[Help] Proximal mapping
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- Properties of Proximal mapping
- 1 L1 Lipschitz and monotone
- 2 Projection Property
- 3 Scaling and translation argument
1. Properties of Proximal mapping
For a convex function
From the fact that the objective function is strictly convex, we know that
1.1 L1 Lipschitz and monotone
Theorem: Proximal mapping for any convex function
- Lipschitz:
∥Proxh(x)−Proxh(y)∥2≤∥x−y∥2. - Monotone:
(Proxh(x)−Proxh(y))T(x−y)≥0.
proof: From the definition of proximal mapping, if
Then from the convexity,
we have
which means that
Then,
1.2 Projection Property
Theorem: Proximal mapping for any convex function
proof: If
from which, we have
So
1.3 Scaling and translation argument
Theorem: Let
proof: Assume
Then
so we have
so
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