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observable-matrix-dynamics-ml

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Code for Halperin & Martin (2026): OMD paper and grokking-probe paper. Private during peer review.

observable-matrix-dynamics-ml 최신버젼 다운로드

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# Observable Matrix Dynamics (OMD): code for the OMD paper

Python code, scripts, figures, and animated visualisations supporting
the manuscript by Igor Halperin on a matrix-valued observable
formulation of neural-network training dynamics.

## Paper

- **Igor Halperin (2026).** *Learning as Observable Matrix Dynamics:
  Diffusive Relaxations versus Phase Transitions.*
  [arXiv:2606.29679](https://arxiv.org/abs/2606.29679). The compiled
  manuscript is [`omd/OMD_BBS_inference.pdf`](./omd/OMD_BBS_inference.pdf),
  with the online appendix in
  [`omd/OMD_BBS_inference_online_appendix.pdf`](./omd/OMD_BBS_inference_online_appendix.pdf)
  and the long-form companion in
  [`omd/ml_distance_matrices.pdf`](./omd/ml_distance_matrices.pdf).

## Related research papers

The OMD framework builds on a series of distance-matrix and frustrated-particle
papers by Igor Halperin (the **FBP**, **F2**, and **FDM** trilogy):

- **FBP (foundational).** Igor Halperin (2026). *Order Out of Noise and
  Disorder: Fate of the Frustrated Manifold.* `arXiv:2601.18653`.
- **F2.** Igor Halperin (2026). *Frustrated Fields: Statistical Field Theory
  for Frustrated Brownian Particles on 2D Manifolds.* `arXiv:2605.05366`.
- **FDM.** Igor Halperin (2026). *Frustrated Dynamics of Distance Matrices.*
  `arXiv:2605.05376`.

The OMD paper is the direct adaptation of the FDM framework from the
frustrated-particle physics setting to neural-network training.

## Related code repository

Earlier numerical code supporting the FBP, F2, and FDM trilogy is in a
separate public repository:

- [`ighalp/frustrated-brownian-particles-manifolds`](https://github.com/ighalp/frustrated-brownian-particles-manifolds)
  — Python code for FBP simulations on curved manifolds (sphere, torus)
  and the FDM matrix-side diagnostics on the FBP test bed.

## Repository layout

```
observable-matrix-dynamics-ml/
├── README.md                                  (this file)
└── omd/
    ├── OMD_BBS_inference.pdf                   compiled manuscript
    ├── OMD_BBS_inference_online_appendix.pdf   online appendix
    ├── ml_distance_matrices.pdf                long-form companion
    ├── README.md                              notes on the OMD code
    ├── experiments_snapshot_2026-05-29.md     experiment-status notes
    ├── requirements.txt
    ├── ml_fdm/                  main Python package
    │   ├── distance_matrix.py   arccos and joint distance-matrix code
    │   ├── spectral.py          spectral observables (beta, d_beta, ...)
    │   ├── ibbs.py              I-BBS multiplet / slope inference
    │   ├── product_spheres.py   product-manifold spectra
    │   ├── joint.py             train-test joint object
    │   ├── representations.py   feature extraction
    │   ├── output_level.py      output-level matrix loss
    │   ├── io_utils.py          checkpoint i/o
    │   └── experiments/         training and visualisation scripts
    ├── scripts/                 analysis scripts that produce the figures
    ├── configs/                 experiment configs
    ├── figures/                 PNG figures used in the manuscript
    └── gifs/                    animated GIF visualisations
```

## Reproducing the experiments

Top-level scripts in `omd/ml_fdm/experiments/` correspond to the
experiments of the OMD paper, named so the mapping is direct:

| Section | Script | Purpose |
|---|---|---|
| 5.1, 5.2 | `mnist_collapse.py` | MNIST + MLP feature emergence and high-WD neural collapse |
| 5.3 | `uci_regression.py` | multi-output regression |
| 5.4 | `gan_mode_collapse.py` + `gan_viz.py` | 8-Gaussian GAN |
| 5.5 | `grokking_transformer.py` | modular-arithmetic grokking |
| 5.6 | `sparse_parity.py` + `sparse_parity_viz.py` | sparse-parity learning |
| 5.7 | `task_switch.py` | synthetic supervisory-target switch |
| 5.8 | `input_topology.py` | input-distribution topology change |
| 5.9 | `matrix_loss_mnist.py` | MNIST + matrix loss |
| 5.10 | `matrix_loss_cifar.py` | CIFAR-10 + matrix loss |
| 5.9 follow-up | `matrix_grad_reg.py` | gradient-direction regulariser (chain-rule, Omega^step, Ito) |

Visualisation scripts produce the MDS + r_perp embedding figures and
animated GIFs:

- `phase_transition_viz.py` — task switch and input topology
- `diffusive_viz.py` — MNIST and regression
- `sparse_parity_viz.py` — sparse parity
- `gan_viz.py` — GAN scatter and MDS embeddings

The analysis scripts that produce the manuscript figures live in
`omd/scripts/`.

Setup:

```bash
cd omd
pip install -r requirements.txt
python -m ml_fdm.experiments.grokking_transformer --help
```

## Contact

Igor Halperin — ighalp@gmail.com

## License

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