Vector quantization, a problem rooted in Shannon's source coding theory, aims to quantize high-dimensional Euclidean vectors while minimizing distortion in their geometric structure. We propose TurboQuant to address both mean-squared error (MSE) and inner product distortion, overcoming limitations of existing methods that fail to achieve optimal distortion rates. Our data-oblivious algorithms, suitable for online applications, achieve near-optimal distortion rates (within a small constant factor) across all bit-widths and dimensions. TurboQuant achieves this by randomly rotating input vectors, inducing a concentrated Beta distribution on coordinates, and leveraging the near-independence property of distinct coordinates in high dimensions to simply apply optimal scalar quantizers per each coordinate. Recognizing that MSE-optimal quantizers introduce bias in inner product estimation, we propose a two-stage approach: applying an MSE quantizer followed by a 1-bit Quantized JL (QJL) transform on the residual, resulting in an unbiased inner product quantizer. We also provide a formal proof of the information-theoretic lower bounds on best achievable distortion rate by any vector quan-tizer, demonstrating that TurboQuant closely matches these bounds, differing only by a small constant ( 2. 7) factor. Experimental results validate our theoretical findings, showing that for KV cache quantization, we achieve absolute quality neutrality with 3.5 bits per channel and marginal quality degradation with 2.5 bits per channel. Furthermore, in nearest neighbor search tasks, our method outperforms existing product quantization techniques in recall while reducing indexing time to virtually zero. 1 Introduction Vector quantization (VQ) in Euclidean space is crucial for efficiently handling high-dimensional vectors across a spectrum of computational domains, from training and deploying large-scale AI and deep learning models to powering vector databases for search/retrieval systems. The core objective is to compress high dimensional vectors by quantizing them-converting floating-point coordinate values to low-bitwidth integers-while minimizing distortion, quantified by metrics such as 1 arXiv:2504.19874v1 By preserving these properties, inner product queries can be answered rapidly, with minimal latency, and using reduced computational and communication resources. This problem's roots trace back to Shannon's seminal work on Source Coding theory [48, 49], which established that the least distortion achievable by block source codes, now known as vector quan-tizers, is defined by the Shannon distortion-rate function, determined by the statistical properties of the source and the chosen distortion measure, such as MSE. Today, VQ plays a critical role in fundamental computational domains, including AI, deep learning, and search systems. A key application of VQ is in the deployment of AI models, including large language models (LLMs) [5, 18, 7, 52].

Recent research efforts on semantic communication have mostly considered accuracy as a main problem for optimizing goal-oriented communication systems. However, these approaches introduce a paradox: the accuracy of artificial intelligence (AI) tasks should naturally emerge through training rather than being dictated by network constraints. Acknowledging this dilemma, this work introduces an innovative approach that leverages the rate-distortion theory to analyze distortions induced by communication and semantic compression, thereby analyzing the learning process. Specifically, we examine the distribution shift between the original data and the distorted data, thus assessing its impact on the AI model's performance. Founding upon this analysis, we can preemptively estimate the empirical accuracy of AI tasks, making the goal-oriented semantic communication problem feasible. To achieve this objective, we present the theoretical foundation of our approach, accompanied by simulations and experiments that demonstrate its effectiveness. The experimental results indicate that our proposed method enables accurate AI task performance while adhering to network constraints, establishing it as a valuable contribution to the field of signal processing. Furthermore, this work advances research in goal-oriented semantic communication and highlights the significance of data-driven approaches in optimizing the performance of intelligent systems.

W e contribute to advancing the understanding of Riemannian accelerated gradient methods. In particular, we revisit " Accelerated Hybrid Proximal Extragradient " (A-HPE), a powerful framework for obtaining Euclidean accelerated metho ds [ 29 ]. Building on A-HPE, we then propose and analyze Riemannian A-HPE. The core of our analysis consists of two key components: (i) a set of new insights into Euclidean A -HPE itself; and (ii) a careful control of metric distortion caused by Riemannian g eometry . W e illustrate our framework by obtaining a few existing and new Riemannian acc elerated gradient methods as special cases, while characterizing their accelerat ion as corollaries of our main results.

We propose the first global accelerated gradient method for Riemannian manifolds. Toward establishing our result we revisit Nesterov's estimate sequence technique and develop an alternative analysis for it that may also be of independent interest. Then, we extend this analysis to the Riemannian setting, localizing the key difficulty due to non-Euclidean structure into a certain ``metric distortion.'' We control this distortion by developing a novel geometric inequality, which permits us to propose and analyze a Riemannian counterpart to Nesterov's accelerated gradient method.