Residual Quantization (RQ)

• Residual Quantization (RQ) is a multistage quantization method that represents data as a sum of codewords from successive codebooks to iteratively minimize approximation error.
• It uses recursive algorithms with greedy or beam search assignments to construct exponential-sized codebooks, enhancing high-dimensional approximate nearest neighbor search and compression performance.
• Modern enhancements, including neural scoring, local transformations, and regularized codebook learning, address diminishing returns and improve rate-distortion performance across applications.

Residual Quantization (RQ) is a multistage quantization strategy for representing data—classically data vectors but now also neural weights, activations, and other structured signals—by successively quantizing the residuals left unaccounted by previous quantization stages. Because RQ generalizes single-stage vector quantization to compositions of codebooks capturing increasingly fine approximation error, it is widely used in high-dimensional approximate nearest neighbor (ANN) search, learned data compression, and deep learning model quantization. RQ and its variants form a central technical foundation for contemporary scalable retrieval, discrete representation learning, and low-bit quantization in both traditional and neural methods.

1. Mathematical Formulation and Standard Algorithm

Let $\mathbf{x} \in \mathbb{R}^d$ be a vector to be quantized. RQ represents $\mathbf{x}$ as the sum of $M$ codewords, one from each of $M$ codebooks: $$\mathbf{x} \approx \sum_{m=1}^M \mathbf{c}_m(i_m(\mathbf{x}))$$ where $\mathbf{c}_m(i_m(\mathbf{x}))$ is the codeword selected from the $m^{\text{th}}$ codebook $C_m$. The RQ process is recursive:

• Set residual $r_0 = \mathbf{x}$.
• For each stage $m=1,\ldots,M$:
  • Choose $$i_m(\mathbf{x}) = \arg\min_k \| r_{m-1} - \mathbf{c}_m(k) \|^2$$.
  • Assign $r_m = r_{m-1} - \mathbf{c}_m(i_m(\mathbf{x}))$.

The quantization error for each vector is thus: $$E = \left\| \mathbf{x} - \sum_{m=1}^{M} \mathbf{c}_m(i_m) \right\|^2$$

Learning proceeds by constructing each codebook (typically via k-means on the current residuals) to minimize total error. The process iterates across all data, learning a hierarchy of codebooks.

Decoding is computationally lightweight: reconstruct $\mathbf{x}$ as the direct sum of its indexed codewords. Encoding (assigning codewords) is, however, computationally hard (NP-hard in general), and is typically approximated by greedy stagewise assignment or via more advanced search strategies such as beam search (Liu et al., 2015).

2. Information Theoretic Properties and Codebook Utilization

A key strength of RQ is the effective construction of exponential-sized codebooks: the combined "super-codebook" has $K^M$ clusters when each stage codebook has size $K$, allowing for fine discretization of high-dimensional data while keeping per-stage codebook memory manageable.

Information-theoretic measures, particularly codebook entropy,

$S(C_m) = -\sum_{k=1}^K p_k^m \log_2 p_k^m$

quantify the utilization efficiency of each codebook (where $p_k^m$ denotes usage probability). In practice, as stages progress, residuals become random and their structure degrades, causing codebooks in later stages to become less utilized, degrading rate-distortion performance and limiting practical code lengths (Liu et al., 2015, Yuan et al., 2015).

3. Enhancements and Modern Variants

3.1. Codebook Learning Improvements

Warm-started iterative k-means and subspace clustering, as in Improved Residual Vector Quantization (IRVQ) (Liu et al., 2015), mitigate diminishing entropy by initializing k-means in lower-dimensional PCA subspaces and gradually increasing dimension, maintaining high codebook entropy and prolonging the useful depth of RQ.

Regularized codebooks (e.g., variance regularization per rate-distortion theory) lead to sparsity and improved generalization in high dimensions. In RRQ (Ferdowsi et al., 2017, Ferdowsi et al., 2017), codeword variances are set by soft-thresholding: $\sigma_{C_j}^2 = \max(\sigma_j^2 - \gamma, 0)$ where $\gamma$ is found by "water-filling" for the best distortion-rate allocation, yielding codebooks that are both efficient and robust in high-dimensional settings.

3.2. Improved Encoding Schemes

Multi-path vector encoding (MVE) (Liu et al., 2015) and beam search-based encoding (Vallaeys et al., 6 Jan 2025) maintain multiple candidate partial approximations at each stage—rather than a single greedy path—reducing error propagation and limiting suboptimal choices that arise from greedy assignment.

Codeword pre-selection and neural scoring functions (Vallaeys et al., 6 Jan 2025) further reduce computational overhead, by first filtering codewords with a cheap proxy and then applying a more expressive (possibly neural) quantizer.

3.3. Local Transformations and Alignment

Transformed Residual Quantization (TRQ) (Yuan et al., 2015) applies per-cluster orthogonal transformations to align the geometry of residual clusters before quantization, reducing misalignment and improving quantization fidelity by solving Procrustes problems for each residual cluster.

3.4. Neural and Context-Adaptive RQ

Implicit neural codebooks (Huijben et al., 26 Jan 2024, Vallaeys et al., 6 Jan 2025) enable context-dependent codeword generation by conditioning codebooks at each stage on partial reconstructions (or previous stage selections). This approach leverages compact neural networks (e.g., MLPs) to produce specialized codebooks or codeword adjustments, capturing code dependencies and improving rate-distortion—closely related to neural additive quantization.

4. Applications Across Domains

4.1. High-dimensional ANN Search

RQ forms the basis for several leading approximate nearest neighbor search schemes, where high-dimensional database vectors are compressed into sequences of residual-quantized codes. Enhancements such as TRQ (Yuan et al., 2015) and IRVQ (Liu et al., 2015) yield substantial improvements in recall at fixed code length, outperforming Product Quantization (PQ) and Optimized PQ (OPQ) when feature independence does not hold. Large-scale benchmarks (SIFT1M, GIST1M, SIFT1B) consistently validate these advances.

4.2. Discrete Representation Learning

Neural models such as BRIDLE (Nguyen et al., 4 Feb 2025) integrate RQ into self-supervised training pipelines for audio, image, and video. Hierarchical codebooks allow for fine-grained latent discretization, improved code usage, and enhanced downstream performance, surpassing one-codebook VQ in encoder representation quality.

4.3. Compression and Denoising

Multilayer RQ and regularized codebooks enable VQ-based image compression and joint denoising (Ferdowsi et al., 2017), with efficacy confirmed on CroppedYale-B facial datasets. By avoiding overfitting (using random, regularized codebooks), these schemes outperform JPEG-2000 for low-rate compression and rival leading denoisers (BM3D) at high noise levels.

4.4. Quantization of Neural Networks

Recursive residual expansion strategies (e.g., REx (Yvinec et al., 2022)) and robust scalar quantization (e.g., RFSQ (Zhu, 20 Aug 2025)) integrate RQ ideeas with scalar or vector quantizers, providing enhanced trade-offs in device-specific, low-bit quantization for DNNs. Group sparsity and data-free schemes are especially relevant for efficient adaptation to hardware constraints.

4.5. Signal Coding in Real-Time and Neural Codecs

In neural audio codecs, RQ (often as Residual Vector Quantization, RVQ) enables efficient MDCT-domain coding, as in StreamCodec's RSVQ (Jiang et al., 9 Apr 2025), which combines scalar and vector quantizers for hierarchical and residual coding of causal features in real-time streaming applications.

5. Theoretical Limitations and Open Issues

5.1. Diminishing Returns and Entropy Loss

RQ is subject to diminishing returns; as the number of stages increases, the residual becomes increasingly unstructured and codebook utilization decays, as quantified by information entropy measurements (Figure 1 in (Liu et al., 2015)). This limits practical depth and motivates entropy-preserving learning strategies.

5.2. Computational Complexity

Optimal RQ encoding is NP-hard, as it can be formulated as a high-order Markov Random Field optimization. Most practical systems rely on greedy, multi-path, or approximated encoding.

5.3. Structural Mismatch for Hierarchical Data

RQ in Euclidean space inadequately models exponential growth in hierarchically structured data (e.g., trees), as Euclidean volume grows polynomially with radius. Hyperbolic RQ (HRQ) (Piękos et al., 18 May 2025) resolves this by shifting to hyperbolic geometry, improving discrete representation and downstream task performance for such data by aligning residual computation and codebook operations with hierarchical geometry.

5.4. Codebook Collapse and the Hourglass Phenomenon

Intermediate layers of multi-stage RQ can collapse, particularly in generative retrieval settings, resulting in underutilization and path sparsity (the "hourglass" effect (Kuai et al., 31 Jul 2024)). Remedies include layer removal, variable-length codes, or adaptive token pruning.

6. Experimental Validation and Comparative Results

Method	Dataset	Key Metric	RQ	Enhanced RQ (e.g., IRVQ/TRQ/QINCo2)
ANN Search	SIFT1M	Recall@4 (64-bit)	50.4%	58.3% (IRVQ) (Liu et al., 2015)
ANN Search	GIST1M	Recall@4	10.6%	16.2% (IRVQ)
ANN Search	SIFT1B	Recall@1 (10K)	0.359 (OPQ)	0.426 (TRQ) (Yuan et al., 2015)
Compression	BigANN1M	MSE (8B/16B)	2.49	1.12 (QINCo) (Huijben et al., 26 Jan 2024)
Audio Codec	LibriTTS	ViSQOL (1.5 kbps)	—	4.30 (RSVQ) (Jiang et al., 9 Apr 2025)
Video Perception	JHMDB	PCK/GBOPs	—	94.1/176 (ResQ) (Abati et al., 2023)

These figures illustrate the consistent pattern: RQ and its direct descendants typically outperform PQ/OPQ at the same or lower code budget, and further gains are obtainable via codebook adaptation, local transformation, or contextual/neural codebook constructions.

7. Specialized Adaptations and Future Directions

Modern research trends include:

• Neural codebooks and implicit networks for context-adaptive quantization (Huijben et al., 26 Jan 2024, Vallaeys et al., 6 Jan 2025).
• Hyperbolic quantization for structured/hierarchical data (Piękos et al., 18 May 2025).
• Plug-and-play, data-free expansion and adapter designs for fast post-training quantization (Yvinec et al., 2022, Luo et al., 1 Aug 2024).
• Dynamic code allocation and variable-rate coding in neural codecs leveraging importance maps and bit allocation strategies (Chae et al., 8 Oct 2024).
• Integration with generative models and LLM-based retrieval frameworks (Kuai et al., 31 Jul 2024).

Residual Quantization has evolved into a highly flexible, extensible framework that serves as the backbone of scalable, accurate, and resource-efficient discrete representation and search in modern machine learning and data compression pipelines. The continued convergence of algorithmic, information-theoretic, and neural modeling approaches will further extend RQ's reach and practical utility.