[SOLVED] ECE509 Homework5-Non-smooth function

1. Consider the following non-smooth function on the real line:

f(x) = max!

Describe the subdifferential ∂f(x) at every point x ∈R.

2. Prove or disprove: the subdifferential ∂f(x) of a convex function is a convex set at every x ∈R.
3. Recall the subdifferential for the nuclear norm for an n₁ × n₂ matrix X.

∂$X$” = #UV ^T + W : UW = 0,WV = 0,$W$ ≤ 1$

In the expression above, X has rank r and its SVD is X = UΣV ^T, where U is n₁ × r, Σ is r × r and V is n₂ × r. Recall that

x⁺ = prox_t!c_dot!!(X) = argmin

Z

if and only if

X−X⁺ ∈ t∂’’X⁺’’“

Show that we can compute the prox operator above by singular value thresholding:

X max(σ_it,0).

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