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So far my understanding about kernel methods is that they are ways to map our features to a higher dimension space - allowing us to fit non-linear data using linear models.

I don't understand much more than that. I am not understanding how it is computationally faster nor how is it much different than just including polynomial or interaction terms in your basic regression model. Or just some other engineered feature. I am also not understanding when would you use what method to model non-linearity.

Is there a good resource to explain kernels from the ground up?

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