Robust Residual FSQ (RFSQ) Overview

• RFSQ is a set of techniques that blend robust estimation, residual/predictive modeling, and fixed scalar quantization, applicable across neural compression, regression, and digital fault detection.
• It employs learnable scaling and invertible layer normalization to maintain dynamic range and improve quantization effectiveness, achieving up to 45% perceptual loss improvement in experiments.
• RFSQ methods have been validated through diverse applications, offering robustness against outliers and quantization artifacts in both software algorithms and hardware implementations.

Robust Residual FSQ (RFSQ) encompasses a family of estimation and quantization techniques uniting robustness to outliers, residual/predictive modeling, and fixed scalar quantization. The term arises in several contexts: neural compression with scalar quantizers and conditioning (Zhu, 20 Aug 2025), robust functional regression via principal component basis (Boente et al., 2022), and digital signal residual generation for fault detection on reconfigurable hardware (Kim, 2023). RFSQ methodologies are distinguished by the systematic integration of robustness (to small signals, outliers, or quantization artifacts) and residual structures (either through multi-stage signal removal or equation-based residue analysis), augmented by fixed or finite scalar quantization or quantizer-like elements. This article provides an integrated and comparative account of the key RFSQ formulations across these research domains.

1. FSQ and Residual Quantization in Neural Compression

Finite Scalar Quantization (FSQ) quantizes each dimension of a real vector $z \in \mathbb{R}^d$ independently, i.e., $$Q_i(z_i) = \mathsf{round}\Big(\frac{z_i(L_i-1)}{2}\Big) \cdot \frac{2}{L_i-1}$$ for $L_i$ scalar levels per dimension, yielding bitrates of $\sum_i \log_2 L_i$. Residual quantization constructs a multi-stage sequence: set $r_0 = x$, and for $k = 1 \ldots K$,

$q_k = Q_k(r_{k-1}),\qquad r_k = r_{k-1} - q_k$

producing a residual sequence $r_k = x - \sum_{i=1}^k Q_i(r_{i-1})$. Empirically, $\|r_k\| \ll \|r_{k-1}\|$ after each stage ("residual-magnitude decay"), rendering later FSQ stages ineffective as residual values fall within negligible quantization bins.

RFSQ for neural compression (Zhu, 20 Aug 2025) resolves this limitation via: (a) learnable scaling factors, introducing $\alpha_k$ per stage so $q_k = \mathrm{FSQ}_k(\alpha_k r_{k-1})$ and $r_k = r_{k-1} - q_k / \alpha_k$; and (b) invertible layer normalization, normalizing the residual input before quantization, then reversing after quantization (with $r_k = r_{k-1} - \sigma_k \tilde{q}_k - \mu_k$). Both mechanisms maintain dynamic range for effective quantization per stage, with the parameter updates permitting differentiable, end-to-end optimization (using straight-through estimators for the quantization steps).

A summary table of experiment results on ImageNet (128×128, 12 bits/token) is as follows:

Method	$L_1$	LPIPS	$\Delta$LPIPS vs FSQ
FSQ	0.143	0.182	–
LFQ	0.241	0.361	–98%
VQ-EMA	0.355	0.489	–168%
RFSQ-4×1024-Scale	0.103	0.101	+44.5%
RFSQ-4×1024-LayerNorm	0.102	0.100	+45.1%
RFSQ-4×1024-None	0.113	0.121	+33.5%

These outcomes indicate that RFSQ with layer normalization and scaling achieves up to 45% improvement in perceptual loss and nearly 29% reduction in $L_1$ error over baseline FSQ; layer normalization in particular demonstrates consistent advantage across configurations.

2. Conditioning Mechanisms: Scaling and Invertible LayerNorm

Learnable scaling preserves residual magnitude at each stage, enabling more informative quantization even at late stages. The forward and backward flows are:

• $q_k = \mathrm{FSQ}_k(\alpha_k r_{k-1})$
• $r_k = r_{k-1} - q_k/\alpha_k$

The backward pass with a straight-through estimator computes

$$\frac{\partial \mathcal{L}}{\partial \alpha_k} \approx \left\langle\frac{\partial \mathcal{L}}{\partial q_k}, r_{k-1}\right\rangle + \left\langle\frac{\partial \mathcal{L}}{\partial r_k}, \left[-\frac{q_k}{\alpha_k^2} - \frac{r_{k-1}}{\alpha_k}\right]\right\rangle$$

Invertible layer normalization removes mean and rescales variance, with

$$\mu_k = \frac{1}{d}\sum_{i=1}^d [r_{k-1}]_i\,,\quad \sigma_k = \sqrt{\frac{1}{d}\sum_{i=1}^d ([r_{k-1}]_i - \mu_k)^2 + \epsilon}$$

$$y_k = \frac{r_{k-1} - \mu_k}{\sigma_k},\quad \tilde{q}_k = \mathrm{FSQ}_k(y_k),\quad q_k = \sigma_k \tilde{q}_k + \mu_k$$

The invertibility and straightforward backpropagation through normalization, with quantization approximated via straight-through gradients, allow the entire quantization stack to be trained efficiently.

Both mechanisms result in superior performance compared with unconditioned FSQ or standard VQ approaches. Four-stage cascades show optimal trade-off, with diminishing returns beyond $K = 4$.

3. Algorithmic and Training Details

The RFSQ end-to-end algorithm is fully differentiable except for the quantization steps, which employ straight-through estimators. Typically, ImageNet experiments employ:

• $K = 4$ stages of FSQ with $L_i = 4$ (symmetric quantization levels).
• Initialization: $\alpha_k = 1.0$, $\epsilon = 1\!\times\! 10^{-6}$ in LayerNorm, no learned affine parameters.
• Adam optimizer ($\mathrm{lr}=8\!\times\! 10^{-4}$, linearly decayed, weight decay $5\!\times\!10^{-5}$, batch size 2048 across 16 GPUs).
• Loss combines $L_1$ and LPIPS perceptual terms; an optional regularizer penalizes deviation of $\alpha_k$ from $1$.
• Gradient clipping at $1.0$ and linear warmup during startup.
• For reconstructive quality, LayerNorm proves superior in maintaining texture and color fidelity on qualitative assessment grids.

Ablation indicates that the benefits of RFSQ are robust to number of stages and quantizer levels, but most pronounced with adequate depth and careful per-stage dynamic range management.

4. Robust Residual–Functional Quadratic Estimation in Regression

In functional data analysis, the RFSQ estimator (Boente et al., 2022) refers to robust MM-estimation within a truncated quadratic polynomial regression over robust principal component (PCA) basis functions. Observing $(y_i, X_i)$ with $X_i$ a function in $L^2[0,1]$, the quadratic regression is

$$y_i = \alpha_0 + \int X_i(t) \beta_0(t)\,dt + \int\int X_i(s) X_i(t)\upsilon_0(s,t)dtds + \sigma_0\epsilon_i$$

which is reformulated after robust functional PCA as a regression on principal scores and their quadratics

$$y_i \approx \alpha + \sum_{j=1}^p b_j \xi_{ij} + \sum_{1\leq j\leq \ell \leq p} u_{j\ell} \xi_{ij} \xi_{i\ell} + \sigma\epsilon_i$$

The estimation proceeds as follows:

1. Robust centering and PCA (e.g., sign-scatter or projection-pursuit robust estimators) yield directions $\hat\phi_j$ and scores $\hat\xi_{ij}$.
2. A design matrix $\mathbf{Z}_i$ of scores and their quadratic products is formed.
3. An S-scale estimator minimizes the scale $s$ satisfying

$$\frac{1}{n-(p+q)} \sum_{i=1}^n \rho_0\left(\frac{r_i(\theta)}{s}\right) = b_0$$

1. An MM-estimation step fixes $s$ and minimizes

$$\sum_{i=1}^n \rho_1\left(\frac{r_i(\theta)}{s}\right)$$

Algorithmically, the two-step procedure yields estimators with a 50% breakdown point and tunable efficiency; Fisher-consistency and convergence is established in both finite and infinite-rank regimes.

5. RFSQ in Robust Residual Generation for Digital Fault Detection

In the digital residual generator context (Kim, 2023), RFSQ methodology is realized by embedding robustness in the creation and processing of residuals for fault detection in linear dynamical systems, particularly when implemented on FPGAs. The model employs a Kalman innovations approach, recasting the system into Vector AutoRegressive with eXogenous inputs (VARX) and generating residuals between observed and predicted sequences: $$r_{k,L} = Y_{k,L}^{\text{meas}} - (H_{L,p}Z_{k-L,p} + T_u U_{k,L})$$

The covariance of the residuals, built by accumulating both identification error and noise, enables computation of a whitened test statistic $T(k) = \tilde{r}_{k,L}^T \tilde{r}_{k,L}$, which is chi-squared under the no-fault null hypothesis. Thresholding $T(k)$ at level $Y_\alpha$ controls the false alarm rate.

Implementation steps:

• Offline system identification with sufficient SNR, model order selection, and estimation of all necessary Markov and whitening matrices.
• Online real-time processing using memory-efficient rolling buffers, matrix–vector multiplications, and dot products.
• Floating-point to fixed-point conversion uses quantization step optimization (via, e.g., MATLAB HDL Coder) and validation of false-alarm rate invariance through Monte Carlo simulation.
• On hardware (e.g., Xilinx Zynq PYNQ-Z2), dataflow leverages efficient pipelining, BRAM for model storage, DSP block arithmetic, and design trade-offs between latency, resource use, and throughput.

The design achieves sub-microsecond latencies ($0.26\;\mu$s for fixed-point), preserves the design false-alarm budget, and supports dynamic reconfiguration and resource sharing.

6. Comparative Summary and Domain-Specific Considerations

Domain	RFSQ Mechanism	Robustness Target	Key Advantages
Neural Compression	Multi-stage FSQ w/ scaling/layernorm	Small-residual signal suppression	45% gain in perceptual loss, stable training
Functional Regression	Robust PCA + MM quadratic regression	Outliers/high leverage	Fisher-consistency, 50% breakdown
Digital Residual Generation	Covariance-corrected VARX residuals, fixed-point quant	Identification noise, quant error	FPGA throughput, controlled FAR

Robust Residual FSQ approaches demonstrate that integrating residual mechanisms with robust, adaptive conditioning—be it through scaling, normalization, or covariance compensation—substantially improves performance in compression, regression, and signal integrity. Each instantiation leverages domain-appropriate strategies for managing dynamic range compression, outlier suppression, or hardware-induced quantization, while maintaining statistical or operational guarantees such as efficiency, breakdown point, or real-time fault detection. The efficacy of conditioning layers, robust pre-processing, and model-aware residual construction are key unifying elements.

7. Implementation, Performance, and Practical Recommendations

• RFSQ neural compression: Prefer invertible layer normalization per quantization stage for consistent gains and codebook efficiency; optimize number of stages ($K=4$) and initialize dynamically scaled parameters.
• Robust functional regression: Employ robust S-scale and MM estimation post robust FPCA to achieve high breakdown and consistency; ensure PCA methodology matches data ellipticity.
• FPGA residual generation: Carry out systematic fixed-point format selection; verify empirical false-alarm rate against floating-point baseline before hardware deployment; pipeline matrix operations and store model matrices in on-chip memory.

Further, across all RFSQ variants, direct gradient manipulation through straight-through estimators or analytic differentiability within conditioning transforms is essential for tractable and effective optimization or real-time operation in complex pipelines. The architecture-agnostic nature of residual-robust-quantization schemes enables flexible application across machine learning, signal processing, and statistical modeling domains.