1. Geometric Optimization via Composite Majorization
ANNA SHTENGEL, Weizmann Institute of Science
ROI PORANNE and OLGA SORKINE-HORNUNG, ETH Zurich
SHAHAR Z. KOVALSKY, Duke University
YARON LIPMAN, Weizmann Institute of Science
ACM Transactions on Graphics (SIGGRAPH) 2017
Presenter: Chuan-Wei Sun

2. Introduction Nonlinear minimization problem of the general form
Product a sequence of approximations, 𝑋 𝑛 until reach
minimum
of approximations, Xn for each round
And make it converge to a solution of the function

3. Introduction Construct an osculating convex quadric to f at 𝑋 𝑛
Determines the next approximation 𝑋 𝑛+1
Various choice of and osculating quadric (Hessian)

4. Newton’s method Archetypal Newton’s algorithm For non-convex functions
Hessian of f itself
Quadratic order of convergence for strictly convex functions
For non-convex functions
Hessian is indefinite

5. Related works Mesh Laplacian Reweight the mesh Laplacian
Shahar Z. Kovalsky, Meirav Galun, and Yaron Lipman Accelerated Quadratic Proxy for Geometric Optimization. ACM Trans. Graph. 35, 4, Article 134 (July 2016), 11 pages.
Ligang Liu, Lei Zhang, Yin Xu, Craig Gotsman, and Steven J Gortler A local/global approach to mesh parameterization. In Computer Graphics Forum, Vol Wiley Online Library, 1495–1504.
Olga Sorkine and Marc Alexa As-rigid-as-possible surface modeling. In Symposium on Geometry processing, Vol. 4.
Reweight the mesh Laplacian
Michael Rabinovich, Roi Poranne, Daniele Panozzo, and Olga Sorkine-Hornung Scalable Locally Injective Maps. ACM Transactions on Graphics (TOG) 36, 2 (2017), 16:1–16:16.
First order methods
Do not directly use second order derivatives of the energy
Enforcing positive semidefinitness heuristically

6. Goal Devise a second order optimization approach
Applicable to generic class of composite nonlinear energies

7. Majorization-Minimization
An optimization framework
Finding a fitting convex surrogate upper bound at each iteration (majorizer)
Replace optimization problem with a sequence of simpler optimization problems

8. Convex-Concave procedure
A subfamily of MM algorithms
Decompose the function: r = 𝑟 𝑟 −
𝑟 + : convex
𝑟 − : concave
Construct convex majorizer of r
replacing 𝑓 − with it’s linear approximation at 𝑋 0

9. Meta-algorithm Construct a convex majorizer to f centered at 𝑋 𝑛
Compute its Hessian at 𝑋 𝑛
Consider unconstrained case only

10. Summary of main results
Energies of the form
Where
Each decompose as
Construct a convex majorizer to f at 𝑋 0
Hessian

11. Derivation of majorizer and Hessian
Preliminaries:
For a function r with its convex-concave decomposition
Majorizer
Minimizer
𝑟 ( 𝑟 ) is:
convex (concave)
Coincide with r up to first order at x = 𝑥 0
Global majorizer (minimizer) of r:

12. Derivation of majorizer and Hessian
Omit summation
Fix 𝑥 0 , set 𝑢 0 = g( 𝑥 0 ), and let
Define the function [g] :
Proposed majorizer for f at 𝑥 0 :
Define
Majorizing domain

13. Proof
Proposition: Function is a convex majorizer of f at 𝑥 0 over the set
Is PSD
𝑓 coincides with f up to first order at x = 𝑥 0

14. Proof ℎ 𝑗 𝑢 𝑗 [ 𝑔 𝑗 ](x; 𝑥 0 ) 𝑔 𝑗 (x) for all x ∈ Ω (x0). Case -1:
< 0
for all x ∈ Ω (x0).
𝑢 𝑗
[ 𝑔 𝑗 ](x; 𝑥 0 )
𝑔 𝑗 (x)

15. Energies on triangular meshes
Piecewise linear mapping, from N vertices into vector x ∈
For each triangle i in the mesh
Energy
Edward Chien, Renjie Chen, and Ofir Weber Bounded distortion harmonic shape
interpolation. ACM Transactions on Graphics (TOG) 35, 4 (2016), 105.

16. Energies on triangular meshes
Symmetric Dirichlet energy
Symmetric as-rigid-as-possible (ARAP) energy

17. Demo

18. Iteration count

20. Discussion Pros Cons or limitation in this paper High convergence rate
Avoid enforcing positive semidefinitness heuristically
Cons or limitation in this paper
Majorizer depends on the particular decomposition
Proposed Hessian is not coincide with the true Hessian
Optimality or Quality of H are ill-defined