FEAT - Add quadratic datafit with no access to the target

[QB3]

Description of the feature

Exact feature

Solve the following optimization problem

$$\arg \min_{\beta} \frac{1}{2n} || X \beta ||^2 + \frac{1}{n} \beta^\top X^\top y + \text{penalty} \enspace,$$

with no access to $y$, but with access to $X^\top y$.

Additional context

Context, I have been discussing with @shz9 to implement a specific datafit for genomic applications (@shz9 is finishing his PhD on statistical analysis of genomics data). From what I understood, genomics data are sensitive: one does not have access to the target $y$: one only has access to the design matrix $X$, and an estimation of $X^\top y$ (usually estimated from another dataset).

Steps

I guess we have to add the datafit $$\frac{1}{2n} || X \beta ||^2 + \frac{1}{n} \beta^\top X^\top y \enspace ,$$ and handle the fact there is no $y$ provided.