Hi,
Let me explain it step by step.

Here is kaiming_uniform_.
Wherenegative_slope=sqrt(5)
so thegain=sqrt(2/6)=1/sqrt(3)
for kaiming.
If we replace this inbound
formula, we getbound = [1/sqrt(3) ] * [sqrt(3/ fan_in)]
which with a little simplification, it will bebound = 1/sqrt(fan_in)
which can be represented bybound^2 = 1 / fan_in
. 
In linear implementation code you referenced:
So what we have here is that k= 1/in_feautres
which in case of kaming it can be represented k=1/fan_in
. Also, we want a boundary of [sqrt(k), sqrt(k)]
where k = bound^2= 1 / fan_in
from step 1.
For simplcity, just replace sqrt(5)
in gain
formula then optain bound
in kaiming_uniform_
and replace the bound
as k
in linear.
Edit: Add some related posts
 Kaiming init of conv and linear layers, why gain = sqrt(5) · Issue #15314 · pytorch/pytorch · GitHub
 Why the default negative_slope for kaiming_uniform initialization of Convolution and Linear layers is √5?
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