Geometric Vector Quantification Method

• Geometric vector quantification is a framework that maps complex, high-dimensional data into interpretable real vector spaces using geometric and statistical tools.
• It employs projective, tomographic, and scattering techniques to extract latent features from systems such as language models, photonic states, and quantum ensembles.
• Its implementations reduce computational overhead and improve interpretability in fields like AI, optics, and quantum state analysis through targeted measurement protocols.

The geometric vector quantification method provides a framework for extracting, characterizing, and reconstructing latent high-dimensional vectorial information embedded within complex physical, neural, or mathematical systems. Rooted in geometric principles, this class of methods converts the problem of quantifying abstract high-dimensional states—such as those arising in LLMs, photonic systems, quantum ensembles, or graph-structured data—into tractable measurements or computations amenable to geometric and statistical analysis. Approaches operating under this paradigm are central in fields spanning interpretable AI representations, quantum tomography, nonlinear optics, and geometric deep learning, using projective, tomographic, scattering, or interferometric constructs to achieve a direct mapping to structured, interpretable vector spaces.

1. Mathematical Foundations and General Principles

At its core, geometric vector quantification involves defining an explicit real vector space, each axis of which is identified with a physically or semantically interpretable property. For a domain $D$, a typical instantiation is an $n$-dimensional real vector space: $$\mathcal{V} = \mathrm{span}\{e_1,e_2,\dots,e_n\} \subset \mathbb{R}^n$$ where $e_i$ correspond to ontologically distinct properties or measurement features. Each entity, state, or concept is mapped to a real vector $v\in\mathbb{R}^n$, with $v_i$ quantifying the entity along property $b_i$. This geometric embedding enables the use of all standard metric and statistical tools for analysis, including distance, similarity, and subspace structure (Rothenfusser et al., 16 Jun 2025).

Complementarily, the geometric vector quantification method extends to the reconstruction of Hilbert-space states, for instance in quantum or photonic systems. The goal is to obtain a complete specification of the state (up to global phase) by measuring a reduced set of geometrically defined observables, thereby avoiding full-field reconstructions (Fang et al., 28 Jun 2025).

2. Domain-Specific Implementations

a. Vector Ontologies in World Model Extraction

In machine learning, especially for interpretability of neural representations, a "vector ontology" is constructed by selecting domain-relevant features (e.g., audio properties for musical genres) and assigning each as a basis vector. LLMs are then prompted so as to output values along each dimension, which are aggregated into an interpretable geometric vector. For instance, a musical genre might be represented by its projections along Spotify's audio features: danceability, energy, speechiness, acousticness, instrumentalness, liveness, valence, and tempo. These integer-valued indices can be mapped via bin-centering to real coordinates, formalizing the LLM's latent representation as a point in an 8D vector space (Rothenfusser et al., 16 Jun 2025).

b. Photonic 4D Spin-Orbit State Quantification

For high-dimensional photonic states, arbitrary pure states in a 4D Hilbert space (spanned by two OAM and two SAM basis states) are quantified by mapping the state to the SU(4) Poincaré hypersphere. Instead of iterative modal decomposition, three distinct 2D centroid ellipses are fitted from simple interferometric measurements. The salient geometric features (ellipticity and orientation) of each ellipse correspond directly to the spherical coordinates parameterizing the constituent SU(2) subspaces, enabling extraction of all nine Stokes-like parameters uniquely specifying the state up to global phase (Fang et al., 28 Jun 2025).

c. Tomographic Portraits of Quantum Spin Systems

In quantum tomography, normalized non-redundant vector tomograms are obtained by measuring expectation values of dequantizer operators $D_k$ (linear combinations of identity and Pauli matrices parameterized by non-coplanar unit vectors). The resulting vector of measurement outcomes forms a normalized tomogram, from which the full quantum state can be linearly reconstructed using a dual set of quantizer operators (Korennoy, 2018).

d. Equivariant Geometric Scattering on Graphs

For vector fields defined on nodes of a geometric graph, geometric vector quantification is performed by applying equivariant scattering transforms. These transforms use vector diffusion wavelets that propagate vector features across the graph in a symmetry-respecting manner, yielding invariant or equivariant descriptors suitable for downstream learning and analysis (Johnson et al., 1 Oct 2025).

3. Measurement, Projection, and Inference Protocols

A distinguishing feature of geometric vector quantification methods is the use of application-dependent schemes to project hidden or latent states into the interpretable vector space:

• Projection via Prompting (LLMs): LLMs are queried with systematically varied natural language prompts, and their outputs along pre-determined axes are discretized, centered, and embedded as integer or real vectors (Rothenfusser et al., 16 Jun 2025).
• Interferometric Ellipse Fitting (Photonic States): The unknown 4D state is interfered with a fixed reference, with spin/polarization projections measured as centroid trajectories as a function of optical delay, yielding three ellipses whose parameters are algebraically mapped to the desired spherical coordinates (Fang et al., 28 Jun 2025).
• Tomographic Linear Mapping (Quantum States): Measurement outcomes corresponding to specifically constructed dequantizer observables are collected and linearly inverted to recover the full quantum state or the Bloch vector (Korennoy, 2018).
• Wavelet Scattering (Geometric Graphs): Multi-scale wavelet transforms parameterized by graph topology and local tangent frames are applied to vector fields; the scattering moments or coefficients at various scales are used as compact, invariant descriptors (Johnson et al., 1 Oct 2025).

4. Quantification Metrics and Geometric Analysis

Evaluation and comparison within geometric vector quantification frameworks typically involve geometric metrics that capture both internal consistency and external validity:

• Cosine Similarity and Euclidean Distance: Used to quantify alignment between vectors or inferred centroids and ground-truth references (Rothenfusser et al., 16 Jun 2025).
• Centroid and Pairwise Distances: To assess consistency across multiple measurements or prompt formulations.
• Hypersphere-Volume Coverage: Fraction of the total vector space volume occupied by measurement clusters, providing an interpretable measure of spatial concentration or dispersion.
• Stokes Parameters and Generalized Coordinates: Complete sets of orthogonal projections (e.g., the nine coordinates on the SU(4) Poincaré hypersphere in photonics) serve as a unique characterization of each state (Fang et al., 28 Jun 2025).
• Tomogram Normalization and Non-Redundancy: Ensuring that vectorized measurement outcomes are linearly independent and normalized supports invertibility and robustness in reconstruction (Korennoy, 2018).

5. Experimental Realizations and Computational Protocols

Experimental and computational instantiations vary by domain:

• Photonic Experiments utilize wave plates, $q$-plates, and polarization beam splitters to prepare, interfere, and measure requisite states, with high-speed imaging to extract centroid trajectories in real time. The method has been successfully applied to free-space beams as well as higher-order mode groups in few-mode fibers, tracking modal evolution under perturbations without full-field tomography (Fang et al., 28 Jun 2025).
• LLM World View Extraction is conducted by prompt engineering, repeated querying, and aggregation of outputs to form vectorial concept embeddings, validated against large music databases (Rothenfusser et al., 16 Jun 2025).
• Geometric Scattering on Graphs is implemented by constructing diffusion operators, computing local frames via SVD, and efficiently assembling sparse wavelet transforms to quantify graph-structured data (Johnson et al., 1 Oct 2025).
• Quantum State Tomography employs measurement settings prescribed by closure and non-redundancy constraints on geometric vectors, ensuring complete specification with minimal redundancy (Korennoy, 2018).

6. Generalizations, Applications, and Extensions

The geometric vector quantification paradigm generalizes to any system where (i) the domain possesses interpretable, preferably orthogonal, properties, (ii) sufficient measurement or elicitation protocols exist for state projection, and (iii) geometric or statistical methods can be exploited for analysis and verification (Rothenfusser et al., 16 Jun 2025).

Application areas include:

• Multi-dimensional optical metrology
• Structured-light communication in classical and quantum regimes
• Interpretable world model extraction from LLMs
• Quantitative analysis of spin-orbit and orbital degrees of freedom in photonic and quantum systems
• Vector-valued invariant feature extraction in geometric machine learning

A plausible implication is that as high-dimensional experimental and computational systems become pervasive, geometric vector quantification offers a principled, operational approach for state characterization and analysis, often circumventing the need for high-complexity inversion or full state tomography.

7. Comparative Advantages and Limitations

Key advantages include:

• Direct geometric interpretability, mapping latent or abstract high-dimensional states to concrete, physically or semantically meaningful coordinates.
• Reduction of experimental and computational overhead by focusing on high-information observables (such as centroids or specific projection dimensions) instead of full field or density matrix reconstructions.
• Robustness to measurement imperfections due to normalization and redundancy-minimization in the quantification schemes.
• Amenability to generalization for $N$-dimensional internal spaces via construction of appropriate interference or measurement channels (cycling to SU($N$) representations in optics or generalized ontologies in AI) (Fang et al., 28 Jun 2025, Rothenfusser et al., 16 Jun 2025).

Limitations include potential requirements for domain-specific calibration, the need for accurate binning or discretization in many practical settings, and constraints arising from measurement device imperfections or the design of suitable projective measurements.

---

References:

• "Vector Ontologies as an LLM world view extraction method" (Rothenfusser et al., 16 Jun 2025)
• "Geometric quantification of photonic 4D spin-orbit states" (Fang et al., 28 Jun 2025)
• "Normalized non-redundant vector tomographic portraits of spin states" (Korennoy, 2018)
• "Equivariant Geometric Scattering Networks via Vector Diffusion Wavelets" (Johnson et al., 1 Oct 2025)