# Connection Between Orthogonal Projection onto the Unit Simplex and the Softmax Function

Referring to papers Softmax to Sparsemax and Efficient Projections onto the L1-Ball, what is the relationship between a euclidean projection onto the probability simplex and applying the Softmax function? Both resulting vectors $\boldsymbol{w}$ will satisfy the constraint $\sum_{i}w_{i}=1$, but clearly Softmax is not idempotent and therefore not a projection.

I am also interested in projecting to the L1-ball, where $||\boldsymbol{w}||_{1}\leq 1$. Is there an equivalent function (even if not a projection) that can be applied in the same sense as the Softmax in the first part of this question?

Also, in the context of constrained optimisation, is $||\boldsymbol{w}||_{1}\leq 1$ a more relaxed constraint to $\sum_{i}w_{i}=1$?

I can see one direct connection between the actual projection onto the unit simplex and the softmax function. The softmax function is the result of the first iteration of applying the Mirror-Descent algorithm with the Bregman Distance $D_\phi$ generated by $\phi(x) = \sum_{i=1}^n x_i \ln(x_i)$ to the problem $\min_x \{ ||y - x||_2^2 : \sum_{i=1}^n x_i = 1, x \geq 0 \}$, which defines the projection onto the unit simplex.