Quantization-Based Distances

• Quantization-based distances are metrics and pseudometrics that measure the error from approximating complex or infinite objects by finite representations.
• They use variational formulations and optimal transport principles to formalize structural discrepancies in data, geometry, and quantum states.
• Applications span machine learning, signal processing, and quantum information, enabling efficient resource quantification and task-specific adaptations.

Quantization–based distances are a class of metrics and pseudo-metrics arising from or intrinsically linked to the process of quantizing data, measures, or geometric objects. Such distances formalize the error or structural discrepancy induced by replacing a complex or infinite object by a finite or discrete approximation, with definitions and analytic tools varying across probability theory, geometry, signal processing, statistical learning, and mathematical physics. These structures serve as the mathematical backbone for optimal vector quantization, metric learning, kernel approximations, quantum/classical correspondence, and resource quantification in quantum information.

1. Foundational Principles and Definitions

Quantization–based distances generally measure the cost (geometrically or algebraically) of substituting a rich object (measure, vector, function, operator, sheaf) with a finite, more tractable representative. This is exemplified by the canonical distortion for vector quantization, branched optimal transport costs, MMD–based quantization, quantum pseudometrics, and resource distances on quantum measurement assemblages.

Common foundational features include:

• Variational or infimal formulations: Distances are defined as the minimal cost over a space of feasible couplings, plans, or codes, e.g., optimal quantizers in branched transport (Pegon et al., 2023), or optimal atomic approximations in MMD (Mehraban et al., 14 Mar 2025).
• Pseudometric structures: Many quantization–based distances are only pseudometrics, not genuine metrics, due to invariances or physical constraints (e.g., in quantum pseudometrics (Golse et al., 2021, Golse et al., 2017)).
• Induced metrics by environment or task: The Canonical Distortion Measure (CDM) defines a task–adapted distance by integrating the output loss over an environment of target functions (Baxter, 2019).

These formalizations encode the substantive effect of quantization not only on pointwise errors but directly in functionals, distances, or resource monotones.

2. Quantization–Based Distances in Optimal Transport and Measure Approximation

Optimal transport provides several fundamental settings for quantization–based distances:

• Branched Transport Quantization: In (Pegon et al., 2023), the branched optimal transport distance $\mathbf d_\alpha$ is defined via Lagrangian traffic plans, with cost structures favoring mass aggregations and merging. The optimal $N$–point quantization problem seeks atomic measures minimizing $\mathbf d_\alpha$ to a target measure. Distinct from the classical ($\alpha=1$) semi-discrete OT where partition regions are Voronoi cells, for $\alpha<1$ the corresponding cell boundaries become fractal and no explicit half-space structure is available.
• Asymptotic Results: The “branched transport Zador theorem” provides the scaling of optimal costs and the limit empirical distributions of quantizer supports, extending the classical Zador’s theorem to the branched setting.
• Rate–Distortion and Uniformity: Uniform Delone–type bounds in the Ahlfors–regular case ensure both covering and separation radii scale optimally with $N^{-1/d}$.

For algorithmic applications, quantization–based approximation accelerates computation of Wasserstein distances between measures by “compressing” empirical samples with k–means or similar clustering, then computing OT over the quantized supports with controlled bias and variance tradeoffs (Beugnot et al., 2021).

3. Quantization–Induced and Task–Specific Metrics

Quantization–based metrics can be constructed to encode task-adaptive similarity:

• Canonical Distortion Measure (CDM): Introduced by Baxter (Baxter, 2019), the CDM for a space $X$ and function “environment” $(\mathcal{F},\mathcal{Q})$ is

$$d_{CDM}(x,x') = \int_{\mathcal{F}} o(f(x), f(x'))\, d\mathcal{Q}(f)$$

where $o$ is an output–space distortion. This metric is canonical: optimizing quantization with respect to $d_{CDM}$ yields Voronoi partitions simultaneously minimizing expected approximation error for all $f\in \mathcal{F}$. For linear, thresholded-linear, and quadratic environments, CDM reduces to squared Euclidean, angular, and function–specific metrics, respectively.

• Learning CDMs: When $\mathcal{F}$ is unknown but samplable, neural networks can be trained (via empirical CDM–labels) to learn $d_{CDM}$, making quantization task-informed.
• Contrast to Standard Metrics: Hamming and Euclidean are output–agnostic; CDM directly encodes relevance for downstream tasks.

4. Quantization in Kernel, Quantum, and Sheaf-Theoretic Frameworks

Quantization–based distances are prominent in advanced frameworks:

• MMD–based Quantization: For kernel-based distances, the Maximum Mean Discrepancy (MMD) between a continuous target $P$ and quantized $P^n = \sum p_i\delta_{x_i}$ is given by RKHS norms of embeddings (Mehraban et al., 14 Mar 2025). Explicit expressions for the optimal weights (subject to simplex constraints) and joint minimization over supports reduce the problem to convex or stochastic-gradient regimes, and admit efficient deterministic solvers for Gaussian kernels. The resulting minimal MMD encodes quantization fidelity in the chosen kernel topology.
• Quantum Pseudometrics: Extensions of Wasserstein/Monge–Kantorovich distances to quantum settings proceed via positive quantizations and operator-valued couplings (Golse et al., 2021, Golse et al., 2017). The semiquantum pseudometric $\mathcal{E}_\hbar(f,R)$ infimalizes over couplings between a classical and a quantum state (partial quantization), with a dual Kantorovich formula incorporating operator constraints. The fully quantum pseudo-distance $\mathrm{MK}_\hbar$ utilizes operator-valued cost terms. Quantum quantization–based distances are essential for linking mean-field quantum dynamics to classical transport and provide metrics sensitive to semiclassical regime phenomena, unlike Schatten norms.
• Sheaf-Theoretic Quantization and Interleaving: In the microlocal and symplectic setting, derived interleaving distances define a quantization–based metric for sheaves (Asano et al., 2022). This structure is compatible with Hamiltonian flows via sheaf quantizations, and is metrically complete. The interleaving distance encodes stability under Hamiltonian perturbations, and the quantization of Hamiltonian homeomorphisms leverages this metric for Arnold-type theorems.

5. Quantization–Based Distances in High-Dimensional Signal Processing and Machine Learning

Several high-impact methods leverage efficient quantization–based distance computation in large-scale data analysis:

• Pairwise Quantization: Direct minimization of pairwise distortions (dot–product or squared distance) yields quantization transformations (via a SQRT-GRAM or SVD–based linear transform) such that standard quantizers (PQ, OPQ) can be repurposed for relation–centric objectives (Babenko et al., 2016). This approach eliminates traditional bias, notably for inner product and squared distance tasks, and achieves substantial improvements in retrieval and recommendation despite potential drawbacks for nearest-neighbor search accuracy.
• Binary Embeddings via Noise-Shaping Quantization: Distance–preserving binary encodings using ΣΔ quantization on sparse Gaussian projections achieve $\ell_1$–recoverability of Euclidean distances with polynomially decaying quantization error, and outperform traditional memoryless sign-based binary embeddings (Zhang et al., 2020). The time and memory complexity is optimal up to log factors for well–spread data.
• Graph–Quantization Synergy for ANN Search: Advanced ANN indices (e.g., SymphonyQG) tightly couple quantization (e.g., RaBitQ, a bi-valued rotated codebook with unbiased estimator) with SIMD–accelerated FastScan on graph structures, optimizing both graph layout and batch SIMD memory access to expediently estimate distances without expensive re-ranking (Gou et al., 2024).

6. Quantization Artifacts and Error Analysis

The process of quantization induces structural and analytic artifacts in computed distances and functionals:

• Banding in Distance Transforms: Digital quantization in distance transforms introduces quantization error independent of grid spacing, manifesting as gradient banding and discrete plateau effects in the transform, which are robust to grid refinement (Besler et al., 2020). Dithering and PDE-based reinitialization can remove artifacts when exact representations are not available.
• Asymptotic and Uniformity Results: In branched optimal transport quantization (Pegon et al., 2023), uniform separation and covering bounds for quantizer support are established under regularity hypotheses, and partition regions may exhibit highly irregular (possibly fractal) structure as a direct result of quantization cost non-additivity and traffic merging.

7. Applications Across Mathematics, Physics, and Information Theory

Quantization–based distances undergird a wide array of advanced applications:

• Machine Learning: Efficient pairwise–preserving quantization and kernel–based quantization for scalable kernel mean approximations, density estimation, and SVM acceleration (Babenko et al., 2016, Mehraban et al., 14 Mar 2025).
• Quantum Information and Resource Theory: Distance metrics on quantum measurements (diamond–norm–based) induce operationally meaningful resource monotones (e.g., informativeness, coherence, incompatibility), with analytical bounds and implications for quantum resource hierarchies and Bell–type experiments (Tendick et al., 2022).
• Symplectic and Microlocal Geometry: Interleaving–based distances for sheaves are deployed for Hamiltonian homeomorphism quantization and topological invariants (Asano et al., 2022), with completeness, stability, and spectral interpretations.
• Semiclassical and Mean–Field Quantum Analysis: Quantum pseudometrics relate classical and quantum mean–field models, providing convergence rates sharp in $\hbar$ as well as phase–space–sensitive measures, unmatched by operator norms (Golse et al., 2021, Golse et al., 2017).

These multiply intersecting threads reveal the central role of quantization–based distances wherever finite, discrete, or transformed representations replace or approximate richer mathematical structures, dictating the attainable precision, computational efficiency, and theoretical guarantees across disciplines.