Is Sum of Absolute Value / $ {L}_{1} $ Norm of Differences Convex? I'm not sure how to approach this exercise.
One idea is to derive it w.r.t z, show that there is a min-extremum at $z=f_k$ and then show that for each value from the right and the left of the loss function it is positive which will prove that it is convex.
I never worked with Dirichlet functions, I'm not sure how should I show that it is convex analytically. Can somebody give me a clue/solution path ?
It is easier to prove it by using atoms.
The 1st atom is the Absolute Value function $ \left| \cdot \right| $ which is convex.
Then you have linear operation by the subtraction which is convex (Also concave).
Then you linear combination which is also Convex.
Hence the function is Convex.