Subjective Conceptual Spaces: Frameworks & Methods

• Subjective conceptual spaces are mathematical models that represent individual bias and context-dependence through multidimensional, agent-centric coordinates.
• They integrate diverse methods—quantum-inspired Hilbert spaces, embodied affordance measures, topological structures, and LLM-based embeddings—to capture subjective evaluations.
• These frameworks have practical implications for understanding personalized cognition, explainable AI, and the mapping between experience and measurable behavior.

Subjective conceptual spaces are mathematical frameworks designed to characterize individual and group-level representations of concepts, categories, and preferences as they arise through subjective, agent-centered, and often context-dependent processes. These spaces generalize classical conceptual spaces—real-valued, multidimensional coordinates for perceptual categories—by integrating formalisms capable of expressing personal bias, context effects, agency, embodiment, and observer-dependent geometry. Diverse lines of research operationalize subjective conceptual spaces using tools ranging from quantum-inspired Hilbert space models, action-centric and embodied feature spaces, topological and algebraic models of qualia, and individual-centric learning mechanisms such as chunking. This entry surveys the principal formalisms, empirical methodologies, and core findings across these traditions.

1. Quantum-Inspired and Entropic Models in Hilbert Space

Quantum-inspired frameworks model subjective evaluations using pure states in a two-dimensional Hilbert space, spanned by orthonormal basis vectors |like⟩ and |dislike⟩. An individual's instantaneous evaluative attitude is represented as a state vector

$$|\psi\rangle = \cos(\theta/2)\,|like\rangle + e^{i\varphi}\sin(\theta/2)\,|dislike\rangle,$$

with $\theta\in[0,\pi]$ encoding the degree of liking and $\varphi\in[0,2\pi)$ capturing contextual or affective phase bias. The geometric representation employs the Bloch sphere: each pure state corresponds to a point with coordinates $$(x, y, z) = (\sin\theta\cos\varphi,\,\sin\theta\sin\varphi,\,\cos\theta)$$. Unlike real-valued classical conceptual spaces, quantum models naturally accommodate superposition, interference (phase effects), and reversible state evolution under a time-dependent Hamiltonian.

Subjectivity is quantified using two entropic measures:

• Individual indecisiveness is measured by the Shannon entropy $$H_{Shannon} = -[p_{like}\log p_{like} + p_{dislike}\log p_{dislike}]$$, where $p_{like}=|\langle like|\psi\rangle|^2$ and $p_{dislike}=|\langle dislike|\psi\rangle|^2$.
• Group diversity is captured by the Von Neumann entropy $S_{VN} = -\mathrm{Tr}(\rho\log\rho)$ of the empirical density matrix $$\rho = \frac{1}{N} \sum_{i=1}^N |\psi_i\rangle\langle\psi_i|$$.

Maximal individual entropy signifies complete indecisiveness (equatorial states), while maximal group entropy (maximally mixed $\rho$) signifies complete population dispersion. These measures, together with the collective Bloch vector $R = \mathrm{Tr}(\rho\,\vec{\sigma})$, detect coherence, consensus, and polarization in subjective evaluations. Quantum-inspired modeling provides a formal apparatus for studying context effects, temporal evolution, and group alignment in subjective conceptualization, extending the classical geometric theory of concepts (Soodchomshom, 2 Jun 2025).

2. Embodied, Action-Oriented, and Agent-Centered Conceptual Spaces

The embodied approach posits that subjective conceptual spaces must encode affordances—action possibilities and spatial relations between the perceiver and objects—not merely feature vectors of shape or semantics. Each object is characterized by an “action-atom” feature vector at three levels: (1) intrinsic affordances, (2) situatedness (extrinsic spatial context), and (3) empirically elicited low-level contact and kinematic features. Empirical studies collect comprehensive action-atom vectors for each object (e.g., 141 dimensions for mugs), and subjective similarity $S(i, j)$ is assessed via triplet odd-one-out tasks.

Statistical modeling regresses subjective dissimilarity against both visual (deep net) and embodied feature distances, quantifying unique and shared variance (R², ΔR²). Xu et al. find that embodied features alone explain the majority of subjective variance—up to 57% for mugs—while visual features account for significantly less (≈17%). Dimensionality reduction (MDS, PCA) reveals a compact, agent-centered subjective space in which embodied axes (e.g., grasp type, palm orientation) define the core geometry. Procrustes alignment shows that as few as three embodied axes can match the subjective manifold's structure (Xu et al., 2024).

The embodied framework emphasizes that subjective similarity is grounded not only in what an object is, but in what it enables for a specific observer, with effective dimensionality expanding or contracting based on bodily capacities, prior experience, and current spatial relations.

3. Topological and Algebraic Structures for Qualia and Subjectivity

Topological approaches model the space of qualia (subjective experiences) as a sober topology $\tau$ over a set $Q$ of qualia, with open sets interpreted as pure concepts—physically realizable, finitely verifiable properties. The resulting space $(Q, \tau,$ forms an involutive quantale: sequential composition $$, disjunction $\vee$, and an involution $*$ (time-reversal), plus a zero element. The structure supports logical abstraction, subjective time, and coarse-graining in measurement.

Pure concepts correspond to open sets; their conjunction and disjunction to set intersection and union, respectively. For each individual, the accessible qualia $A\subset Q$ constitute a classical measurement space, isomorphic as a topological lattice to the open sets of some locally compact, second-countable space $X$. Thus, the agent's mental model arises as a derived geometric representation based on their available purifications and closure properties (Resende, 2022).

Intersubjective comparison proceeds via structure-preserving maps between subspaces corresponding to different observers, often resembling fuzzy or superposed assignments. This logical superposition yields overlapping clusters of states, affording a mathematically rigorous account of observer dependence and subjective context in conceptual modeling.

4. Methodologies for Extracting Subjective Dimensions with LLMs

Modern LLMs offer empirically tractable means of deriving subjective conceptual spaces via fine-tuned embeddings and ranking strategies. Conceptual spaces are operationalized as $\mathbb{R}^n$ metric spaces, where each axis is a human-interpretable quality dimension (e.g., sweetness, roughness, scary). Methodologies include:

• Conditional-probability probing (e.g., GPT-3): Model assigns conditional probabilities to prompts ("[FOOD] tastes [SWEET]") to obtain the coordinate value along each dimension.
• Fine-tuning classifiers (e.g., DeBERTa-v3, BERT bi-encoders): Models are trained on (concept, property) pairs, predicting binary or continuous property presence to define conceptual-space embeddings.
• Pairwise ranking with SVM aggregation: Pairwise or pointwise model outputs are aggregated into a total order via a linear SVM, requiring only $O(n)$ queries per dimension.

Empirical benchmarks use human-annotated datasets for taste, mass, size, and subjective movie/book features. General findings are:

• LLMs contain significant perceptual and subjective knowledge pre-trained, but fine-tuning on a mix of objective, perceptual, and subjective features is required to achieve robust, transferable rankings (Kumar et al., 2024, Chatterjee et al., 2023).
• Smaller fine-tuned models can match or exceed much larger GPT-3 models, highlighting the role of constraining model capacity and alignment.
• Prompt sensitivity and annotation noise remain important sources of variance.

These techniques extend classical conceptual spaces to real-world, agent-specific, and subjective conceptualizations, albeit with the constraints of annotation quality and the alignment of model representations with experiential reality.

5. Cognitive and Learning-Based Subjectivity: The CogAct Approach

The CogAct cognitive model formalizes subjective conceptual spaces as evolving chunk trees—directed graphs encoding chunked patterns (sequences of primitives) linked with labels (categories) and built by individual experience. The state of a subject’s conceptual space at time $T$ is the tree of chunks and associated labels constructed from their unique input sequence, mediated by STM and chunk-formation routines.

• Chunk formation is governed by discrimination and familiarization routines; retrieval similarity is computed via prefix-based match and difference operations.
• Confidence in category assignment is computed as

$C(c|x) = \frac{a_c(x)}{\sum_{c'} a_{c'}(x)},$

where $a_c(x)$ is the maximal chunk size for category $c$ present in the input $x$.

• Subjectivity arises from idiosyncratic learning histories, short-term memory bounds, and attention constraints.

CogAct outperforms deep learning baselines, particularly in low-data, highly personalized regimes, and adapts model complexity via autonomous chunk creation (Bennett et al., 21 Dec 2025). Subjective conceptual spaces in this framework are agent-relative, adaptive, and transparently interpretable, providing a mechanistic link between cognitive processes and emergent category structure.

6. Subjective Conceptual Spaces in Projective Geometry and Conscious Perspective

The Projective Consciousness Model (PCM) mathematically represents a subject’s field of consciousness (FoC) as a three-dimensional projective space, mapping external Euclidean coordinates $(x, y, z)$ into subject-centered coordinates via projective transformations:

$$\phi(x, y, z) = \left(\frac{x}{cz+1},\,\frac{y}{cz+1},\,\frac{z}{cz+1}\right).$$

Here, $c$ parameterizes the “lens” of subjective perspective; apparent size decays with inverse distance (Stevens’s law). Within the FoC, attentional weighting, multimodal properties, and affective preferences $\mu_{s,e}$ are integrated, with individual uncertainty $\sigma_{s,e}$ modulating value via peripheral certainty models.

The generative model encompasses affective and epistemic drives through Kullback–Leibler divergences between subjective appraisals and idealized expectations, enabling active-inference control of agent behavior. PCM generalizes naturally to Theory of Mind, with nested FoCs and preference tensors for simulating the attributed perspectives of other agents, thus capturing key subjectivity effects in social and clinical contexts (Rudrauf et al., 2020).

7. Theoretical Advances, Limitations, and Implications

Subjective conceptual spaces unify geometric, algebraic, information-theoretic, and learning-based approaches to account for observer- or agent-centric conceptualization. Each framework introduces specific mathematical and empirical strengths:

• Quantum-inspired models introduce superposition and entropy-based signals absent in classical spaces.
• Embodied models extend the feature basis to agent-object affordances and spatial relations, aligning with ecological and neuroscientific predictions.
• Topological formalizations provide a rigorous infrastructure for relating qualia, logical abstraction, and the emergence of observer geometry.
• LLM-based extraction demonstrates practical means of constructing high-dimensional spaces from language data, contingent upon finetuning, annotator bias, and prompt engineering.
• Chunking-based cognitive models offer transparent, mechanistically plausible accounts of how subjective concept spaces emerge and adapt on minimal data.

Limitations include: the complexity of fitting quantum parameters in practice; the necessity of extensive, high-quality annotated datasets in LLM extractions; the challenge of capturing semantic abstraction and generalization in symbolic models; and the constraints of current embodied feature taxonomies for non-manipulable objects.

Subjective conceptual spaces are central to advancing individualized models of cognition and perception, explainable AI, personalized recommendation, and the theoretical understanding of consciousness, measurement, and intersubjectivity. Ongoing research continues to refine these mathematical and empirical tools and to clarify their scope, explanatory power, and integration with neural and behavioral data.