Comment by imurray

Here is a pytorch optimizer that can maintain a matrix as orthogonal throughout optimization:

https://github.com/adrianjav/pogo — POGO: A Proximal One-step Geometric Orthoptimizer

https://arxiv.org/abs/2602.14656 — An Embarrassingly Simple Way to Optimize Orthogonal Matrices at Scale; Adrián Javaloy, Antonio Vergari