PrismQuant: Rate-Distortion-Optimal Vector Quantization for Gaussian-Mixture Sources

Classical rate-distortion (RD) theory has long established the fundamental limits of lossy compression by quantifying the minimum number of bits required to represent a source under a prescribed distortion constraint. However, widely used distortion measures such as mean-squared error often fail to capture perceptual quality or semantic validity, which are increasingly central in modern learning-driven applications. Rate-distortion-perception (RDP) theory extends the RD framework by introducing perception as a third fundamental axis, quantified via distributional similarity between the source and reconstructed signals, leading to the rate-distortion-perception function (RDPF). This tutorial provides a structured overview of the coding principles underlying perception-aware lossy compression and surveys recent achievability results under different randomness assumptions. It then presents a unifying optimization viewpoint for computing the RDPF as defined by Blau and Michaeli, for both discrete and continuous sources under broad families of perceptual constraints, including f-divergences, alpha-divergences, and Wasserstein-based metrics. Special attention is given to computational tools such as alternating minimization schemes, Newton-based methods, and convex optimization formulations, as well as to analytically tractable cases such as Gaussian sources and the perfect-realism regime. Unlike recent broad surveys that emphasize generative architectures and AI-empowered communication systems, this tutorial focuses on the coding-theoretic and computational machinery needed to characterize, compute, and interpret the RDP limits. Finally, the tutorial outlines promising research directions at the intersection of information theory, neural compression, robust source coding, and perception-aware networked control systems.

Uniform random rotations (URRs) are a common preprocessing step in modern quantization approaches used for gradient compression, inference acceleration, KV-cache compression, model weight quantization, and approximate nearest-neighbor search in vector databases. In practice, URRs are often replaced by randomized Hadamard transforms (RHTs), which preserve orthogonality while admitting fast implementations. The remaining issue is the performance for worst-case inputs. With a URR, each coordinate is individually distributed as a shifted beta distribution, which converges to a Gaussian distribution in high dimensions. Generally, one RHT is not suitable in the worst case, as individual coordinates can be far from these distributions. We show that after composing two RHTs on any $d$-sized input vector, the marginal distribution of every fixed coordinate of the normalized rotated vector is within $O(d^{-1/2})$ of a standard Gaussian both in Kolmogorov distance and in $1$-Wasserstein distance. We then plug these bounds into the analyses of modern compression schemes, namely DRIVE and QUIC-FL, and show that two RHTs achieve performance that asymptotically matches URRs. However, we show that two RHTs may not be sufficient for Vector Quantization (VQ), which often requires weak correlation across fixed-size blocks of coordinates (as opposed to only marginal distribution convergence for single coordinates). We prove that a composition of three RHTs leads to decaying coordinate covariance. This ensures that any fixed, bounded, multi-dimensional VQ codebook optimized for URRs has the same expected error when using three RHTs, up to an additive term that vanishes with the dimension. Finally, because practical inputs are rarely adversarial, we propose a linear-time ${O}(d)$ check on the input's moments to dynamically adapt the number of RHTs used at runtime to improve performance.

We derive the optimal quantizer of a real-valued random variable $W$ with distribution $P_W$ such that 1) the distribution of the quantization output $X$ that can take $k$ values follows any specified distribution $P_X$ over $\{1,\ldots,k\}$, and 2) the minimum mean squared error (MMSE) of estimating $W$ from $X$ is minimized. It is shown that the optimal quantizer takes the form $X=σ\big(F_{σ^{-1}(X)}^{-1}(F_W(W))\big)$, where $σ$ is the optimal permutation of $\{1,\ldots,k\}$ among all permutations to minimize the MMSE, and $F$ is the cumulative distribution function. When $P_W$ is uniform over an interval or $P_X$ is uniform over $\{1,\ldots,k\}$, the quantizer takes a simple form $X=F_{X}^{-1}(F_W(W))$. The concept of majorization plays a key role in the optimality proof. Specifying the output distribution is useful for designing quantizers with explicitly controlled output entropy, maximized mutual information between input and output, tailored output distribution to match channel input requirements for communication, and data anonymization.