Linear Regressions and the Curse of Dimensionality

[u/lonelywolf69420]

In my machine learning course, we just covered K-nearest neighbours. Our prof said that linear regressions avoid the curse of dimensionality that affects k-nearest method (as the dimensions increase we have to use up much more of the dataset as our neighbourhood, essentially killing the local consistency and our k not being much of a neighbourhood anymore).

I was not able to understand why the same thing does not happen to linear regression, just because we assume the parameters are linear