Table 3.
Solvers for finding a minimizer of the generalized PWC noise-removal functional in table 2. The first column is the solver algorithm, the second is the different PWC methods to which the technique can be applied in theory.
| solver | can apply to | notes |
|---|---|---|
| analytical convolution | linear diffusion | problems with only square loss functions are analytical in a similar way |
| linear programming (Boyd & Vandenberghe 2004) |
robust total variation regularization |
direct minimizer of functional; also all piecewise linear convex problems |
| quadratic programming (Boyd & Vandenberghe 2004) |
total variation regularization convex clustering shrinkage |
direct minimizer of functional; also all problems that combine square likelihood with absolute regularization loss |
| stepwise jump placement (Gill 1970; Kerssemakers et al. 2006; Kalafut & Visscher 2008) |
step-fitting objective step-fitting jump penalization robust jump penalization |
greedy spline fit minimizer of functional |
| finite differencing (Mrazek et al. 2006) |
total variation regularization total variation diffusion convex clustering shrinkage mean shift clustering likelihood mean shift clustering soft mean shift clustering soft K-means clustering |
finite differences are not guaranteed to converge for non-differentiable loss functions |
| coordinate descent (Friedman et al. 2007) |
total variation regularization robust total variation regularization |
|
| iterated mean replacement (Cheng 1995) |
mean shift clustering likelihood mean shift clustering |
obtainable as adaptive step-size forward Euler differencing |
| weighted iterated mean replacement (Cheng 1995) |
soft mean shift clustering soft likelihood mean shift clustering |
obtainable as adaptive step-size forward Euler differencing |
| piecewise linear regularization path follower (Rosset & Zhu 2007; Hofling 2009) |
total variation regularization convex clustering shrinkage |
|
| least-angle regression path follower (Tibshirani & Taylor 2010) |
total variation regularization | reverse of piecewise linear regularization path follower |