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

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

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)

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

Related works Mesh Laplacian Reweight the mesh Laplacian Shahar Z. Kovalsky, Meirav Galun, and Yaron Lipman. 2016. 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. 2008. A local/global approach to mesh parameterization. In Computer Graphics Forum, Vol. 27. Wiley Online Library, 1495–1504. Olga Sorkine and Marc Alexa. 2007. 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. 2017. 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

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

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

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

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

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

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:

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

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

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)

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. 2016. Bounded distortion harmonic shape interpolation. ACM Transactions on Graphics (TOG) 35, 4 (2016), 105.

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

Demo https://www.youtube.com/watch?v=zXF2_dEjQXI

Iteration count

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