Sum penalty naming

[idc9]

What should we name the following two classes of penalties for problems of the form

min_{coef} L(coef) + p(coef)

Perhaps someone has named these already?

Additive/Infimal sum

p(coef) = min_{b_1, ..., b_q s.t. coef = sum_{j=1}^q b_j}  sum_{j=1}^q b_j p(b_j)

which is equivalent to solving

min_{b_1, ..., b_q}  L(sum_{j=1}^q b_j) + sum_{j=1}^q b_j p(b_j)

E.g. this is the low rank + sparse of (Candes et al., 2011), row sparse plus entrywise sparse of (Tan el al., 2014) or overlapping group lasso of (Jacob et al., 2009)

Ideas

1. Additive sum
2. Infimal sum

Infimal sum feels most accurate e.g. it is close to the "infimal convolution" operation -- see Section 2.3.2 of FOMO

Fully overlapping sum

p(coef) = sum_{j=1}^q b_j} p_j (b_j)

e.g. this is the sparse group or sparse fused lasso

Ideas:

1. Fully overlapping sum