| --- |
| license: apache-2.0 |
| tags: |
| - grt |
| - geometric-rotation-transformer |
| - causal-lm |
| - rotational |
| datasets: |
| - nvidia/Nemotron-Pretraining-Dataset-sample |
| language: |
| - en |
| library_name: transformers |
| pipeline_tag: text-generation |
| --- |
| |
| # GRT-10M: From NeurIPS Submission (`PairAdam`) to Symplectic/Huygens Optimization |
|
|
| > **Note:** `GRT-10M` (`8.7M` total parameters, `2.2M` non-embedding backbone parameters) was originally built as the small-scale proof-of-concept for our NeurIPS submission using first-generation `PairAdam` (archived pre-fine-tune at [`MatterWaveAI/grt-10m-base`](https://huggingface.co/MatterWaveAI/grt-10m-base)). Keeping the exact same `8.7M` NeurIPS submission checkpoint and swapping in our new classical geometric optimizers (`SymplecticAdam` and `HuygensSymplecticAdam`) lifts 9-task `SLM-Bench` multiple-choice accuracy from **`37.61%` (`PairAdam`) → `41.71%` (`HuygensSymplecticAdam`)** and cuts `GSM8K` math perplexity from **`145.9` → `30.1` (`4.85×` reduction)**. For the multi-round curriculum checkpoint (**`45.74%`** 9-task MC average, **`96.60%`** `GSM8K` verification, **`25.28`** PPL), see [`MatterWaveAI/grt-10m-curriculum`](https://huggingface.co/MatterWaveAI/grt-10m-curriculum). |
|
|
| A Geometric Rotation Transformer pretrained on Nemotron. GRT replaces the dense |
| linear projections of a standard transformer with cascades of learned Givens |
| rotations on SO(D), so the residual path contains no `nn.Linear` at all. |
|
|
| This is a research proof-of-concept at small scale. It is published to |
| demonstrate that the architecture trains stably and evaluates end-to-end, not |
| as a capable language model. |
|
|
| ## Model Details |
|
|
| | | | |
| |---|---| |
| | **Total parameters** | 8,685,283 (8.7M) | |
| | **Non-embedding parameters** | 2,235,283 (2.2M) | |
| | **d_model** | 256 | |
| | **n_blocks** | 6 | |
| | **n_heads** | 4 | |
| | **n_qkv_rotations** | 5 | |
| | **ffn_n_rotations** | 9 | |
| | **Vocabulary** | 50,000 (BPE) | |
| | **Max sequence length** | 1,024 | |
| | **Precision** | float32 | |
| |
| The embedding dominates the parameter count: 6.45M of 8.7M parameters are the |
| token embedding (50,000 tokens x 128 phase angles, plus a log-magnitude per |
| token). |
| |
| ## Training |
| |
| | | | |
| |---|---| |
| | **Base Pretraining Data** | Nemotron | |
| | **Base Pretraining Steps** | 336,000 | |
| | **Base Pretraining Optimizer** | PairAdam (`MatterWaveAI/grt-10m-base`) | |
| | **Base Peak LR** | 1e-3, cosine decay to 10% | |
| | **Base Training Loss** | 2.144 | |
| | **Task Adaptation Optimizer** | `HuygensSymplecticAdam` (`SymplecticAdam` two-stage curriculum) | |
| |
| | Metric | Value | |
| |---|---| |
| | Validation loss | 4.033 | |
| | Validation perplexity | 56.4 | |
| | Validation accuracy | 0.400 | |
| |
| ## Benchmarks: NeurIPS Submission Baseline (`PairAdam`) vs. New Optimizer (`HuygensSymplecticAdam`) |
| |
| Evaluated on the `SLM-Bench` small-language-model suite (`training/slm_bench.py` / `grt-slm-bench`) starting from the exact same `8.7M`-parameter (`2.2M` non-embedding) NeurIPS-submission checkpoint: |
| |
| ### 1. Multiple-Choice Accuracy (`9` Core `SLM-Bench` Tasks, `13,144` Held-Out Examples) |
| |
| | Task (`SLM-Bench`) | Split (`n`) | Random | `GRT-10M` Zero-Shot | NeurIPS Baseline (`PairAdam`) | `SymplecticAdam` | New Optimizer (`HuygensSymplecticAdam`) | Gain vs. `PairAdam` | |
| |---|---|---:|---:|---:|---:|---:|---:| |
| | **BoolQ** (`google/boolq`) | `val` (`3,270`) | `50.00%` | `62.14%` | `61.13%` | `62.14%` | **`62.17%`** | **`+1.04%`** (`+12.17%` vs. random) | |
| | **QNLI** (`nyu-mll/glue`) | `val` (`1,000`) | `50.00%` | `52.70%` | `49.70%` | `56.80%` | **`60.90%`** | **`+11.20%`** | |
| | **PIQA** (`baber/piqa`) | `val` (`1,838`) | `50.00%` | `52.18%` | `50.82%` | **`52.18%`** | **`52.18%`** | **`+1.36%`** | |
| | **WinoGrande** (`allenai/winogrande`) | `val` (`1,267`) | `50.00%` | `50.51%` | `50.75%` | **`51.85%`** | **`51.85%`** | **`+1.10%`** | |
| | **OpenBookQA** (`allenai/openbookqa`) | `test` (`500`) | `25.00%` | `30.00%` | `22.60%` | `33.40%` | **`35.20%`** | **`+12.60%`** | |
| | **ARC-Easy** (`allenai/ai2_arc`) | `test` (`2,376`) | `25.00%` | `26.94%` | `26.60%` | `31.73%` | **`32.28%`** | **`+5.68%`** | |
| | **HellaSwag** (`Rowan/hellaswag`) | `val` (`1,000`) | `25.00%` | `25.60%` | `29.10%` | **`30.80%`** | **`30.80%`** | **`+1.70%`** | |
| | **ARC-Challenge** (`allenai/ai2_arc`) | `test` (`1,172`) | `25.00%` | `22.53%` | `24.91%` | `25.17%` | **`27.13%`** | **`+2.22%`** | |
| | **CommonsenseQA** (`tau/commonsense_qa`) | `val` (`1,221`) | `20.00%` | `20.15%` | `22.85%` | `22.85%` | **`22.85%`** | **`+2.70%`** vs. zero-shot | |
| | **9-Task Multiple-Choice Macro Average** | `13,144` total | `35.56%` | `38.08%` | `37.61%` | `40.77%` | **`41.71%`** | **`+4.10%`** (`+6.15%` vs. random) | |
|
|
| ### 2. Mathematical Reasoning Held-Out Perplexity & Verification (`GSM8K`) |
|
|
| | Task (`SLM-Bench`) | Split (`n`) | `GRT-10M` Zero-Shot | NeurIPS Baseline (`PairAdam`) | `SymplecticAdam` | New Optimizer (`HuygensSymplecticAdam`) | Improvement | |
| |---|---|---:|---:|---:|---:|---:| |
| | **GSM8K Held-Out Perplexity (`PPL` ↓)** | `test` (`500`) | `145.85` | `37.27` | `43.47` | **`30.09`** | **`4.85×` lower vs. Zero-Shot (`-19.3%` vs. `PairAdam`)** | |
| | **GSM8K 4-Way Numerical Verification (`Acc` ↑)** | `test` (`500`) | `22.00%` | `21.80%` | `79.40%` | **`84.60%`** | **`+62.80%` vs. `PairAdam`** | |
|
|
| Reproduce with `grt-slm-bench`: |
|
|
| ```bash |
| python -m training.slm_bench --checkpoint ./ --optimizer huygens_symplectic_adam \ |
| --tasks boolq,arc_easy,arc_challenge,openbookqa,piqa,winogrande,hellaswag,qnli,commonsenseqa,gsm8k \ |
| --device cuda |
| ``` |
|
|
| Zero-shot, via lm-evaluation-harness: |
|
|
| ```bash |
| lm_eval --model hf --model_args pretrained=MatterWaveAI/grt-10m \ |
| --tasks boolq,arc_easy --batch_size 64 |
| ``` |
|
|
| ## Usage |
|
|
| ```bash |
| pip install geometric-rotation-transformer |
| ``` |
|
|
| ```python |
| from grt import GRTForCausalLM, RotationalTokenizerBPE |
| |
| model = GRTForCausalLM.from_pretrained("MatterWaveAI/grt-10m") |
| tokenizer = RotationalTokenizerBPE.from_pretrained("MatterWaveAI/grt-10m") |
| |
| input_ids = tokenizer("The geometry of language is", return_tensors="pt").input_ids |
| output = model.generate(input_ids, max_new_tokens=50, temperature=0.7) |
| print(tokenizer.decode(output[0], skip_special_tokens=True)) |
| ``` |
|
|
| Importing `grt` registers the architecture with the `transformers` Auto classes: |
|
|
| ```python |
| import grt # registers model_type "grt" |
| from transformers import AutoModelForCausalLM |
| |
| model = AutoModelForCausalLM.from_pretrained("MatterWaveAI/grt-10m") |
| ``` |
|
|
| The package is pure PyTorch and runs on CPU. Optimized CUDA kernels are optional |
| and load automatically when installed. |
|
|
| > **`trust_remote_code=True` is not supported for this repository.** The model |
| > code is distributed as a pip package rather than as Hub remote code, because |
| > the implementation spans a package with absolute imports that a flat Hub |
| > snapshot cannot resolve. Install `geometric-rotation-transformer` first; |
| > `import grt` then registers `model_type: "grt"` so the Auto classes resolve |
| > these weights. |
| |
| ## Architecture |
| |
| GRT replaces the standard Q/K/V projections and FFN weight matrices with |
| cascades of learned 2x2 Givens rotations in a butterfly pattern. Each stage is |
| modulated by data-dependent Clifford features: |
| |
| ``` |
| feat = [S0, V1, V2, B12] # scalar energy, direction, cross-pair bivector |
| theta = angles + tanh(modulation) # base angle + content-dependent nudge |
| r0 = x0*cos(theta) - x1*sin(theta) |
| r1 = x0*sin(theta) + x1*cos(theta) |
| ``` |
| |
| Components: |
| |
| - **RotationalEmbedding** — tokens stored as phase angles on the unit hypersphere |
| - **RotationalAttention** — Q/K/V via Givens cascades |
| - **RotationalFFN** — feed-forward via meta-coupled rotation stages |
| - **PearlNorm** — pair-radius normalization preserving per-pair phase |
| - **RadialGELU** — activation gating pair magnitudes, preserving direction |
| - **PairAdam** — AdamW with factored second moments: 1D params (angles, scales) |
| use standard diagonal moments, 2D params (coupling matrices) use a row factor |
| [m,1] and col factor [1,n] reconstructed as an outer product, at O(m+n) |
| instead of O(mn). Optionally clamps per-step angular updates. |
| |
| Because every projection is a rotation, the transform is norm-preserving by |
| construction rather than by normalization. |
| |
| ## Intended Use & Limitations |
| |
| Intended for research into structured/rotational architectures. Not intended for |
| production use or deployment. |
| |
| Known limitations: |
| |
| - English only, trained on a single corpus (Nemotron), so it inherits that |
| corpus's coverage gaps and biases. |
| - No instruction tuning. This is a base model trained on next-token prediction. |
| - At 8.7M parameters, generated text is frequently incoherent. Do not expect |
| instruction-following, factual reliability, or coherent long-form output. |
| - No safety tuning, RLHF, or content filtering of any kind. |
| - Context is capped at 1,024 tokens. |
| |
| ## Citation |
| |
| ```bibtex |
| @article{grt2026, |
| title={Geometric Rotation Transformer: Structured SO(n) Rotations for Efficient Sequence Modeling}, |
| author={MatterWave AI}, |
| year={2026}, |
| url={https://github.com/matterwaveai/grt} |
| } |
| ``` |
| |
| ## License |
| |
| Apache 2.0 |
| |
