Implementation of Fused Ridge ( "squared l2norm.linear" ) / rr.squared_error()

[sehoff]

How can I implement a "squared l2norm.linear" ?

Example implementation for Fused Lasso:

    import regreg.api as rr
    loss = rr.squared_error(X,Y)
    sparsity = rr.l1norm(p, lagrange=1)
    fused = rr.l1norm.linear(D, lagrange=10)
    problem = rr.container(sparsity, fused, loss)

, where D is the difference operator that penalizes first differences in the coefficient vector.
--> So, what I would need is to replace fused in the example above with something along the lines fused = rr.l2norm.linear(D, lagrange=10)^2, where I put the "^2" here for exposition but I am aware that this does not work.

Workaround:
I found a workaround, where I map the sparse fused ridge problem into a lasso problem, by artificially expanding my X and Y matrices (Lemma 1 in Hebiri, van de Geer (2011)). Now I can use the normal lasso to solve the problem.
However, to create said artificial matrices I need to know whether the rr.squared_error() indeed 'only' minimizes 1/2 (Y-Xbeta)^2, as documented, or internally controls for the sample size like 1/2N (Y-Xbeta)^2?

EDIT:
Is the following a possible solution:

    import regreg.api as rr
    loss = rr.squared_error(X,Y)
    sparsity = rr.l1norm(p, lagrange=1)
    fused = rr.quadratic_loss(p, Q=D, coef=lambda)
    problem = rr.container(sparsity, fused, loss)

where D is the difference operator that penalizes first differences in the coefficient vector. I then control the amount of penalization by choosing different values for coef ?!
Thanks in advance!