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qmm 0.18.2

Q-MM: A Python toolbox for Quadratic Majorization-Minimization

Q-MM is a Python implementation of Majorize-Minimize Quadratic
optimization algorithms. Algorithms provided here come from
[1] C. Labat and J. Idier, “Convergence of Conjugate Gradient Methods with a
Closed-Form Stepsize Formula,” J Optim Theory Appl, p. 18, 2008.

and
[2] E. Chouzenoux, J. Idier, and S. Moussaoui, “A Majorize–Minimize Strategy
for Subspace Optimization Applied to Image Restoration,” IEEE Trans. on
Image Process., vol. 20, no. 6, pp. 1517–1528, Jun. 2011, doi:
10.1109/TIP.2010.2103083.

See documentation for more
background. If you use this code, please cite the references above and a
citation of this toolbox will also be appreciated, see below. You
can also click ⭐ on the repo.
Quadratic Majorize-Minimize
The Q-MM optimization algorithms compute the minimizer of objective
function like
J(x) = ∑ₖ μₖ ψₖ(Vₖ·x - ωₖ)
where x is the unknown vector, Vₖ a linear operator, ωₖ a fixed data, μₖ
a scalar, ψₖ(u) = ∑ᵢφₖ(uᵢ), and φₖ a function that must be
differentiable, even, coercive, φ(√·) concave, and 0 < φ'(u) / u <
+∞.
The optimization is done thanks to quadratic sugorate function. In
particular, no linesearch or sub-iteration is necessary, and close form
formula for the step are used with guaranteed convergence.
A classical example, like in the figure below that show an image
deconvolution problem, is the resolution of an inverse problem with the
minimization of
J(x) = ||² + μ ψ(V·x)
where H is a low-pass forward model, V a regularization operator that
approximate gradient (kind of high-pass filter) and ψ an edge preserving
function like Huber. The above objective is obtained with k ∈ {1, 2},
ψ₁(·) = ||², V₁ = H, ω₁ = y, and ω₂ = 0.

Features

The mmmg, Majorize-Minimize Memory Gradient algorithm. See
documentation and [2] for details.
The mmcg, Majorize-Minimize Conjugate Gradient algorithm. See
documentation and [1] for details.
No linesearch: the step is obtained from a close form formula
without sub-iteration.
No conjugacy choice: a conjugacy strategy is not necessary
thanks to the subspace nature of the algorithms. The mmcg
algorithm use a Polak-Ribière formula.
Generic and flexible: there is no restriction on the number of
regularizer, their type, ..., as well as for data adequacy.
Provided base class for objectives and losses allowing easy and fast
implementation.
Just one file if you like quick and dirty installation, but
available with pip.
Comes with examples of implemented linear operator.

Installation and documentation
Q-MM is essentially just one file qmm.py. We recommend using poetry
for installation
poetry add qmm

The package can also be installed with pip. More options are described
in the documentation.
Q-MM only depends on numpy and Python 3.6.
Example
The demo.py presents an example on image deconvolution. The first step
is to implement the operators V and the adjoint Vᵀ as callable
(function or methods). The user is in charge of these operators and
these callable must accept a unique Numpy array x and a unique return
value
(partial
in the functools module in the standard library is usefull here).
There is no constraints on the shape, everything is vectorized
internally.
After import of qmm, user must instantiate Potential objects that
implement φ and Objective objects that implement μ ψ(V·x - ω)
import qmm
phi = qmm.Huber(delta=10) # φ

data_adeq = qmm.QuadObjective(H, Ht, HtH, data=data) # ||y - H·x||²
prior = qmm.Objective(V, Vt, phi, hyper=0.01) # μ ψ(V·x) = μ ∑ᵢ φ(vᵢᵗ·x)

Then you can run the algorithm
res = qmm.mmmg([data_adeq, prior], init, max_iter=200)

where [data_adeq, prior]{.sourceCode} means that the two objective
functions are summed. For more details, see
documentation.
Contribute

Source code: https://github.com/forieux/qmm
Issue tracker: https://github.com/forieux/qmm/issues

Author
If you are having issues, please let us know
orieux AT l2s.centralesupelec.fr
More information about me here. F. Orieux and
R. Abirizk are affiliated to the Signal and Systems Laboratory
L2S.
Citation
Q-MM has a DOI with Zenodo
.
Specific version can also be cited. Citation can be
François Orieux, & Ralph Abirizk. (2022). Q-MM: The Quadratic Majorize-Minimize
Python toolbox (v0.12.0). Zenodo. https://doi.org/10.5281/zenodo.6373070

A example of bibtex is
@software{francois_orieux_2022_6373070,
author = {François Orieux and Ralph Abirizk},
title = {Q-MM: The Quadratic Majorize-Minimize Python toolbox},
month = mar,
year = 2022,
publisher = {Zenodo},
version = {0.12.0},
doi = {10.5281/zenodo.6373069},
url = {https://doi.org/10.5281/zenodo.6373069}
}

Acknowledgement
Author would like to thanks J.
Idier, S.
Moussaoui
and É. Chouzenoux. É.
Chouzenoux has also a Matlab package that implements 3MG for image
deconvolution that can be found on her
webpage.
License
The project is licensed under the GPLv3 license.