A Unified Hessian-Guided Framework for Post-Training Quantization

What it claims

HZ-PTQ is an integer post-training quantization method for weights and activations, from Qualcomm AI Research and Aarhus University (arXiv, 29 October 2026). It chooses transforms, weight scales, weight rounding and activation clipping by one shared objective. That objective is a Kronecker-factorized approximation of the layer-output Fisher, which serves as a quadratic proxy for the KL divergence between full-precision and quantized model outputs. The authors argue that weight-Hessian methods such as YAQA (Tseng et al., 2026) cannot see activation quantization error. An output-side curvature can see it.

How it works

The output Fisher is approximated as (G_Z G_Zᵀ ⊗ G_Zᵀ G_Z) / ‖G_Z‖²_F, where G_Z is the gradient with respect to the layer output. The approximation is exact when G_Z is rank 1. Three statistics are estimated and stored from a small calibration set: E[G_Zᵀ G_Z], E[G_W G_Wᵀ] and E[Diag(G_Z G_Zᵀ)]. Input-feature curvature is derived at runtime as W E[G_Zᵀ G_Z] Wᵀ. The procedure then runs in three stages:

1. Transform. A symmetric positive-definite matrix M is the matrix geometric mean of E[G_W G_Wᵀ]⁻¹ and E[G_Xᵀ G_X]. In practice M is block-diagonal and followed by a Hadamard rotation, giving T = M_block·H. If output gradients are assumed isotropic, this reduces to the CAT transform (Federici et al., 2026).
2. Weight scales and rounding. Input-side and output-side curvatures are block-LDL decomposed per quantization tile [k_i, k_o]. Each block's scale is set by grid search under D_i. Its rounding error is then propagated to the remaining blocks through L_i and L_o, in anti-diagonal order from the bottom-right. Blocks with larger Fisher trace go first. The paper says it differs from YAQA in using D_i to set scales and in its Fisher proxy.
3. Activation clipping. A shared factor γ_x ∈ (0, 1] is grid-searched on calibration data against the same proxy, using the already-rounded weights.

Stages 2 and 3 are separated by an approximate split of the error into weight and activation terms that assumes they are independent.

Evidence

The fetched text gives only Figure 1. It covers Llama 3.2 1B Instruct under W4A4 and measures, across linear layers, how well each layer-local proxy correlates with the model's output KL:

• Output Frobenius-norm error: r = −0.310
• Fisher-weighted weight error: r = 0.733
• Full output-Fisher-weighted error: r = 0.982
• The proposed Kronecker-factorized output Fisher: r = 0.989

The paper claims to beat state-of-the-art transform and rounding methods across model families in channel-wise and block-wise settings. Those comparison numbers are not in the fetched text.

Limits

The page cuts off partway through Section 3.6. No results tables, model list, bit-width sweep, runtime or calibration cost, or code release appear in it. Activation scales are computed per input at runtime, so deployment needs dynamic activation quantization. It also needs an online transform (block-diagonal matrix times Hadamard) applied to activations. The method rests on several stated approximations: a rank-1 gradient structure, independence between weight and activation error, standard rounding-noise assumptions, and a transform that is fixed before the other parameters are optimized.