HuggingFace.Quantization_in_Depth/Transcript/6_Per Channel Quantization_HFQD.txt

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Now let's
move on to the per channel quantization.

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We need to store the scales
and the zero point for each row

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if we decide to quantize along the rows
and we need to store them

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along each column, if we decide
to quantize along the columns.

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The memory needed to store all these
linear parameters is pretty small.

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We usually use per channel quantization
when quantizing models in eight-bit.

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You will see that in the next lesson
we use units.

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Now let's call
the per channel quantization.

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And don't worry about this slide.

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We'll do it in the notebook.

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So let's code the per channel quantization.

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To simplify the work we will just restrict

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ourselves to the symmetric mode
of the linear quantization.

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So the function will be called linear q
symmetric per channel.

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So we expect this function
to take as arguments the tensor.

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The dimension.

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So whether we want to quantize
along the rows

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or the columns,
if we are talking about the 2D matrix,

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we set the default value
to be equal to torch.int8.

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And at the end we expect
to get the quantized tensor and the scale.

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We don't need the zero point since we are
trying to do it with the symmetric mode.

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So let's define a test tensor

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so that we can work through the code.

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We will use the test tensor
that we defined previously.

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The first step is to know
how big the scale tensor will be.

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Since we are doing per channel,
we will be having more than one scale

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and we need to create a tensor
to store these values.

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So the shape of the tensor scale
would be equal

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to that.

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Tensor.

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shape.

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And with a specific dimension we need
to set the dimension to be equals to zero.

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If we want to quantize along the rows.

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Otherwise we need to set it to one.

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If we want to quantize along the columns.

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So let's check the output dimension.

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As you can see we get three.

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And indeed we need three scales value.
One for these numbers.

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This one. And the last one is this one.

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Now we can create the scale
tensor using torch.zeros.

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This will create a tensor
with the shape output dimension

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and each element will be equal to zero.

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Let's have a look.

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And indeed we get that.

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Now what we need to do is to iterate

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through each one of these rows

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and calculate the scale
for each one of them.

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To do that,
we will loop over the output dimension.

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Now we need to get a sub row.

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For example the first row,
the second row or the third row.

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To do that we will use the select method

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and we need to set two arguments.

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First, the dimension and second, the index.

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And now just to be sure let's
check out what the sub tensor looks like.

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We should have a tensor for each row.

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And indeed we were able

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to extract each row in a tensor.

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Now that we manage to get the sub tensor,
all we need to do is to apply

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the get q scale
symmetric function to that sub tensor

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in order to get the scale
related to that row

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and store it inside the scale tensor.

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So we need to set

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in the index position of the tensor scale

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the scale for that particular sub tensor.

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To do that, we will use the get q scale
symmetric function and we just pass

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the set answers.

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Let's check now what scale look like.

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We did manage to

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store the scales
related to each row inside this tensor.

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Now that we have stored all scales,
we need to do a little bit of processing

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in order to reshape the scale
so that when we divide the original tensor

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by the tensor scale, each column
is divided by the correct scale.

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To do that, we define the shape
that the scale tensor should have.

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Let's have a look at this scale shape.

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It's full of one.

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Then we need to set the scale shape

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at index in to be equal to minus one,
which will give us that.

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And the last thing we need to do
is to reshape the scale using the view

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method, using the scale shape
that we just defined.

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And we get the following scale.

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And this is the scale
we need in order to be able

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to divide the original tensor
by the tensor scale.

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So that each row is divided
by each value of the scale.

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I know this is a bit complex,
since it involves how to divide tensor

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by tensors in PyTorch.

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Let's have a look at an example.

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In order to understand how a view works
and how to divide tensor by tensor

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in such way that you divide
each rows or each columns.

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Let's say we have the following matrix.

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And we have the following scale.

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Just like in the previous example,

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the shape of the scale is three.

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We can reshape that tensor in such a way
that the first dimension is of size one,

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and the second dimension
can contain the rest.

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To do that, we can use the view function.

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The shape of the of the scale is three.

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We can reshape that tensor
using the view function.

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For example, we can reshaped
so that the first dimension

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is one and the second dimension is three.

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And as expected,
we get a tensor of size one by three.

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An alternative way to do that
is just to replace the three by minus one.

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What this does is that view will be able
to find the right shape

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where you put the minus one.

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You can also reshape S
so that the first dimension

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will end up with being three,
and the last dimension to be one.

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By doing that.

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Now let's try to divide the matrix
M along the row.

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So, the scale we need in order to divide
each row is this one.

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As you can see,
the scale shape is the following.

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We have three as the first dimension
and one as the second dimension.

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And let's perform the division.

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And as you can see,

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we managed to, divide along the rows.

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You can see that this rule was untouched
since it's always divided by one.

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The second one was divided by five,

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and the last one,
the third one was divided by ten.

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If we use the following scale instead.

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With the following shape.

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One by three

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and we divide the matrix
by the specific scale, we see that

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in this case
we divided each column by the scales.

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So here
as you can see this column is untouched.

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We have one four and seven.

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The second column was divided by five
and the last column was divided by ten.

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Now let's go back to quantizing our tensor.

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If you remember well the scale
that we got at the end was the following.

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And if we check

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the shape of the scale,

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this is the right shape for the scale
in order to quantize each row.

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Now all we need to do is to quantize

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the tensor by,

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using the linear shape
q scale and zero point

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function
that we called it in the previous lab.

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And we just need to pass the test
tensor, the scale,

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and the zero point
which should be equal to zero.

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Since we are doing symmetric quantization.

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And as you can see, we end up
with the following quantized tensor.

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Now let's put everything we did
in a function

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called linear q symmetric per channel.

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Okay.

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As you can see here
we get the output dimension.

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We create the scale
tensor with the output dimension shape.

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We iterate through the output dimension.

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And for each index we get the sub tensor

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and we store the scale
in the index position.

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Then we reshape the scale

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here.

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Lastly, we get the quantized tensor
using the linear query

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scale and zero point function.

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And that's it.

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We get the quantized tensor and the scale.

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Now that we have our function, let's check

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if we were indeed able to quantize
along a specific dimension.

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So I replaced the test tensor
that we defined earlier.

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And this time
we will quantize along the first dimension

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and the second dimension.

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So we'll have the quantized tensor zero
and the scale zero.

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We get that by using the linear
symmetric per channel function.

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And we need to pass the test tensor.

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And we need to precise

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that the dimension that we are
quantizing is zero.

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Let's do the same for the other dimension.

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So we'll call it

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quantized in cell
one and scale underscore one.

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To get the summary
we need also to dequantize each tensor.

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So let's first do the case
where the dimension is equal to zero.

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We have the dequantized
tensor underscore zero

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which is equals to linear dequantization.

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And we need to precise

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the quantized tensor on a scale zero

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its scale and zero.

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Since the zero point is equal to zero.

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Now we have everything to get the summary

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using the plot quantization
error function.

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And that's it.

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As you can see,
we indeed quantized along the rows.

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You can see that
we have the maximum quantized value here

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127 here, here and here.

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And the quantization was pretty good.

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As you can see, the original tensor
is pretty close to the dequantized tensor.

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And that the quantization error
tensor is not so bad.

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Let's have a better metric
by computing the quantization error.

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And we get a quantization error of 1.8.

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If we remember well when we did
the potential symmetric linear

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quantization,
we had the quantization error around 2.5.

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Now let's check what happens
if we quantize along the columns.

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We'll do the same thing as we did as a

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but with the quantized tensor
underscore one.

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So as you can see here
we define the dequantized tensor

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underscore one
by using the linear dequantization.

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And we passed the quantization
into underscore one and scale one.

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And then we plot the quantization error.

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This will give us the following summary.

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And as you can see here, we managed
indeed to quantize along the columns.

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This time
the quantization error is even lower.

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You see that we get a lower quantization
error in both cases

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compared to tensor quantization.

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This is because outlier values
will only impact the channel

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it was in, instead of the entire tensor.