There does not seem to be a lot of literature, but here is a nice geometric illustration of what is going on with negative ridge.

I will consider estimators of the form $$\hat{\boldsymbol\beta}_\lambda = (\mathbf X^\top \mathbf X + \lambda \mathbf I)^{-1}\mathbf X^\top\mathbf y$$ arising from the loss function $$\mathcal L_\lambda = \|\mathbf y - \mathbf X\boldsymbol\beta\|^2 + \lambda \|\boldsymbol\beta\|^2.$$ Here is a rather standard illustration of what happens in a two-dimensional case with $\lambda\in[0,\infty)$:

[![enter image description here][1]][1]

Now consider what happens when $\lambda\in(-\infty, -s^2_\max)$, where $s_\mathrm{max}$ is the largest singular value of $\mathbf X$. For very large negative lambdas, $\hat{\boldsymbol\beta}_\lambda$ is of course close to zero. When lambda approaches $-s^2_\max$, the term $(\mathbf X^\top \mathbf X + \lambda \mathbf I)$ gets one singular value approaching zero, meaning that the inverse has one singular value going to minus infinity. This singular value corresponds to the first principal component of $\mathbf X$, so in the limit one gets $\hat{\boldsymbol\beta}_\lambda$ pointing in the direction of PC1 but with absolute value growing to infinity.

What is really nice, is that one can draw it on the same figure: betas are given by points where circles touch the ellipses *from the inside*. **I do not really understand why this works** (the loss function interpretation is lost when $\lambda<0$), but it does:

[![enter image description here][2]][2]

When $\lambda\in[-s^2_\mathrm{min},0]$, a similar logic applies, allowing to continue the ridge path on the other side of the OLS estimator. Now the circles touch the ellipses from the inside. In the limit betas approach the PC2 direction (it happens far outside this sketch):

[![enter image description here][3]][3]

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The $\lambda\in(-\infty, -s^2_\max)$ range has an interpretation in terms of constrained ridge regression, see https://stats.stackexchange.com/questions/340099. This is related to what is known as "continuum regression", see my answer in the linked thread. I am currently not aware of any loss function interpretation for the $\lambda\in[-s^2_\mathrm{min},0]$ range.


  [1]: https://i.sstatic.net/aYCv8.png
  [2]: https://i.sstatic.net/VuIJs.png
  [3]: https://i.sstatic.net/bOZTl.png