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Gram-matrix-parameterization.md

@ -110,11 +110,7 @@ $\operatorname{End}(\mathbb{R}^n)$. We apply the cotangent vector $d\operatornam
#### Finding minima
We minimize the loss function using a cheap imitation of Ueda and Yamashita's regularized Newton's method with backtracking.
* Kenji Ueda and Nobuo Yamashita. ["Convergence Properties of the Regularized Newton Method for the Unconstrained Nonconvex Optimization,"](https://doi.org/10.1007/s00245-009-9094-9) 2009.
The minimization routine is implemented in [`engine.rs`](../src/branch/main/app-proto/src/engine.rs). (In the old Julia prototype of the engine, it's in [`Engine.jl`](../src/branch/main/engine-proto/gram-test/Engine.jl).) It works like this.
We minimize the loss function using a cheap imitation of Ueda and Yamashita's uniformly regularized Newton's method with backtracking [[UY](https://code.studioinfinity.org/StudioInfinity/dyna3/wiki/Numerical-optimization#uniform-regularization)]. The minimization routine is implemented in [`engine.rs`](../src/branch/main/app-proto/src/engine.rs). (In the old Julia prototype of the engine, it's in [`Engine.jl`](../src/branch/main/engine-proto/gram-test/Engine.jl).) It works like this.
1. Do Newton steps, as described below, until the loss gets tolerably close to zero. Fail out if we reach the maximum allowed number of descent steps.
1. Find $-\operatorname{grad}(f)$, as described in ["The first derivative of the loss function."](Gram-matrix-parameterization#the-first-derivative-of-the-loss-function)
@ -138,14 +134,12 @@ The minimization routine is implemented in [`engine.rs`](../src/branch/main/app-
#### Other minimization methods
Ge, Chou, and Gao have written about using optimization methods to solve 2d geometric constraint problems.
- Jian-Xin Ge, Shang-Ching Chou, and Xiao-Shan Gao. ["Geometric Constraint Satisfaction Using Optimization Methods,"](https://doi.org/10.1016/S0010-4485(99)00074-3) 1999.
Instead of Newton's method, they use the BFGS method, which they say is less sensitive to the initial guess and more likely to arrive at a solution close to the initial guess.
Ge, Chou, and Gao have written about using optimization methods to solve 2d geometric constraint problems [GCG]. Instead of Newton's method, they use the BFGS method, which they say is less sensitive to the initial guess and more likely to arrive at a solution close to the initial guess.
Julia's [Optim](https://julianlsolvers.github.io/Optim.jl) package uses an [unconventional variant](https://julianlsolvers.github.io/Optim.jl/v1.12/algo/newton/) of Newton's method whose rationale is sketched in this [issue discussion](https://github.com/JuliaNLSolvers/Optim.jl/issues/153#issuecomment-161268535). It's based on the Cholesky-like factorization implemented in the [PositiveFactorizations](https://github.com/timholy/PositiveFactorizations.jl) package.
- **[GCG]** Jian-Xin Ge, Shang-Ching Chou, and Xiao-Shan Gao. ["Geometric Constraint Satisfaction Using Optimization Methods,"](https://doi.org/10.1016/S0010-4485(99)00074-3) 1999.
### Reconstructing a rigid subassembly
Suppose we can find a set of vectors $\{a_k\}_{k \in K}$ whose Lorentz products are all known. Restricting the Gram matrix to $\mathbb{R}^K$ and projecting its output orthogonally onto $\mathbb{R}^K$ gives a submatrix $G_K \colon \mathbb{R}^K \to \mathbb{R}^K$ whose entries are all known. Suppose in addition that the set $\{a_k\}_{k \in K}$ spans $V$, implying that $G_K$ has rank five.