What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications

@article{Lucet2009WhatSI,
  title={What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications},
  author={Yves Lucet},
  journal={SIAM Rev.},
  year={2009},
  volume={52},
  pages={505-542}
}
  • Y. Lucet
  • Published 1 March 2009
  • Computer Science, Mathematics
  • SIAM Rev.
Computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolic-numeric algorithms. Our objective is to disseminate widely the most efficient numerical algorithms useful for… 

Computing the conjugate of convex piecewise linear-quadratic bivariate functions

TLDR
A new algorithm to compute the Legendre–Fenchel conjugate of convex piecewise linear-quadratic (PLQ) bivariate functions and compose these operators to compute classical convex analysis operators such as the Moreau envelope, and the proximal average.

Graph-Matrix Calculus for Computational Convex Analysis

  • Bryan GardinerY. Lucet
  • Computer Science, Mathematics
    Fixed-Point Algorithms for Inverse Problems in Science and Engineering
  • 2011
We introduce a new family of algorithms for computing fundamental operators arising from convex analysis. The new algorithms rely on the fact that the graph of the subdifferential of most convex

Computing the partial conjugate of convex piecewise linear-quadratic bivariate functions

TLDR
This work modifications a recent algorithm for computing the convex (Legendre-Fenchel) conjugate of convex PLQ functions of two variables, to compute its partial conjugates, which is more flexible and simpler than the original full conjugating algorithm.

Convex Hull Algorithms for Piecewise Linear-Quadratic Functions in Computational Convex Analysis

TLDR
This work presents two algorithms, one based on maximum and conjugate computation that is easy to implement but has quadratic time complexity, and another based on direct computation that requires more work to implementbut has optimal (linear time) complexity.

Tropical Geometry and Piecewise-Linear Approximation of Curves and Surfaces on Weighted Lattices

TLDR
This chapter provides the optimal solution of max-plus and min-plus equations using morphological adjunctions that are projections on weighted lattices, and applies it to optimal piecewise-linear regression for fitting max-$\star$ tropical curves and surfaces to arbitrary data that constitute polygonal or polyhedral shape approximations.

Tropical Geometry and Machine Learning

TLDR
This article summarizes introductory ideas and objects of tropical geometry, providing a theoretical framework for both the max-plus algebra that underlies tropical geometry and its extensions to general max algebras, and provides optimal solutions and an efficient algorithm for the convex regression problem.

Computing the Subdifferential of Convex Piecewise Linear-Cubic Functions

TLDR
A method is introduced that manipulates the PLCVC data structure in the CCA toolbox to plot an unbounded domain within a specified window and a class that implements a method computing the subdifferential at any point of a PLC function is introduced.

Techniques and Open Questions in Computational Convex Analysis

  • Y. Lucet
  • Computer Science, Mathematics
  • 2013
We present several techniques that take advantage of convexity and use optimal computational geometry algorithms to build fast (log-linear or linear) time algorithms in computational convex analysis.

Mathematical Morphology and Its Applications to Signal and Image Processing

TLDR
This paper outlines the optimal solution of maxequations using weighted lattice adjunctions, and applies it to optimal regression for fitting maxtropical curves on arbitrary data.

Hybrid symbolic‐numeric algorithms for computational convex analysis

  • Y. Lucet
  • Mathematics, Computer Science
  • 2007
TLDR
Developing efficient tools to compute fundamental transforms arising in convex analysis and allowing more insight into the calculation of the Fenchel conjugate and related transforms are focused on.
...

References

SHOWING 1-10 OF 345 REFERENCES

Differential morphology and image processing

  • P. Maragos
  • Computer Science, Mathematics
    IEEE Trans. Image Process.
  • 1996
TLDR
The analysis of the multiscale morphological PDEs and of the eikonal PDE solved via weighted distance transforms are viewed as a unified area in nonlinear image processing, which is called differential morphology, and its potential applications to image processing and computer vision are discussed.

Computational convex analysis : from continuous deformation to finite convex integration

TLDR
A computational framework for computer-aided convex analysis is introduced and a method to find primal-dual symmetric antiderivatives from cyclically monotone operators is discussed, which can be formulated as shortest path problems.

Convex Hull Algorithms for Piecewise Linear-Quadratic Functions in Computational Convex Analysis

TLDR
This work presents two algorithms, one based on maximum and conjugate computation that is easy to implement but has quadratic time complexity, and another based on direct computation that requires more work to implementbut has optimal (linear time) complexity.

New sequential exact Euclidean distance transform algorithms based on convex analysis

  • Y. Lucet
  • Computer Science, Mathematics
    Image Vis. Comput.
  • 2009

The Hamilton-Jacobi skeleton

TLDR
A new algorithm for simulating the eikonal equation is introduced, which offers a number of computational and conceptual advantages over the earlier methods when it comes to shock tracking and a very efficient algorithm for shock detection.

PDEs for Morphological Scale-Spaces and Eikonal Applications

Minimizing within Convex Bodies Using a Convex Hull Method

TLDR
This paper presents numerical methods to solve optimization problems on the space of convex functions or among convex bodies and gives approximate solutions better than the theoretical known ones, hence demonstrating that the minimizers do not belong to these classes.

The piecewise linear-quadratic model for computational convex analysis

TLDR
The main result states the existence of efficient (linear time) algorithms for the class of piecewise linear-quadratic functions for which such class is closed under convex transforms.

A Linear Time Algorithm for Computing Exact Euclidean Distance Transforms of Binary Images in Arbitrary Dimensions

TLDR
The algorithm, which is based on dimensionality reduction and partial Voronoi diagram construction, can be used for computing the DT for a wide class of distance functions, including the L/sub p/ and chamfer metrics.

Mathematical morphology: The Hamilton-Jacobi connection

TLDR
It is asserted that mathematical morphology operations with a convex structuring element are captured by a differential deformation of the boundary along the normal, governed by a Hamilton-Jacobi partial differential equation (PDE).
...