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

Chethan Reddy G. P

machine learning

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

arXiv:2607.13511

summary

The paper presents ExTernD, a post‑training factorization that expands the rank of ternary matrix decompositions to correct quantization errors, enabling large language models to achieve accuracy close to bf16 across arbitrary quantization levels.

Abstract

We introduce ExTernD (Expanded-rank Ternary Decomposition), a post-training factorization of each LLM weight matrix $A \in R^{m \times n}$ into $A \approx B diag (D) C$ with ternary factors $B \in {- 1, 0, + 1}^{m \times k}$ , $C \in {- 1, 0, + 1}^{k \times n}$ and a real scale vector $D \in R^{k}$ . The inner rank $k = I^¼ min (m, n)$ is deliberately expanded beyond full rank ( $I^¼ > 1$ ), 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 $ε > 0$ : ExTernD approaches bf16 accuracy arbitrarily closely, which no ternary scheme with a fixed plane count can do. Memory and compute scale continuously with $I^¼$ , and factor sparsity continuously with a threshold $I ¨ $ , 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 $I^¼ = 3$ 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.

Topics & keywords

#quantization #large language models #ternary decomposition #post-training quantization #model compressionexpanded-rank ternary decompositionpost-training factorizationeffective bits per weightbf16 accuracyQwen3.5-4B