Non-Riemannian optimisation

This package contains Rust codes for the manuscript “Forward-backward methods in bilaterally bounded Alexandrov spaces” (arXiv:2503.24126) by Heikki von Koch and Tuomo Valkonen ⟨tuomov@iki.fi⟩. It concerns the solution of problems of the type $$ \min_{x ∈ M} F(x) + G(x) $$ where $F$ is a smooth function and $G$ a possibly nonsmooth convex function on a manifold $M$, which we do not assume to be Riemannian. It may, for example, be an embedded manifold in $ℝ^3$ with sharp edges. We have implemented the cube and the cylinder, as well as squared and non-squared distance functions.

Installation and usage

Installing dependencies

Most dependencies are managed by the Cargo build system of Rust. You will only need to manually install the “stable” Rust system itself. This can be done with rustup.

Building and running the experiments

To compile and install the program, use:

cargo install --path=.

When doing this for the first time, several dependencies will be downloaded. After this, you may run the experiment with

non-riemannian-opt

Alternatively, you may build (without installing) and run the experiments with

cargo run --release

Generating the graphical illustrations

We include LaTeX/PGFPLOTS files for graphically illustrating the numerical results. To generate cube.pdf and cylinder.pdf that visualise the computations, after running non-riemannian-opt as indicated above, you can use:

cd visualisations
pdflatex cube
pdflatex cylinder

Internals and auxiliary results

If you are interested in the program internals, the integrated source code documentation may be built and opened with

cargo doc              # build dependency docs
misc/cargo-d --open    # build and open KaTeX-aware docs for this crate

The cargo-d script ensures that KaTeX mathematics is rendered in the generated documentation through an ugly workaround. Unfortunately, rustdoc, akin to Rust largely itself, is stuck in 80's 7-bit gringo ASCII world, and does not support modern markdown features, such as mathematics.

In the doc/ subdirectory, you can find aux_results.pdf (and .tex) that includes additional detailed geodesic formulas that are not included in the main manuscript.