通过扩展秩的三值分解，让大模型量化精度逼近任意水平。
ExTernD: Expanded-Rank Ternary Decomposition Ternary LLM PTQ with Accuracy Approaching Any Quantization Level
- 将权重矩阵分解为三值因子与实数缩放向量，扩展秩以纠正量化误差。
- 在Gemma和Qwen3.5上实现接近bf16精度，5.2-5.5有效bpw即达Q4_K水平。
- 精度可精确匹配目标，无需硬性跳到固定比特位宽，适合高效部署。
我们提出ExTernD（扩展秩三值分解），将大模型权重矩阵$A \in \mathbb{R}^{m \times n}$分解为$A \approx B \mathrm{diag}(D) C$，其中三值因子$B \in \{-1,0,+1\}^{m \times k}$、$C \in \{-1,0,+1\}^{k \times n}$与实数缩放向量$D \in \mathbb{R}^k$构成。内秩$k = μ\min(m,n)$被刻意扩大至超过满秩（$μ>1$），使超出部分修正前期量化误差。理论上证明残差随$k$单调下降，可低于任意$\varepsilon > 0$：ExTernD可任意逼近bf16精度，而固定平面数的三值方案无法做到。内存与计算量随$μ$连续增长，因子稀疏度随阈值$τ$连续调整，因此可精确达到目标精度，而非四舍五入至下一比特位。在Gemma-4-E2B与Qwen3.5-4B上，其每矩阵精度达5.2-5.5有效bpw（加重要性加权后为5.1-5.5），媲美Q4_K；全模型转换在$μ=3$时，维基文本-2困惑度为10.10，优于bf16的9.78（+3.2%），接近Q4_K/Q5_K精度带（约5.7有效bpw）。
原文摘要 · Abstract (English)
We introduce ExTernD (Expanded-rank Ternary Decomposition), a post-training factorization of each LLM weight matrix $A \in \mathbb{R}^{m \times n}$ into $A \approx B \mathrm{diag}(D) C$ with ternary factors $B \in \{-1,0,+1\}^{m \times k}$, $C \in \{-1,0,+1\}^{k \times n}$ and a real scale vector $D \in \mathbb{R}^k$. 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 Q4_K'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 Q4_K/Q5_K accuracy band at ~5.7 effective bpw.
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