Thematic Representation Set (TRS)

• Thematic Representation Set (TRS) is a structured abstraction that formalizes local and global thematic structures in a corpus.
• It employs term extraction, semantic indexing, and ontology-driven annotation to construct navigable semantic networks.
• TRS enables enhanced information retrieval, mathematical knowledge mapping, code optimization, and thematic portfolio construction in finance.

A Thematic Representation Set (TRS) is a structured abstraction for the representation, analysis, and navigation of themes within a heterogeneous dataset or corpus. The TRS formalizes both local (per-document or entity) and global (corpus- or domain-level) thematic structures, supporting the quantification of thematic pertinence, the mapping of inter-theme associations, and the construction of navigable semantic networks. Applications span information retrieval, scientific knowledge mapping, computational linguistics, mathematical knowledge representation, qualitative research methodology, and semantic portfolio management in finance.

1. Formalization and Construction of TRS

A TRS is typically produced via a multi-phase process that distills the thematic structure embedded in a corpus:

• Term and Concept Extraction: Initial phases employ automatic summarization (e.g., with TextTiling) and semantic indexing (e.g., Latent Semantic Indexing), followed by construction of a term co-occurrence network. Each term pair $(t_1, t_2)$ is weighted by an association confidence score, $conf(t_1 \to t_2) \in [0,1]$, quantifying local semantic proximity (Chabi et al., 2011).
• Concept Vector Enrichment: The term vector is enhanced by the detection of compound terms and the inclusion of semantically or statistically related terms derived from the global co-occurrence network.
• Ontology-Driven Theme Annotation: Disambiguated terms (by similarity measures such as Wu-Palmer or Resnik against a domain ontology) are mapped to concepts, and in turn, to structured theme hierarchies. Formally, theme pertinence in document $d$ is computed as:

$$\text{pertin}(Th_x) = \frac{\sum_{i=1}^{n_{Th_x}} \text{weight}(c_i)}{n_{c}}$$

where $c_i$ are concepts affiliated with theme $Th_x$ and $n_c$ is the total number of concepts in the document (Chabi et al., 2011).

• Corpus-Level Association Matrix: Thematic associations across the corpus are captured in a matrix, with entries computed via

$$AD(Th_1, Th_2) = \frac{Doc(Th_1, Th_2)}{Doc(Th_1) + Doc(Th_2) - Doc(Th_1, Th_2)}$$

where $Doc(Th_1, Th_2)$ counts documents that instantiate both themes (Chabi et al., 2011).

• Path and Graph Construction: Themes and subthemes are represented as nodes in a semantic graph or hypergraph; edges encode degree of association. Traversal defines “thematic paths” for corpus exploration.

2. Mathematical and Computational Models Underpinning TRS

The TRS formalism is instantiated via computational structures:

• Co-occurrence Networks: Graphs where nodes are terms/concepts, edges have association weights, and confidence is a normalized empirical statistic.
• Ontology-Guided Disambiguation: Using scores such as

$score(cc) = \sum_{j=1}^{n_{cv}} Simil(cc, CV_j)$

for candidate concept $cc$ with respect to validated concepts $CV_j$ (Chabi et al., 2011).

• Theme Pertinence and Weighting: Both local (per-document) and global (corpus) pertinence is computed; the latter as

$weight(Th_x) = \frac{nbDoc(Th_x)}{nbDocTot}$

which formalizes the statistical weight of a theme across the corpus (Chabi et al., 2011).

• Association Degree Matrices: For every theme pair, the association degree (as above) allows construction of a square matrix suitable for further spectral or graph-based analytical methods.
• Navigable Graphs/Hypergraphs: Thematic paths, i.e., sequences of theme nodes traversed via association-weighted edges, facilitate pathfinding, cluster detection, and knowledge discovery.

3. TRS Across Domains: Semantics, Mathematics, Code Optimization, and Finance

The TRS abstraction has domain-specific incarnations:

Domain	TRS Elements	Methodological Basis
Text Analysis	Themes, association relations, thematic paths	Ontology, LSI, co-occurrence, path analysis
Mathematical Theory	Definitions, axioms, thematic clusters	Semantic nets, production rules
Code Optimization	Rewrite rules, equivalence classes, proof sets	E-graph TRS, equality saturation
Financial Thematics	Themes (ETFs, sectors), stocks, profiles	Hierarchical contrastive, temporal refinement

Contextualization: In mathematics, the TRS parallels a semantic net constructed through deductive production rules from atomic axioms, with nodes corresponding to formulae and edges defined by logical transformations (Luxemburg, 2016). In code optimization, a TRS encodes all reachable forms of an expression under term rewriting, where e-graphs store equivalence classes and facilitate selection of optimal forms (Kourta et al., 2021). In thematic investing, a TRS explicitly connects investment themes to asset profiles, with semantic overlap and temporal dynamics encoded in the representation (Lee et al., 23 Aug 2025).

4. Applications: Retrieval, Search, Knowledge Discovery, and Visualization

Principal applications of TRS frameworks include:

• Enhanced Indexing and Retrieval: Document themes are indexed at multiple granularities, facilitating semantic search and greater recall-precision trade-offs in information retrieval systems (Chabi et al., 2011).
• Thematic Path Navigation: Users can navigate semantic graphs by traversing thematic paths—sequences of thematically-associated nodes—enabling exploratory search and discovery of indirect associations.
• Visualization: Node-link (hypergraph) representations visualize the thematic landscape; edge thickness or labels denote association strengths, and pie charts may quantify theme pertinence per document (Chabi et al., 2011).
• Trend and Relationship Discovery: The association matrix and thematic graphs support analysis of trend emergence, cross-disciplinary linkages, and latent connections otherwise invisible in keyword-based retrieval (Chabi et al., 2011).
• Portfolio Construction: In finance, candidate assets are retrieved by their semantic alignment with themes and further filtered via temporal return dynamics, supporting robust thematic portfolio design (Lee et al., 23 Aug 2025).

5. Methodological Extensions and TRS Refinement

Sophisticated TRS systems employ advanced algorithmics and optimizations:

• Automated Term Reduction and Duplication Resolution: In qualitative analysis, cumulative codebooks are pruned using chain-of-thought comparison modules (e.g., with DSPy), yielding a Unique Cumulative Codebook (UCC). The Inductive Thematic Saturation (ITS) metric,

$ITS = \frac{|UCC|}{|TCC|}$

quantitatively measures redundancy and saturation (Paoli et al., 6 Mar 2025).

• Dynamic Code and Metadata Assignment: As new documents are added, codes/themes are iteratively checked against the UCC, reducing variance and order effects relative to zero-shot approaches.
• Integration of Temporal Dynamics: Especially in financial applications, stock representations are temporally refined via recent asset returns, implemented through adapters trained with triplet loss to align short-term performance with semantic fit (Lee et al., 23 Aug 2025).

6. Limitations and Challenges

Key challenges in TRS construction and application include:

• Ontology and Label Opaqueness: For linguistic/thematic applications, reliance on opaque theme/role labels (e.g., PropBank Arg0/Arg1) can limit model adherence and user interpretability unless prompts or interface are carefully engineered (Alshemali et al., 19 Oct 2024).
• Scaling and Saturation: For very large or ambiguous corpora, theme association graphs and co-occurrence networks can become dense, necessitating early stopping conditions or heuristics to manage computational resource requirements (Kourta et al., 2021).
• Semantic Coherence Assessment: Ensuring that themes and associated content are both semantically coherent and contextually relevant can require domain-specific filtering, especially with automatically generated content or open-ended prompts (Alshemali et al., 19 Oct 2024).

7. Impact and Outlook

The TRS construct has demonstrated substantial impact in multiple fields:

• In information science, TRS enables corpus-level semantic overviews, refines document retrieval, and facilitates trend spotting and cross-corpus discovery (Chabi et al., 2011).
• In formal knowledge engineering, TRS-inspired semantic nets ensure mathematical concepts are systematically linked, supporting both strict derivation and thematic grouping (Luxemburg, 2016).
• In compiler theory, TRS using e-graphs and equality saturation support expressive code simplification and optimization pipelines, with applications in symbolic reasoning engines and proof systems (Kourta et al., 2021).
• In finance, formal TRS linked with hierarchical contrastive learning and temporal adapters provide robust instruments for thematic asset screening and adaptive portfolio construction, exceeding the capabilities of baseline retrieval and investing systems (Lee et al., 23 Aug 2025).

TRS frameworks continue to evolve, integrating more nuanced semantic, structural, and temporal signals, and extend their applicability to emergent domains and increasingly complex datasets. With growing interest in explainable AI and semantic search, TRS methodologies offer a scalable and interpretable means to bridge content analysis, knowledge representation, and applied decision-support systems.