Regularized Residual Quantization (RRQ)

• Regularized Residual Quantization (RRQ) is a multi-layer quantization framework that extends classical residual quantization using rate-distortion-based regularization to achieve efficient signal representation.
• It employs reverse water-filling to allocate codeword variances and enforce sparsity, preventing overfitting in high-dimensional, variance-decaying data.
• RRQ demonstrates superior performance in image compression, denoising, and super-resolution, offering both low distortion and computational efficiency.

Regularized Residual Quantization (RRQ) is a multi-layer quantization framework designed to extend traditional residual quantization for high-dimensional, variance-decaying data. RRQ utilizes rate-distortion-based regularization—specifically, reverse water-filling—to control codeword variances, enabling deep quantization hierarchies while preventing overfitting and fostering sparsity. The approach is particularly suitable for domains with strong dimension-wise variance decay, such as decorrelated images, and finds applications in signal compression, denoising, and super-resolution.

1. Theoretical Foundations and Rate-Distortion Regularization

RRQ formalizes residual quantization within a rate-distortion theory framework for independent Gaussian sources. Given data vector components $X_j \sim \mathcal{N}(0, \sigma_j^2)$, the classical reverse water-filling solution provides the optimal distortion allocation:

$$D_j = \begin{cases} \gamma &\text{if }\sigma_j^2 \geq \gamma \ \sigma_j^2 &\text{if }\sigma_j^2 < \gamma \end{cases}$$

This leads to the optimal codeword variance for dimension $j$:

$\sigma_{C_j}^2 = \max(0,\, \sigma_j^2 - \gamma)$

In RRQ, this soft-thresholding is imposed as a regularization during dictionary (codebook) learning, ensuring that only dimensions with significant variance are allotted nonzero codeword variance. The regularization objective enforces this optimal variance structure, penalizing deviations from the theoretical prescription. Specifically, the variance-regularized K-means (VR-Kmeans) objective for codebook learning is:

$$\min_{C,A}\ \frac{1}{2}\|X - CA\|_F^2 + \frac{1}{2}\lambda \left\| \sum_{j=1}^{n} P_j C C^\top P_j - S \right\|_F^2$$

where $$S = \operatorname{diag}(\sigma_{C_1}^2, \ldots, \sigma_{C_n}^2)$$, $P_j$ selects the $j$-th dimension, and $\lambda$ is a regularization weight.

2. Multi-Layer Quantization and Sparsity

Classic RQ applies multiple quantization layers to the input by successively quantizing the residual error between input and reconstructed signal. RRQ extends this cascade: after each quantization stage, the regularized dictionary is used to quantize the current residual. Codeword variance regularization induces sparsity—dimensions with variance below threshold $\gamma$ do not contribute to later codebooks. This sparsity is advantageous in high-dimensional settings, reducing computational burden and overfitting. The multi-layer structure enables RRQ to achieve low distortion without losing generalization, as demonstrated by experiments showing minimal gap between training and test errors when compared to vanilla K-means.

Layer-wise update equations are:

• Compute the residual $X^{(l)} = X^{(l-1)} - \widehat{X}^{(l)}$,
• Assign codeword variances via $$\sigma_{C_j}^{(l)} = \max(0,\, D_j^{(l-1)} - \gamma^*)$$,
• Generate codewords from $\mathcal{N}(0, S^{(l)})$ or via VR-Kmeans.

The cumulative reconstruction after $L$ layers is $\hat{X} = \sum_{l=1}^{L} \widehat{X}^{(l)}$.

3. Codebook Construction: Regularized Random Sampling vs. Clustering

RRQ can use random codeword generation, guided by the regularized variances, or iterative VR-Kmeans clustering. The random codeword approach samples from a Gaussian distribution with zero mean and diagonal covariance $S^{(l)}$, avoiding the potential for overtraining when using classic K-means in very high-dimensional and variance-decaying spaces. VR-Kmeans, on the other hand, can closely match theoretical optimal reconstruction distortion but is more computationally intensive.

Both approaches depend on robust decorrelation of the input signals (see below), ensuring approximate Gaussianity and strong variance decay.

4. Preprocessing for Effective RRQ in Image Domains

Effective RRQ on images (e.g., facial images) requires decorrelated inputs with variance decay. The recommended pipeline is:

1. Apply a global 2D discrete cosine transform (2D-DCT) to each image, spreading out energy and separating frequency bands.
2. Segment DCT coefficients by zig-zag scan into $M$ sub-bands.
3. Apply local PCA (without dimensionality reduction) within each sub-band.

This step-wise decorrelation results in input vectors suitable for RRQ—variance decay across dimensions leads directly to the theoretical regularization prescription.

5. Empirical Performance and Applications

Experiments with synthetic variance-decaying data show RRQ achieves lower test distortion compared to classic K-means and vanilla RQ, particularly when the signal is high-dimensional. In image domains, RRQ demonstrates superior reconstruction quality for both compression and denoising tasks. On the CroppedYale facial dataset, RRQ outperforms JPEG-2000 at lower bitrates and BM3D at high noise levels, yielding sharper reconstructions and better preservation of perceptual details.

For super-resolution, RRQ-trained codebooks on high-resolution images were effective for reconstructing high-frequency details in low-resolution images from the test set.

Applications include:

• Domain-specific image compression (e.g., facial images, MRI scans),
• Joint compression and denoising (using clean image priors for noisy test images),
• Efficient representation in high-dimensional signal domains,
• Potential extension to other modalities (audio, remote sensing, medical imaging) with variance-decaying components.

6. Connections to Other Regularized Quantization Methods

The regularization principle in RRQ—imposing statistical optimality and promoting sparsity—has parallels in other frameworks. For instance, Regularized Classification-Aware Quantization (RCAQ) (Severo et al., 2021) regularizes quantization schemes by combining classification loss and reconstruction error to improve generalization. PARQ (Jin et al., 19 Mar 2025) uses convex piecewise-affine regularizers to induce hard quantization in large-scale models, providing theoretical last-iterate convergence guarantees. Modern neural compression approaches, such as Robust Residual Finite Scalar Quantization (RFSQ) (Zhu, 20 Aug 2025), use conditioning and normalization strategies to maintain residual signal throughout multiple FSQ stages—addressing a key challenge also found in RRQ layered cascades.

A plausible implication is that RRQ, with its strong theoretical regularization and practical sparsity mechanism, can be readily integrated with or extended by other convex regularization frameworks or advanced neural quantization methods, especially for further reducing overfitting, improving robustness, and achieving competitive rate-distortion trade-offs in diverse signal domains.

7. Prospects and Broader Impacts

The success of RRQ depends on variance decay and decorrelation in input data. Its mathematically principled regularization—anchored in rate-distortion theory—offers insight not only for quantization, but for unsupervised dictionary learning and sparse representation. In deep learning and neural compression, RRQ-inspired structures could inform new sparse quantization architectures, combining theoretical insights with practical efficiency.

Broader implications include:

• Adaptive quantization strategies for scalable high-dimensional compression,
• Modular quantization for generative modeling (see ResGen (Kim et al., 13 Dec 2024)),
• Integration with neural codebook prediction or convex regularization for robust signal representation.

Given its demonstrated utility for compression, denoising, and image restoration, RRQ continues to influence design principles in high-dimensional vector quantization, sparse coding, and neural network signal processing where optimal rate allocation and model generalization are paramount.