Scalar-Vector Combined Quantization

• Scalar–vector–combined quantization is an encoding strategy that decomposes high-dimensional signals into scalar gains and vector shapes, ensuring efficient rate–distortion performance.
• It integrates adaptive scalar quantization for magnitude features with residual-based vector quantization to preserve perceptual attributes and reduce bitrate.
• Implementations like Daala PVQ and RWKVQuant demonstrate improved coding efficiency, robust adaptation, and perceptually aligned reconstruction in video, audio, and neural network compressions.

Scalar–vector–combined quantization strategies encode high-dimensional signals with both scalar and vector quantization operators, exploiting structure and perceptual models to achieve near-optimal rate–distortion performance and adaptivity. This composite approach has been used in video coding (Daala PVQ), online vector quantization (TurboQuant), neural model PTQ (RWKVQuant), and neural audio coding (StreamCodec RSVQ). These systems improve coding efficiency, preserve perceptual features such as texture or signal envelope, and enable low-bitrate deployment across diverse architectures.

1. Mathematical Principles of Scalar–Vector Quantization

Scalar–vector–combined quantization decomposes the input signal $x\in\mathbb{R}^N$ into components suitable for separate quantization modalities. A typical factorization is gain–shape, as in Daala PVQ (Valin et al., 2016), or sequential residuals as in StreamCodec RSVQ (Jiang et al., 9 Apr 2025). The core mathematical framework is as follows:

• Gain–Shape Decomposition: $x = g\,u$ with $g = \|x\|_2$ (scalar gain, energy) and $u = x/\|x\|_2$ (shape, direction).
• Sequential Residual Quantization: For $z\in\mathbb{R}^D$, $r_0 = z - Q_s(z)$, $r_j = r_{j-1} - Q_v^{(j)}(r_{j-1})$, reconstructing $\hat{z} = Q_s(z) + \sum_{j} Q_v^{(j)}(r_{j-1})$.

Scalar quantizers operate on magnitude components (energy, singular value, scalar projection), while vector quantizers operate on shape, direction, or residuals, using either algebraic codebooks (PVQ pyramid), deterministic K-means, or learned codebooks.

2. Workflow: Quantizer Integration and Selection

Constructing a scalar–vector–combined quantizer entails:

• Scalar Quantization: Uniform or non-uniform quantizers for gain, bias, or individual coordinates; e.g., Daala: $\hat{g} = \gamma Q$; RWKVQuant: $Q_{\mathrm{SQ}}(x)$ with scale and zero-point.
• Vector Quantization: Normalized integer codebooks (Daala), cluster-based per-block codebooks (RWKVQuant), improved vector quantizers with codebook balancing and clustering (RSVQ), or product/online codebooks (TurboQuant).

Adaptive selection mechanisms choose the quantization mode per block, layer, or signal domain. RWKVQuant applies a coarse-to-fine proxy (Xu et al., 2 May 2025):

• Interval entropy proxy $P_c(G)$ for uniformity.
• Moment-based proxy $P_f(G)$ for outlier detection.
• Decision logic: If $P_c < T_c$ and $P_f < T_f$, use scalar; else use vector quantization.

This enables local adaptation to signal statistics and structural heterogeneity.

3. Advanced Features: Perceptual Modeling and Entropy Coding

Scalar–vector–combined strategies exploit perceptual models and codebook structure for gains in rate–distortion efficiency:

• Contrast masking via companding: Daala modifies scalar gain quantization to match perceptual masking curves: $\hat{g} = Q_g\, \gamma^{\beta}$ with $\beta = 1/(1-\alpha)$ (Valin et al., 2016).
• Shape codebook resolution: The number of PVQ pulses $K$ matches angular and gain quantization error, adapting deterministically to the gain.
• Entropy coding: Daala PVQ conditions magnitude/runlength models on pulses, improving coding sharpness. StreamCodec RSVQ uses codebook balancing loss and clustering to force uniform codebook utilization (Jiang et al., 9 Apr 2025).

These mechanisms maximize signal fidelity at fixed bitrate and avoid the inefficiencies of independent scalar or vector quantization.

4. Residual and Hierarchical Vector Quantization

Several approaches extend scalar–vector–combined quantization to hierarchical or residual encoding:

• Residual Scalar–Vector Quantization (RSVQ): StreamCodec (Jiang et al., 9 Apr 2025) encodes first the scalar features, then refines acoustic details residually via two improved vector quantizers.
• TurboQuant’s staged quantization: Scalar-quantize post-rotation, then apply 1-bit Quantized JL on the residual for unbiased inner-product estimation (Zandieh et al., 28 Apr 2025).
• Householder-reflected PVQ prediction: In Daala, the predicted signal is energy-conserved via reflection, maintaining the link between the encoded gain and the actual signal, exploiting correlation without discarding perceptual masking (Valin et al., 2016).

Hierarchical schemes enable flexible budget allocation, codebook balancing, and multi-bit accuracy control.

5. Algorithms and Complexity

All papers provide efficient implementation strategies:

• Daala PVQ: Algebraic pyramid codebooks allow on-the-fly vector search with no training.
• TurboQuant: Fully online, codebook-free, vectorizable via random rotation and per-coordinate lookups; time complexity dominated by rotation ($O(d^2)$), quantization ($O(db)$), and QJL ($O(d^2)$), but can be reduced by structured rotation.
• RWKVQuant: Proxy evaluation $O(N\log N)$ per block, codebook optimization via weighted K-means.
• StreamCodec RSVQ: Scalar projection and trainable vector codebooks; clustering and balancing loss for codebook efficiency.

These techniques deliver memory-saving ratios and inference-time speedups that scale with bits-per-weight and codebook adaptivity.

6. Rate–Distortion and Experimental Outcomes

Scalar–vector–combined quantization achieves substantial improvement over conventional scalar-only or vector-only schemes in both objective and subjective metrics.

Method & Domain	Main Quantization Strategy	Key Metrics	Empirical Results
Daala PVQ	Gain-shape, companded masking	PSNR, bitrate, texture preservation	+0.9 dB / –25% bitrate (image), +0.83 dB / –14% bitrate (video) (Valin et al., 2016)
TurboQuant	Scalar + 1-bit QJL (residual)	MSE, inner-product distortion	Near-Shannon rate, recall-neutrality at 3.5 bits/channel (KV cache) (Zandieh et al., 28 Apr 2025)
RWKVQuant	Proxy-guided blockwise hybrid	Perplexity, zero-shot, speed/mem	<1% loss, 2.14× speedup, 2.83× mem save at 3.275 bits (RWKV-6 14B) (Xu et al., 2 May 2025)
StreamCodec RSVQ	Scalar → IVQ+IVQ (residual)	ViSQOL, bitrate-efficiency, codebook CUR	4.30 ViSQOL, 98.5% BE, 100% CUR (LibriTTS 1.5 kbps) (Jiang et al., 9 Apr 2025)

Across modalities, these strategies eliminate representational redundancy, optimize perceptual masking, and maximize codebook utilization.

7. Practical Deployment and Architectural Considerations

Deployment strategies leverage architecture-specific proxying and codebook updates:

• For models with highly uniform or outlier-rich weights (RWKV vs LLaMA), adapt thresholding to achieve optimal SQ/VQ mixes.
• Weighted K-means codebooks for element-wise kernels deliver up to +0.5% accuracy.
• Memory and speed savings scale with percentage of SQ blocks and bit-width; e.g., hybrid quantization with $\delta=0.6, b_{\text{sq}}=3.25, k=3.5$ yields 2× GPU speedup (Xu et al., 2 May 2025).
• StreamCodec’s RSVQ attains high codebook utilization and bitrate-efficiency, and employs balancing losses and clustering for robust codebook maintenance.

Threshold parameters are chosen for desired bpw; proxy and codebook parameters enable smooth accuracy–performance mixes and robust adaptation to changing signal statistics.

---

In summary, scalar–vector–combined quantization strategies (gain–shape factorization, staged residual encoding, blockwise proxies) provide efficient, theoretically grounded solutions for high-fidelity, low-bitrate signal representation and are widely adopted in modern video, audio, and neural network compression systems (Valin et al., 2016, Zandieh et al., 28 Apr 2025, Xu et al., 2 May 2025, Jiang et al., 9 Apr 2025).