A Useful properties of the Moreau envelope and proximal map 14 A.1 Relation between proximal map and Moreau envelope . . . . . . . . . . . . . . . . 14 A.2 Relation between proximal map and derivative . . . . . . . . . . . . . . . . . . . . 15 A.3 Inverse of the Moreau envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.4 Moreau envelope for additive noise model . . . . . . . ...

In this paper, we give a fundamental convexity preserving for spectral functions. Indeed, we investigate infimal convolution, Moreau envelope and proximal average for convex spectral functions, and show that this properties are inherited from the properties of its corresponding convex function. This results have many applications in Applied Mathematics such as semi-definite programmings and eng...

We present several sequential exact Euclidean distance transform algorithms. The algorithms are based on fundamental transforms of convex analysis: The Legendre Conjugate or Legendre–Fenchel transform, and the Moreau envelope or Moreau-Yosida approximate. They combine the separability of the Euclidean distance with convex properties to achieve an optimal linear-time complexity. We compare them ...

Recent work has shown that stochastically perturbed gradient methods can efficiently escape strict saddle points of smooth functions. We extend this body to nonsmooth optimization, by analyzing an inexact analogue a method applied the Moreau envelope. The main conclusion is variety algorithms for optimization envelope at controlled rate. technical insight many proximal subproblem yield directio...

We prove that the projected stochastic subgradient method, applied to a weakly convex problem, drives the gradient of the Moreau envelope to zero at the rateO(k−1/4).

In the recent paper [3], it was shown that the stochastic subgradient method applied to a weakly convex problem, drives the gradient of the Moreau envelope to zero at the rate O(k−1/4). In this supplementary note, we present a stochastic subgradient method for minimizing a convex function, with the improved rate Õ(k−1/2).

2 Deriving inference error from the statistical physics of disordered systems 2 2.1 Replica equations at finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Replica equations in the low temperature limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Moreau envelope formulation of the replica equations . . . . . ....