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Justin Solomon
MIT
Sebastian Claici
MIT
Justin Solomon
MIT
Sebastian Claici
MIT
Justin Solomon Sebastian Claici MIT MIT Justin Solomon Sebastian - - PowerPoint PPT Presentation
Justin Solomon Sebastian Claici MIT MIT Justin Solomon Sebastian - - PowerPoint PPT Presentation
Justin Solomon Sebastian Claici MIT MIT Justin Solomon Sebastian Claici MIT MIT Client Which optimization tool is relevant? Designer Can I design an algorithm for this problem? Patterns, algorithms, & examples common in
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Client
Which optimization tool is relevant?
Designer
Can I design an algorithm for this problem?
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Optimization is a huge field.
Patterns, algorithms, & examples common in geometry processing.
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Image from “Isometry-Aware Preconditioning for Mesh Parameterization” Claici, Bessmeltsev, Schaefer, & Solomon
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Difficult because:
- Distortion is nonlinear/nonconvex
- Parameterization must be one-to-one
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One contribution per triangle
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Vocabulary Simple examples Unconstrained optimization Equality-constrained
- ptimization
Part I (Justin)
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Inequality constraints Advanced algorithms Examples Conclusion
Part II (Sebastian)
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Vocabulary Simple examples Unconstrained optimization Equality-constrained
- ptimization
Part I (Justin)
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Objective (“Energy Function”)
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Equality Constraints
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Inequality Constraints
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Gradient
https://en.wikipedia.org/?title=Gradient
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Jacobian
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
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Hessian
http://math.etsu.edu/multicalc/prealpha/Chap2/Chap2-5/10-3a-t3.gif
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Critical point
(unconstrained)
Saddle point Local min Local max
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Image from “Interactive Exploration of Design Trade-Offs” Schulz, Wang, Grinspun, Solomon, & Matusik
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Vocabulary Simple examples Unconstrained optimization Equality-constrained
- ptimization
Part I (Justin)
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How effective are generic
- ptimization tools?
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How effective are generic
- ptimization tools?
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Try the
simplest solver first.
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(assume A is symmetric and positive definite)
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Normal equations (better solvers for this case!)
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- G. Peyré, mesh processing course slides
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𝒙𝒋𝒌 ≡ 𝟐: Tutte embedding 𝒙𝒋𝒌 from mesh: Harmonic embedding
Assumption: 𝒙 symmetric.
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Never construct 𝑩−𝟐 explicitly
(if you can avoid it)
Added structure helps
Sparsity, symmetry, positive definiteness
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Direct (explicit matrix)
- Dense: Gaussian elimination/LU, QR for least-squares
- Sparse: Reordering (SuiteSparse, Eigen)
Iterative (apply matrix repeatedly)
- Positive definite: Conjugate gradients
- Symmetric: MINRES, GMRES
- Generic: LSQR
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Induced by the connectivity of the triangle mesh. Iteration of CG has local effect ⇒ Precondition!
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What if 𝑾𝟏 = {}?
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Mullen et al. “Spectral Conformal Parameterization.” SGP 2008. Easy fix
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Prevents trivial solution 𝒚 ≡ 𝟏. Extract the smallest eigenvalue.
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Roughly:
1.Extract Laplace-Beltrami eigenfunctions: 2.Find mapping matrix (linear solve!):
Ovsjanikov et al. “Functional Maps.” SIGGRAPH 2012.
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Vocabulary Simple examples Unconstrained optimization Equality-constrained
- ptimization
Part I (Justin)
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Unstructured.
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Gradient descent
Line search Multiple optima!
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Accelerated gradient descent
Quadratic convergence on convex problems!
(Nesterov 1983)
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Newton’s Method
1 2 3
Line search for stability
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Quasi-Newton: BFGS and friends
Hessian approximation
(Often sparse) approximation from previous
samples and gradients
Inverse in closed form!
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Examples:
- “Accelerated Quadratic Proxy for
Geometric Optimization” (Kovalsky et al. 2016)
- “Scalable Locally Injective Maps”
(Rabinovich et al. 2016)
- “Isometry-Aware Preconditioning
for Mesh Parameterization” (Claici et al. 2017)
- …several others
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“Accelerated Quadratic Proxy for Geometric Optimization” (Kovalsky et al. 2016)
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Fröhlich and Botsch. “Example-Driven Deformations Based on Discrete Shells.” CGF 2011.
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Roughly:
- 1. Linearly interpolate edge lengths and dihedral
angles.
- 2. Nonlinear optimization for vertex positions.
Sum of squares: Gauss-Newton
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Matlab: fminunc or minfunc C++: libLBFGS, dlib, others
Typically provide functions for function and gradient (and optionally, Hessian).
Try several!
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Vocabulary Simple examples Unconstrained optimization Equality-constrained
- ptimization
Part I (Justin)
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- Decrease f: −𝛂𝒈
- Violate constraint: ±𝛂𝒉
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Want:
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Turns constrained optimization into
unconstrained root-finding.
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Reparameterization
Eliminate constraints to reduce to unconstrained case
Newton’s method
Approximation: quadratic function with linear constraint
Penalty method
Augment objective with barrier term, e.g. 𝒈 𝒚 + 𝝇|𝒉 𝒚 |
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Example: Levenberg-Marquardt
Fix (or adjust) damping parameter 𝝁 > 𝟏.
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Huang et al. “L1-Based Construction of Polycube Maps from Complex Shapes.” TOG 2014.
Align with coordinate axes Preserve area
Note: Final method includes more terms!
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- Start with a feasible point
- Don’t ever enter the infeasible region
- Use a barrier in the objective
Bijective Parameterization with Free Boundaries (Smith & Schaefer, SIGGRAPH 2015)
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Try lightweight options
versus
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Repeatedly solve linear systems
“Geometric median”
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d can be a Bregman divergence
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Decompose as sum of hard part f and easy part g.
https://blogs.princeton.edu/imabandit/2013/04/11/orf523-ista-and-fista/
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Add constraint to objective
Does nothing when constraint is satisfied
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https://web.stanford.edu/~boyd/papers/pdf/admm_slides.pdf
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Want two easy subproblems
Augmented part
Solomon et al. “Earth Mover’s Distances on Discrete Surfaces.” SIGGRAPH 2014.
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https://en.wikipedia.org/wiki/Frank%E2%80%93Wolfe_algorithm