PretopoMD: Pretopology in Mixed-Data Clustering

• PretopoMD is a pretopological algorithm that clusters mixed data using iterative pseudoclosure operations driven by user-defined DNF rules.
• It processes raw numerical and categorical features directly, avoiding dimensionality reduction and enhancing explainability through explicit logical controls.
• Hyperparameter tuning and a quasi-hierarchical workflow enable customizable cluster construction with competitive performance and improved memory efficiency.

Searching arXiv for PretopoMD and related papers. PretopoMD is a term that appears in two distinct arXiv contexts. In the more specific and technically developed 2025 usage, it denotes a pretopology-based algorithm for hierarchical clustering of mixed data that operates directly on raw numerical and categorical variables, without dimensionality reduction, by combining pseudoclosure operations with user-defined logical rules in Disjunctive Normal Form (DNF) (Levy et al., 27 Nov 2025). In an earlier 2017 usage, the same term designates an application concept within the WLIMES program for precision medicine, described as a platform leveraging PRE-topological and Memory Driven models for individualized, holistic diagnosis and therapy (Simeonov et al., 2017). The 2025 formulation is the clearer standalone methodological object; its defining properties are direct handling of heterogeneous data, customizable cluster logic, quasi-hierarchical dendrogram construction, and an explicit emphasis on explainability (Levy et al., 27 Nov 2025).

1. Terminological scope and referential ambiguity

The two documented uses of the term differ in granularity and disciplinary emphasis.

Context	Description	arXiv id
WLIMES / precision medicine	PRE-topological and Memory Driven modeling platform concept	(Simeonov et al., 2017)
Mixed-data clustering	Pretopology-based mixed-data hierarchical clustering algorithm	(Levy et al., 27 Nov 2025)

In the WLIMES context, PretopoMD is presented as part of a broader formal model theory for living systems. It is associated with multi-level, multi-agent, and multi-temporal integration, explainable AI, and an augmented reality interface for medicine (Simeonov et al., 2017). In the clustering context, PretopoMD is presented as a novel pretopology-based algorithm that addresses mixed-data clustering directly from raw data, with hierarchical and interpretable outputs (Levy et al., 27 Nov 2025).

This suggests that the term has evolved from a broad systems-theoretic and biomedical application concept into a specific unsupervised-learning method. The shared lexical core, “Pretopo,” reflects the role of pretopology in both usages, but the 2025 paper gives the term a sharply defined algorithmic meaning.

2. Pretopological formulation for mixed data

In the 2025 formulation, PretopoMD is grounded in pretopology, which is described as a generalization of topology for the analysis and structuring of discrete datasets, suitable for clustering mixed data through pseudoclosure functions (Levy et al., 27 Nov 2025). The central operator is a pseudoclosure function

$a : \wp(U) \to \wp(U),$

with the properties

$$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$

The closure of a set is obtained iteratively by repeated application of $a$ until stabilization, that is, until $a(F)=F$ (Levy et al., 27 Nov 2025).

The method represents each feature, or logical group of features, by a weighted directed graph $G_i$. Different features may use different distance structures, including Euclidean or Hamming distances, depending on the data type. For $n$ features or groups, the pretopological space is written as

$$(G = \{G_1,\ldots,G_n\},\, \Theta = \{\theta_1,\ldots,\theta_n\},\, DNF).$$

Here, $\Theta$ contains feature-specific thresholds, and $DNF$ is a user-defined Boolean formula controlling membership in the pseudoclosure (Levy et al., 27 Nov 2025).

For an element $x$ and a set $$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$0, the predicate $$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$1 is true when the accumulated incoming edge weight from members of $$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$2 in graph $$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$3 reaches threshold $$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$4:

$$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$5

An element belongs to the pseudoclosure exactly when the DNF rule evaluates to true:

$$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$6

This architecture permits combinatorial and logical relationships between heterogeneous features without mapping categorical features to numbers or otherwise transforming the data (Levy et al., 27 Nov 2025).

The role of DNF is central. Examples given for the Boolean structure include strict conjunction,

$$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$7

and mixed conjunction-disjunction patterns such as

$$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$8

Accordingly, cluster membership can express domain-specific semantics, such as requiring similarity in all selected features or similarity in at least one privileged combination of attributes (Levy et al., 27 Nov 2025).

3. Algorithmic workflow and quasi-hierarchical construction

PretopoMD constructs clusters through seed expansion followed by quasi-hierarchical linkage. The pipeline consists of four explicit stages: seed identification, iterative pseudoclosure, attraction matrix construction, and dendrogram creation (Levy et al., 27 Nov 2025).

The first stage selects initial seeds, defined as small sets of nearby elements. A hyperparameterized procedure, seed_Func(.), determines neighbors for each seed, and the seed size is controlled by $$a(\emptyset) = \emptyset, \qquad A \subseteq a(A) \text{ for all } A \subseteq U.$$9 (Levy et al., 27 Nov 2025). The second stage expands each seed by repeatedly applying the pseudoclosure operator until no further enlargement occurs. In effect, the algorithm grows candidate clusters according to the DNF-governed closure logic rather than by minimizing a global geometric objective.

The third stage computes attraction between the stabilized clusters. For two clusters $a$0 and $a$1, the paper defines

$a$2

and then

$a$3

This attraction quantity measures both overlap and relative scale, thereby supporting parent-child relations among partially overlapping clusters rather than only strict nesting (Levy et al., 27 Nov 2025).

The fourth stage builds a directed quasi-hierarchical dendrogram. Directed links are retained when attraction exceeds a threshold $a$4. If two clusters attract each other reciprocally, redundant sets are removed by retaining only the larger cluster (Levy et al., 27 Nov 2025). The result is an adjacency matrix for a directed graph representing the hierarchical organization.

A common misconception would be to read the output as a conventional agglomerative tree. The paper instead describes a quasi-hierarchical structure: nodes are clusters, edges encode parent-child relations, top-level nodes are clusters with no parent or with the universe set as parent, and the representation allows overlaps and non-strict nesting (Levy et al., 27 Nov 2025). The dendrogram is therefore hierarchical in presentation but not restricted to classical disjoint-tree assumptions.

4. Hyperparameters, logical control, and explainability

PretopoMD exposes cluster construction to explicit logical and hyperparameter control. The principal configurable elements are the DNF rule, seed_Func(.), the seed size $a$5, the hierarchy threshold $a$6, and the graph-building and feature thresholds $a$7 (Levy et al., 27 Nov 2025). These parameters may be tuned manually or functionally, and they are intentionally exposed for explainability and adaptability.

The DNF rule determines the strictness and meaning of cluster membership. An AND-type rule imposes similarity across all selected graphs, whereas an OR-type rule allows cluster formation when similarity holds in at least one key attribute or feature combination (Levy et al., 27 Nov 2025). Because the logic is human-readable, the method provides explicit cluster semantics rather than latent geometric groupings whose rationale must be reconstructed post hoc.

Explainability is reinforced by three design choices. First, the method performs no encoding or dimensionality reduction; data remain in their original raw representation (Levy et al., 27 Nov 2025). Second, the DNF itself provides a transparent account of why an element joins a cluster. Third, the dendrogram displays how clusters merge or split across scales according to traceable rules, making the procedure well suited to eXplainable AI (Levy et al., 27 Nov 2025).

The same features define the method’s principal trade-off. Cluster quality is sensitive to the setting of thresholds and logical rules, which requires tuning and domain knowledge, and excessive complexity in the DNF may reduce interpretability (Levy et al., 27 Nov 2025). PretopoMD therefore treats explainability not as a by-product but as a configurable modeling commitment.

5. Empirical behavior, evaluation criteria, and computational profile

The 2025 paper evaluates PretopoMD with standard clustering metrics, specifically the Silhouette Score, the Calinski-Harabasz Index, and the Davies-Bouldin Index (Levy et al., 27 Nov 2025). The method is presented as demonstrating state-of-the-art or comparable performance on challenging datasets while avoiding dimensionality reduction.

The empirical picture is deliberately mixed rather than uniformly dominant. On the Penguins dataset, competing methods are reported to yield Silhouette scores of approximately $a$8–$a$9 and a Calinski-Harabasz index of $a(F)=F$0, whereas PretopoMD obtains Silhouette (FAMD) $a(F)=F$1, Calinski-Harabasz $a(F)=F$2, and Davies-Bouldin $a(F)=F$3; the accompanying interpretation is that these values are slightly lower but come with better interpretability and more balanced clusters (Levy et al., 27 Nov 2025). On the Sponge dataset, which is described as having many categories and weak clustering, PretopoMD achieves Silhouette (FAMD) $a(F)=F$4, the highest among methods, and Davies-Bouldin $a(F)=F$5, the lowest, indicating the best compactness and separation in that case (Levy et al., 27 Nov 2025).

For generated datasets, the base case of $a(F)=F$6 observations and $a(F)=F$7 clusters is reported to yield the main clusters plus about $a(F)=F$8 outliers. In high-dimensional or noisy settings, PretopoMD is described as maintaining performance without dimensionality reduction, although top scores are sometimes achieved by methods that exploit reduction, such as UMAP plus $a(F)=F$9-means (Levy et al., 27 Nov 2025). The paper’s comparative claim is therefore not universal optimality, but robustness, interpretability, and competitive performance under a direct mixed-data regime.

The computational profile is also distinctive. PretopoMD is reported to use the lowest memory among the methods tested, while incurring higher computation time because of iterative pseudoclosure and graph construction (Levy et al., 27 Nov 2025). A further claim is that the computation is linear in the number of data points and independent of the feature count in the current implementation (Levy et al., 27 Nov 2025). This positions the method as memory-efficient but more operationally intensive than simple centroid-based alternatives.

6. Relation to WLIMES, topology-oriented pipelines, and adjacent research

The earlier WLIMES usage situates PretopoMD within a much broader formal and biomedical setting. There, PretopoMD is described as a platform leveraging PRE-topological and Memory Driven models, using categorical structures such as those in Memory Evolutive Systems and Wandering Logic Intelligence, with support for multi-level, multi-agent, and multi-temporal integration in precision medicine (Simeonov et al., 2017). Its stated orientation is deductive, holistic, and relational rather than inductive, fragmentary, and statistical (Simeonov et al., 2017).

That usage differs substantially from the 2025 clustering algorithm. The former concerns holistic modeling of living systems, conceptual landscapes, and explainable human-machine interaction; the latter concerns unsupervised clustering of mixed datasets through pseudoclosure and logical rules. The shared vocabulary indicates a common pretopological sensibility, but the objects are not interchangeable. This suggests that “PretopoMD” should be read contextually rather than assumed to name a single stable framework across all papers.

A later medical-imaging paper, “TopoAgent: An Agentic Framework for Automated Topology Learning in Medical Imaging,” explicitly frames its contribution as useful for practical application or development of topology-based pipelines like PretopoMD (Meng et al., 29 Jun 2026). TopoAgent automates descriptor selection and configuration for topological data analysis, especially persistent-homology descriptors, through a Perception–Reasoning–Action–Reflection loop with domain-specific tools and dual memory (Meng et al., 29 Jun 2026). A plausible implication is that PretopoMD may be situated within a wider ecosystem of topology-aware modeling and analysis pipelines, even though the 2025 PretopoMD paper itself is centered on clustering mixed tabular data rather than medical imaging.

Across these uses, the stable conceptual thread is the attempt to make structure formation explicit. In the 2025 algorithm, that explicitness takes the form of DNF-governed pseudoclosure and quasi-hierarchical cluster construction (Levy et al., 27 Nov 2025). In the WLIMES setting, it takes the form of formal, visual, and relational modeling of organisms and medical reasoning (Simeonov et al., 2017). The term therefore names not a single doctrine, but a family of pretopology-inflected efforts oriented toward interpretability, structured reasoning, and direct representation of complex heterogeneous domains.