Fused Reorder-and-Quantize Operation

• Fused reorder-and-quantize operation is a technique that integrates data reordering with quantization in a single pass to minimize quantization error and reduce computational overhead.
• It employs offline clustering, in-place quantization, and kernel fusion to optimize memory access patterns and hardware efficiency in neural network inference.
• Empirical studies demonstrate significant improvements in power, latency, and memory usage across transformer and CNN models with minimal loss in accuracy.

A fused reorder-and-quantize operation is a computational primitive that integrates a data reordering phase with quantization in one contiguous or pipelined computational pass. Its goal is to optimize neural network inference by minimizing quantization error, reducing memory and compute overhead, and eliminating redundant data movement. This approach is crucial for achieving reliable low-bitwidth quantization, efficient hardware mapping, and power and latency reductions in both transformer-based and convolutional neural networks.

1. Mathematical Foundations and Operator Definitions

All fused reorder-and-quantize methods share a two-step mathematical structure: (1) a reordering transformation, and (2) a subsequent quantization, often grouped or adapted to the reordered structure. In the RPTQ formulation, the input activation tensor $X \in \mathbb{R}^{B\times N \times C}$ is permuted along its channel dimension by a permutation $S$ derived from clustering channels via K-means on per-channel (min, max) statistics, forming $g$ contiguous channel clusters. The permutation operator $R_S$ rewrites the tensor as $\tilde{X} = R_S(X)$, with each channel $i$ routed to the index $S_i$.

Quantization parameters (scale $s_j$, zero-point $z_j$) are computed per cluster by enveloping all member channels' ranges, so a groupwise quantization is applied after reordering:

$$X_q[:, :, \text{start}_j:\text{end}_j] = \text{clamp}\left( \left\lfloor \frac{\tilde{X}[:, :, \text{start}_j:\text{end}_j]}{s_j} \right\rceil + z_j, -2^{k-1}, 2^{k-1}-1 \right)$$

This ensures that activations within each cluster are mapped optimally into the target $k$-bit integer range, minimizing the quantization error that would otherwise result from channel dynamic range variation (Yuan et al., 2023).

In the context of dot products and accumulations, PQS extends the principle by organizing already-quantized (low-bit) partial results so that sorted accumulation eliminates transient overflow. This "fused" sorting, or magnitude reordering, followed by quantized accumulation, enables the use of a very narrow accumulator—substantially reducing hardware costs (Natesh et al., 12 Apr 2025).

2. Algorithmic Workflows and Fused Kernel Designs

The core implementation pattern in modern fused reorder-and-quantize designs is as follows:

1. Data-driven Reordering (Offline): Perform statistical calibration (e.g., collecting per-channel min/max) and cluster the channels or dot-product elements. The permutation $S$ is computed to form contiguous clusters or optimize sum properties (such as magnitude cancellation in PQS).
2. Kernel Fusion (Runtime):
   • In transformer layers (RPTQ):
     • The LayerNorm kernel is modified to write each output channel directly to its permuted address, embedding the reordering into the LN output writeback.
     • The quantization pass follows immediately, quantizing each cluster (now contiguous) in-place.
   • In PQS, quantized vector pairs are multiplied to form partial products, which are then split by sign, sorted, pairwise canceled, and recursively merged, all within a narrow (e.g., 12–16 bit) accumulator.
   • In FROQ (operand reordering for transformers), the entire dequantization/quantization stage is fused with GEMM by reordering the operation graph: integer-only GEMM is run directly on the quantized inputs, with scaling and dequantization deferred to a single per-output post-processing step (Lin et al., 11 Apr 2025).
3. Weight and Bias Management: To ensure correct mapping between permuted (reordered) activations and weight matrices, all weights are permuted offline to match the channel reordering. Bias terms are adjusted to absorb quantization cross-terms, enabling efficient integer-only computation in the main matrix multiplication.

This fused pass eliminates explicit memory copies and extra kernel launches compared to naive, separate reorder and quantize passes (Yuan et al., 2023, Lin et al., 11 Apr 2025, Natesh et al., 12 Apr 2025).

3. Hardware Mapping and Efficiency Gains

The fused reorder-and-quantize paradigm yields significant hardware design benefits, especially on custom accelerators and systolic arrays:

• RPTQ and Clustered Quantization: By clustering channels and quantizing in groups, memory access patterns are optimized for stride-1, streaming writes. No gather/scatter or random access is needed, which improves effective DRAM bandwidth and reduces latency (Yuan et al., 2023).
• FROQ Integerization: In vision transformer accelerators, reordering the computation graph allows all major compute blocks (linear, matrix multiplication) to operate in low-bit integer mode, keeping all MAC units integer-only. Only one final scalar post-multiplier per output channel is required. This architecture delivers a per-PE power reduction from >1 mW/PE (floating point) to ≈0.4 mW/PE (integer), and up to 2× GEMM kernel speedup (Lin et al., 11 Apr 2025).
• PQS Sorted Accumulation: PQS's microarchitecture integrates small sorting networks before accumulation, reducing the required register width from the standard $p=32$ bits to as low as $p=12$–14 bits for $k=5$ quantization and $N=8$ non-zeros—yielding a ≈2.5× register area and energy reduction. Energy per multiply-accumulate (MAC) drops by ≈83%, and register file and scratchpad bandwidth are similarly reduced (Natesh et al., 12 Apr 2025).

4. Quantization Accuracy and Numerical Properties

The principal motivation for fusing reordering and quantization is to suppress the deleterious effects of dynamic range variation and transient accumulation overflows:

• Clustered Quantization vs. Per-channel: Per-channel quantization can be too aggressive at low bitwidths, yielding minimal headroom. Clustered quantization, as enabled by offline reordering, shares quantization envelopes across channels with similar statistics, dramatically reducing error at very low $k$ (e.g., $k=3$ in RPTQ for OPT-175B, with <1.5 PPL loss) (Yuan et al., 2023).
• PQS Sorted Dot Product: Accumulation order matters for avoiding transient overflows. Sorting and pairing large, oppositely signed partial products ensures that the running sum never exceeds the final result's bit-width. For pruned and quantized models, this guarantees that a narrow accumulator suffices without loss of accuracy—empirically, <1% degradation versus full 32b accumulation on sparse ResNet-18 models (Natesh et al., 12 Apr 2025).
• Integer GEMM with Deferred Dequantization: FROQ demonstrates that with proper scaling management, integer-only MAC operations and deferred dequantization maintain near-baseline accuracy (accuracy drop of −0.17% in the 8.3 MB model on DeiT-S, CIFAR-10) while reducing latency and energy (Lin et al., 11 Apr 2025).

5. Scalability, Critical Limits, and Integration Concerns

Several scalability and integration aspects are illuminated by recent work:

• Offline Cost and Integration: The most expensive phase is the calibration and offline reordering (channel clustering or pruning). Its $O(C \cdot g \cdot I)$ cost (for $C$ channels, $g$ clusters, $I$ K-means iterations) is typically negligible at model load time (Yuan et al., 2023).
• Inference Engine Modifications: Fusion requires only minor modifications to standard kernels: for RPTQ, adjusting the LayerNorm's write index and permuting weights once; for PQS, inserting a small sorter network before narrow accumulation; for FROQ, reordering scaling so all integer MAC computation is contiguous (Lin et al., 11 Apr 2025, Natesh et al., 12 Apr 2025, Yuan et al., 2023).
• Sorter Hardware Cost: For short dot products ($N \leq 16$), the incremental area of the sorter is modest; for transformer-scale $K>256$, multi-level tiling and multi-stage sorting may be needed (Natesh et al., 12 Apr 2025). A plausible implication is that the benefits of the fused approach may require further adaptation for very long dot products.
• Persistent Overflow: Even with sorted accumulation, persistent overflow (where the true sum exceeds accumulator capacity) can only be avoided by aggressive pruning or explicit $\ell_1$ bound enforcement on quantized weights (Natesh et al., 12 Apr 2025).

6. Comparative Impact on Several Model Classes

Model / System	Core Fused Operator	Major Gain	Notable Accuracy Delta
OPT-175B LLM (RPTQ)	LN-fused reorder + group quantize	80% memory (KV cache) reduction, 3b activation	<1.5 PPL loss @ 3b act (Yuan et al., 2023)
DeiT-S ViT (FROQ)	Integer-only MAC w/ post-dequant	0.35–0.42 mW/PE, up to 2× kernel speed	−0.17–0.30% (Lin et al., 11 Apr 2025)
ResNet-18 (PQS)	Sorted quantized dot+summation	2.5× accumulator energy reduction	<1% @ 13–14b accumulator (Natesh et al., 12 Apr 2025)

These empirical results underscore that fused reorder-and-quantize methods universally improve hardware efficiency, preserve accuracy, and extend low-bit quantization to regimes previously regarded as impractical (3-bit LLM activations, integerized ViTs, ultra-narrow accumulators in CNNs).

7. Limitations and Open Directions

While highly effective, fused reorder-and-quantize operations manifest certain limitations:

• Applicability to Extremely Long Dot Products: Sorter network complexity grows for large $K$, so multi-level strategies or hardware-friendly approximations may be required for deep transformer or very wide convolutional layers (Natesh et al., 12 Apr 2025).
• Persistent Overflow and Pruning: Ensuring no persistent accumulator overflow relies on model pruning or explicit norm constraint, which may interact with model expressivity (Natesh et al., 12 Apr 2025).
• Hardware-Specific Tuning: Fully capitalizing on the proposed gains (especially for energy and latency) may require ASIC-level engineering and adjustments to accommodate pipeline hazards and dataflow peculiarities.
• Generality Across Model Architectures: Most results to date are on transformers and CNN backbones; extending fused reorder-and-quantize to RNNs, GNNs, or highly irregular architectures remains an open area.

A plausible implication is that the widespread adoption of fused reorder-and-quantize methods will depend on further hardware-software co-design, more generalized calibration and clustering strategies, and standardization of kernel interfaces to accommodate permutation, quantization, and accumulation logic within network compilers and hardware libraries.