ExTernD: Expanded-Rank Ternary Decomposition Ternary LLM PTQ with Accuracy Approaching Any Quantization Level
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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.
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References · 6
- 2022GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformerscited by 1,919
- 2024The Era of 1-bit LLMs: All Large Language Models are in 1.58 Bitscited by 308
- 2016Trained Ternary Quantizationcited by 208
- 2016Ternary Weight Networkscited by 204
- 2023BitNet: Scaling 1-bit Transformers for Large Language Modelscited by 189
- 2025PTQTP: Post-Training Quantization to Trit-Planes for Large Language Modelscited by 9
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- 2026Breaking the Blocks: Continuous Low-Rank Decomposed Scaling for Unified LLM Quantization and Adaptation
- 2026LoPRo: Enhancing Low-Rank Quantization via Permuted Block-Wise Rotation
- 2023Ternary Singular Value Decomposition as a Better Parameterized Form in Linear Mapping
- 2025The Geometry of LLM Quantization: GPTQ as Babai's Nearest Plane Algorithm
- 2021Pruning Ternary Quantization