bits here are preserved.

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So that's exactly what we want to achieve.

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You can try that out on the other values.

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That's just to double check.

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But this mask should be sufficient
to remove all the bits

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before these, all of these two bits
that we are interested in keeping.

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And there is a general rule, where
you can compute the value of this mask.

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so this mask is simply three.

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So three in case of two bits is

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just two to the power
of two number of bits minus one,

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which corresponds
to the maximum value you can,

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reach within this number of bits.

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So if you encode in binary in two bits,
you can't go above three

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because you can only encode zero, one,
two and three.

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And this formula of course is extensible.

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So you can also try it with four eight

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and you can the same logic.

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So we can perform the masking operation
at the end of everything.

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So we're just going to perform
an and operation

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to mask out those values as explained.

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All right.

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So let's try that out on

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let's say this tensor.

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So this is the packed version
of the tensor.

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if we unpack it we should retrieve exactly

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this tensor.

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Perfect.

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It should be the same. Yeah.

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So yeah, if you compare both tensors,
they're exactly the same.

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1032 and 3333.

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Perfect.

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So. Yeah, we just did it now for two bits.

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We also, applied a very naive approach
where we,

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you know, did two, two, four loops.

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we also didn't consider the case
where the tensor is, has multiple shapes.

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Here we just considered a simple vector.

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But yeah, feel free to enhance
this algorithm.

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Make it faster.

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Try to also extend this algorithm

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so that it works on different shapes,
arbitrary shapes as well.

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So yeah, feel free to battle test
everything.

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Make sure that you have understood all
the internals that I have explained here.

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To wrap up the lesson,
we're just going to see other challenges

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when you quantize large
models, such as large language models.

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And yeah, we're going to wrap up
the whole course with these explanations.

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So, yeah let's move on to the slides.