Condition on Wasserstein Critic - Week 3: Wasserstein GANs with Gradient

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At 1:49, the instructors says that staying within the bounds of (-1,1) slope, means that the function is growing linearly.
Then at 2:08, the instructor shows the following function and says that it grows linearly:

This function doesn’t look linear to me as its gradient is not constant, and changes sign as well.

What does the instructor means by the function growing linearly?? Please help.

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Good question! I have to admit I never noticed she said that - and I’m not sure, either, why she used those words. You’re right, the curve is not growing linearly. I did notice that the second time she says, “the function never grows more than linearly” - so I’m guessing that she was thinking about those lines that are drawn with a slope of 1 and how the function should have a slope that’s less than that, and then she just made an unfortunate choice of words to summarize that.

Anyway, the main point is that for 1-L continuity, the slope should change slowly - with absolute value of the slope less than 1. You can ignore that mention of “growing linearly”

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Thank you! I was actually confused as to what that meant and was searching internet as to what linearly meant in this context. I am still learning a bit of mathematics so I wasnt sure what it meant in this context. Thank you for clarifying that the instructor just meant that “the function never grows more than linearly”.