Vector-Objective Categorization

Updated 6 November 2025

Vector-objective categorization is a framework employing vector representations and multi-objective performance criteria to assign entities to categories.
It leverages methods such as vector space models, wavelet scattering, and graph embeddings for robust categorization in texts, images, and scientific data.
The approach unifies clustering, classification, regression, and dimensionality reduction via explicit geometric, metric, and cognitive structures for enhanced decision-making.

Vector-objective categorization refers to a broad family of models, frameworks, and algorithms that use vector-valued representations, objectives, or decision rules to assign entities—such as documents, objects, patterns, or resource configurations—to categories. This paradigm spans classic vector space models of information retrieval, multi-objective optimization with vector-valued performance criteria, categorical resource assignment with structured objectives, and abstraction of categorization as a theoretically grounded process unifying clustering, classification, regression, and dimensionality reduction.

1. Foundational Definitions and Theoretical Frameworks

At its most general, vector-objective categorization is the process of assigning each object or observation to a category, where both the representations and the evaluation objectives are vectorial. Formalizations include:

Vector space models (VSM), where documents or objects are mapped to high-dimensional feature spaces and categorization is achieved via geometric, similarity-based, or algebraic rules (Odeh et al., 2015, Bruna et al., 2010).
Multi-objective frameworks, where model evaluation or optimization considers a vector of objectives, e.g., minimizing error and runtime jointly, and solutions are compared using vector norms or Pareto dominance (Wang et al., 2022, Marcolli, 2022).
Axiomatic and cognitive frameworks, in which categorization is governed by inner (cognitive/prototype-based) and outer (explicit/membership-based) representations, linked through assignment and similarity operators, and formalized with generalized categorization axioms (Yu, 2015).

Key constructs include:

Outer representation: explicit category assignments (e.g., labels or memberships).
Inner representation: latent or prototype-based encodings, similarity functions, or cognitive models.
Operators: assignment (via memberships), and similarity-based mapping to categories.

This dual structure generalizes and unifies standard ML tasks—clustering, classification, regression, and dimensionality reduction—under common theoretical principles (Yu, 2015).

2. Vector Representation and Feature Space Methodologies

A central technique in vector-objective categorization is the explicit vectorization of entities and categories:

Text and Document Categorization: Documents are represented as high-dimensional TF-IDF vectors, with each term providing a dimension and weight determined by its importance (term frequency and corpus inverse document frequency). Categories can be associated with prototypical vectors or lists of discriminative keywords. Decision rules typically use distance, angle, or explicit keyword overlap (Odeh et al., 2015).

| Step | Method | |-------------------|-----------------------------------------------------------------------------------------------| | Feature extraction| TF-IDF/structured frequency analysis; geometric representation for non-text data | | Category model | Prototypes, clusters, or sets of discriminative vector features | | Decision criteria | Maximum similarity, geometric proximity, or match to high-weighted features |

Wavelet Scattering: For signal and image classification, scattering vectors are constructed as cascades of wavelet transforms and complex modulus, forming highly structured local descriptors with strong invariance properties. Supervised assignment uses PCA-based models in the vector space, leveraging geometric closeness in a (potentially reduced) scattering vector space (Bruna et al., 2010).
Object and Affordance Graph Embeddings: Interactions or affordances are encoded as multi-layer graphs, which are embedded into a latent vector space (e.g., via graph2vec). Hierarchical clustering on these vectors enables unsupervised category discovery based on usage or relational patterns, rather than explicit object features or labels (Toumpa et al., 2023).

3. Multi-objective and Vector-valued Optimization in Categorization

Multi-objective categorization frameworks evaluate or optimize models based on vector-valued performance metrics, not single scalars:

Objective Performance Vector Norm: Performance across multiple objectives (error, time, resource usage) is encoded as a normalized vector, often with weights reflecting importance. The overall evaluation function is a vector norm (commonly Euclidean). The optimization procedure (e.g., the Taguchi method) then seeks model configurations minimizing this norm, enabling principled comparison and ranking of models with respect to potentially competing criteria (Wang et al., 2022).

| Step | What/How | |--------------------|----------------------------------------------------| | Vectorization | Performance vector $\mathbf{P} = [p_j]$ | | Normalization | Scaling/weighting to obtain comparable units | | Comparison | Vector norm as objective: $J(\mathbf{P})$ |

Pareto Optimization in Category Theory: Categorical multi-objective optimization avoids scalarization by using resource categories, valuation functors, and categorical relationships instead of real-valued vectors. Pareto dominance and the frontier are characterized via morphisms (convertibility relations), and the search for optimal assignments is conducted over summing functors and objective functors, supporting far more general and structured objectives than traditional vector approaches (Marcolli, 2022).

4. Metric and Geometric Structures on Vector Objectives

Categorization quality and system behavior are often governed by geometric properties and induced metrics on representation or label space:

Curved Label Space and Metric Tensors: The conventional one-hot label scheme uses a flat, orthogonal label space, which ignores class similarity structure. Introducing a metric tensor $g_{\alpha\beta}$ permits arbitrary pairwise distances, allowing the classification loss to penalize confusions in a manner consistent with real-world class similarity or hierarchical structure. This leads to adaptive loss landscapes that modulate gradient signals based on class relations and confusion statistics, and can directly encode semantic or latent similarity via learned or autoencoder-based metrics (Sheehan et al., 2018).

| Feature | One-hot Label Space | Curved Label Space w/ Metric Tensor | |------------------------|------------------------|----------------------------------------------| | Class distance | Uniform | Modulated by $g_{\alpha\beta}$ | | Loss structure | Equal penalty | Proportional to semantic/empirical similarity| | Hierarchy encoding | None | Directly supported by metric tensor |

Categorical Geometric Algebras: Categories can be viewed as geometric spaces, with arrows (morphisms) functioning as "vectors.” Cat-vector spaces support noncommutative partial addition (composition), algebraic norm, inner and wedge (exterior) products, and parallel/orthogonality notions, supporting metric and Clifford algebra structures entirely within categorical formalisms (Majkic, 5 Mar 2024).

5. Algorithmic and Modeling Developments

Algorithmic advances tailored to vector-objective categorization include:

Selective Feature and Keyword Extraction: Techniques that select only the most discriminative features or keywords (e.g., top-2 by weight) achieve high precision while minimizing computational complexity in categorization tasks (Odeh et al., 2015).
Principal Component Reference Construction: In data clustering, reference vectors derived from leading principal components can expose distinct clusters or regimes in high-dimensional experimental data when traditional fixed reference choices fail to do so, as demonstrated in molecular conductance trace analysis (Hamill et al., 2017).
Instance-Geometry Interaction Models: For structured vector extraction (e.g., multiple vector types from images), unified query encodings that tie semantic instance attributes directly to geometric queries enable concurrent categorization and extraction across diverse vector categories (polygons, polylines, line segments) and outperform type-specific models on benchmark tasks (Yan et al., 15 Oct 2025).
Functional Polyhedral Algorithms for Robust Classification: Classification via polyhedral LP methods, constructed using functions on finite sets and recursive operators (generalizations of Uzawa’s method), provides a robust approach for digital object categorization, yielding all extremal solutions for maximal robustness in noise-prone data regimes (Antonets, 8 Jul 2024).

6. Generalized Axiomatic and Unified Perspectives

A rigorous abstract foundation for vector-objective categorization is provided by generalized categorization axioms:

Axiomatic System: Categorization is governed by existence and uniqueness of both outer (membership, explicit label) and inner (prototype, similarity-based) representations, with axioms ensuring assignment, sample separation, and equivalence principles (Yu, 2015).
Unified Objective Functions: Compactness (intra-category similarity), separation (inter-category distinction), and consistency (alignment of inner/outer representations) collectively define optimization principles underlying most supervised and unsupervised learning algorithms.
Task Unification: The framework subsumes clustering, supervised classification, dimensionality reduction, and regression as special cases via its general operatory and metric formalism, thus positioning categorization as the central abstraction underlying all supervised and unsupervised vector-encoded machine learning.

7. Domain-specific and Generalized Applications

Vector-objective categorization is successfully applied across domains:

Natural Language and Text: High-precision categorization in morphologically-rich languages (e.g., Arabic) relying on top-weighted keywords and robust preprocessing (Odeh et al., 2015).
Physical Experiment Data: Reference-based and PCA-based vector analysis reveals clustering structure in scientific measurement traces (Hamill et al., 2017).
Vision and Affordance Recognition: Structured queries and object-agnostic graph embeddings enable unsupervised affordance categorization in robotics and perception tasks with open sets and occlusions (Yan et al., 15 Oct 2025, Toumpa et al., 2023).
Resource Allocation and Programming: Categorical approaches to Pareto optimization extend beyond vectors, accommodating structured, composite objectives without reducing them to scalars, and supporting algorithmic exploration of categorical frontiers (Marcolli, 2022).

These applications demonstrate both the diversity and the unifying reach of vector-objective categorization as a mathematical and algorithmic paradigm.

Aspect	Vector-objective Categorization Characteristic	Reference Examples
Representation	Explicit vectors (features, wavelets, embeddings), sets of objectives	(Odeh et al., 2015, Bruna et al., 2010, Toumpa et al., 2023)
Multi-objective Evaluation	Vector-valued performance or resource metrics, norms or Pareto dominance for ranking	(Wang et al., 2022, Marcolli, 2022)
Axiomatic/cognitive foundation	Inner/outer representations, similarity and assignment operators, category axioms	(Yu, 2015)
Metric/Geometric Structure	Curvature/metric tensors on label space, geometric algebra structures in category theory	(Sheehan et al., 2018, Majkic, 5 Mar 2024)
Algorithmic technique	Selective feature, principal component reference, cross-attention, polyhedral LP	(Odeh et al., 2015, Hamill et al., 2017, Yan et al., 15 Oct 2025, Antonets, 8 Jul 2024)

Vector-objective categorization thus encompasses a spectrum of approaches where vector-valued representations, metrics, and multi-objective functions are fundamental to the accurate, robust, and theoretically principled organization of data, objects, or systems into categories, with applications across information retrieval, scientific measurement, computer vision, resource allocation, and the theoretical foundations of machine learning.