# non-linear tikhonov/ridge regularization?

For traditional ridge regression, the loss function is

$$loss\_function = ||A\mathbf{x}-\mathbf{b}||_2^2 + ||\Gamma\mathbf{x}||_2^2$$

https://en.wikipedia.org/wiki/Tikhonov_regularization

Is there a matrix formalism to extend this such that $$\Gamma\mathbf{x}$$ is instead a non linear function of $$\mathbf{x}$$? That is

$$loss\_function = ||A\mathbf{x}-\mathbf{b}||_2^2 + ||f(r,\mathbf{x})||_2^2$$

Where $$f(r,\mathbf{x})$$ is a non linear function of $$\mathbf{x}$$ with $$r$$ being a set of tunable parameters.