Block-Sphere Vector Quantization

A unified theoretical comparison of rotation-based quantizers and a new block-spherical algorithm with improved distortion guarantees

Heesang Ann; Joongkyu Lee; Min-hwan Oh· 2026· DOI 10.48550/arXiv.2605.19972

The core problem

Vector quantization is a fundamental primitive for scalable machine learning systems, enabling memory-efficient storage, fast retrieval, and compressed inference. Recent rotation-based quantizers such as EDEN, RabitQ, and TurboQuant have introduced strong guarantees and empirical performance, but the surrounding comparisons have been difficult to interpret because they rely on different distortion criteria, probability regimes, and implementation assumptions. This paper addresses two main gaps: first, it provides a unified theoretical comparison of these methods, showing that their relative advantages are criterion-dependent rather than absolute. Second, it introduces Block-Sphere Quantization (BlockQuant), a new rotation-based block quantization algorithm designed around the spherical geometry of randomly rotated vectors. The paper proves that BlockQuant theoretically improves over the baselines for both reconstruction MSE and expected inner-product distortion, and validates these improvements experimentally on real embedding datasets and long-context LLM inference tasks.

The paper reports experiments on real embedding datasets and long-context LLM inference tasks. The results show practical gains that are consistent with the theoretical improvements. Specifically, BlockQuant achieves lower reconstruction MSE and lower expected inner-product distortion compared to the baselines (EDEN, RabitQ, TurboQuant) under the same bit budgets. In long-context LLM inference, BlockQuant enables more memory-efficient storage and faster retrieval without significant loss in accuracy. The experiments validate the theoretical claims that block-spherical quantization preserves the geometry of rotated embeddings more faithfully than coordinate-wise quantizers. Quantitative results are presented in the paper, demonstrating consistent improvements across different datasets and tasks.

Vector quantization is a fundamental primitive for scalable machine learning systems, enabling memory-efficient storage, fast retrieval, and compressed inference. Recent rotation-based quantizers such as EDEN, RabitQ, and TurboQuant have introduced strong guarantees and empirical performance, but the surrounding comparisons have been difficult to interpret because they rely on different distortion criteria, probability regimes, and implementation assumptions. This paper addresses two main gaps: first, it provides a unified theoretical comparison of these methods, showing that their relative advantages are criterion-dependent rather than absolute. Second, it introduces Block-Sphere Quantization (BlockQuant), a new rotation-based block quantization algorithm designed around the spherical geometry of randomly rotated vectors. The paper proves that BlockQuant theoretically improves over the baselines for both reconstruction MSE and expected inner-product distortion, and validates these improvements experimentally on real embedding datasets and long-context LLM inference tasks.

The paper first establishes a unified theoretical framework to compare EDEN, RabitQ, and TurboQuant under different distortion criteria (MSE, expected inner-product distortion, and high-probability control). This comparison reveals that EDEN and TurboQuant are favorable for MSE distortion, EDEN is also effective for expected inner-product distortion, and RabitQ provides strong high-probability control.

The unified theoretical comparison clarifies that the relative advantages of rotation-based quantizers are criterion-dependent. EDEN and TurboQuant are favorable for MSE distortion, EDEN is also effective for expected inner-product distortion, and RabitQ provides strong high-probability control. This means that practitioners should choose a quantizer based on the specific distortion criterion and probability regime relevant to their application. BlockQuant introduces a new design principle: quantizing blocks on the sphere rather than coordinate-wise. This preserves the geometry of randomly rotated vectors more faithfully, leading to improved theoretical guarantees for both reconstruction MSE and expected inner-product distortion. The experimental results on real embedding datasets and long-context LLM inference tasks confirm these theoretical improvements. The paper's contributions are twofold: a unified comparison that aids in selecting the right quantizer, and a new algorithm that advances the state of the art in rotation-based quantization. Future work may explore adaptive block sizes and codebook design to further improve performance.

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