# ExTernD: Expanded-Rank Ternary Decomposition Ternary LLM PTQ with Accuracy Approaching Any Quantization Level

arXiv: https://arxiv.org/abs/2607.13511 · HTML: https://mlsanity.com/paper/2607.13511/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 9 references that are in the map.
- First posted on arXiv 15 Jul 2026.

## Builds on (most cited first, 6 of 6)

- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [The Era of 1-bit LLMs: All Large Language Models are in 1.58 Bits](https://mlsanity.com/paper/2402.17764/) (2024)
- [Trained Ternary Quantization](https://mlsanity.com/paper/1612.01064/) (2016)
- [Ternary Weight Networks](https://mlsanity.com/paper/1605.04711/) (2016)
- [BitNet: Scaling 1-bit Transformers for Large Language Models](https://mlsanity.com/paper/2310.11453/) (2023)
- [PTQTP: Post-Training Quantization to Trit-Planes for Large Language Models](https://mlsanity.com/paper/2509.16989/) (2025)

## Abstract (arXiv)

We introduce ExTernD (Expanded-rank Ternary Decomposition), a post-training factorization of each LLM weight matrix A ∈ ℝm × n into A ≈ B diag(D) C with ternary factors B ∈ -1,0,+1m × k, C ∈ -1,0,+1k × n and a real scale vector D ∈ ℝk. The inner rank k = μ min(m,n) is deliberately expanded beyond full rank (μ > 1), so that components past full rank correct the quantization error of earlier ones. We prove the residual decreases monotonically in k and can be driven below any varepsilon > 0: ExTernD approaches bf16 accuracy arbitrarily closely, which no ternary scheme with a fixed plane count can do. Memory and compute scale continuously with μ, and factor sparsity continuously with a threshold τ, so an accuracy target is hit exactly rather than rounded to the next bit-width. ExTernD matches Q4K's per-matrix accuracy at 5.2-5.5 effective bpw (5.1-5.5 with importance weighting) on Gemma-4-E2B and Qwen3.5-4B, and a full Qwen3.5-4B conversion at μ = 3 reaches 10.10 wikitext-2 perplexity against 9.78 for bf16 (+3.2%), placing it near the Q4K/Q5K accuracy band at ~5.7 effective bpw.