Hierarchical clustering

Overview

Hierarchical clustering arranges groups in a tree, so a user can read the structure at several levels instead of picking one flat partition. The paper motivates its method with exactly this property: hierarchical, explainable outputs matter for decisions 2512.03070 §1 2512.03070 §8.5.

Most clustering methods assume a Euclidean space, so they do not apply directly to tables that mix numeric and categorical columns [arxiv:2512.03070 §1] 2512.03070 §2. The one source on this page targets that case. Its method is the only tree-producing variant with a card here. Hierarchical clustering in general (linkage rules, dendrogram cuts and similar) is not covered by any card yet, so this page will grow as more papers are compiled.

Variants

PretopoMD: pretopology-based hierarchy

The paper's own method. It works on the raw mixed columns, without a shared numeric space. Single rows seed sets, each set is grown to a fixed point under a pseudoclosure operator, and the resulting sets are arranged into a tree by attraction between them 2512.03070 §4.8. The paper gives the pretopological definitions in words, with no numbered equation for the attraction step 2512.03070 §4.8. Card: Mixed Data Clustering Survey and Challenges.

It takes no cluster count k, and the paper's own table of method properties marks it as needing no k 2512.03070 tab. 2 2512.03070 §4.8. Its missing-value handling is not listed in that table [arxiv:2512.03070 tab. 2].

Reported results are mixed. On Penguins it gives CH 105.17 with two clusters 2512.03070 tab. 6 2512.03070 §7.3. On Heart Disease it gives CH 67.88 and a Davies-Bouldin of 1.31, the paper's text reports clusters of 206 and 64 rows 2512.03070 tab. 8 2512.03070 §7.3. On the 1000-row sparse synthetic case it has a negative Silhouette 2512.03070 tab. 12.

Tandem pretopology variants (FAMD, UMAP, PaCMAP)

The paper calls these Pretopo-FAMD, Pretopo-UMAP and Pretopo-PaCMAP. Each first embeds the mixed table in a Euclidean space and then runs the pretopology procedure on that embedding 2512.03070 §7.3 2512.03070 §3.1 2512.03070 §3.3 2512.03070 §3.4. The paper does not state whether these tandem variants return a tree, so this page does not claim one. The tandem variants win some tables and lose others; see the results under "When to use which".

Gower-based hierarchical method (Phillip and Ottaway)

A baseline built on Gower similarity for mixed data 2512.03070 §4.6. This page does not state its linkage rule, because the source does not record one. Across the benchmark it is competitive on Silhouette with Gower distance: on Heart Disease it reaches 0.457 against PretopoMD's 0.416, while PretopoMD has the higher CH and lower Davies-Bouldin [arxiv:2512.03070 tab. 8]. Baselines in that comparison were run with k chosen by the Elbow method 2512.03070 §7.3.

Named but not covered

The paper's related work names DIVCLUS-T and Unsupervised Binary Trees as further tree-based methods for comparison 2512.03070 §4.8. No card exists for either, so this page says nothing about how they work or perform.

When to use which

Implementation sketch

The steps below follow the paper's verbal description of PretopoMD. The paper gives no formula for attraction, no rule for overlapping sets and no parameter values for the pretopology method 2512.03070 §4.8. Any implementation must fill those in from elsewhere and check the result against the paper's tables.

# Pretopology-based hierarchy over the rows of a mixed table
# [arxiv:2512.03070 §4.8]
# Input:  N rows, with numeric and categorical columns
# Output: sets of rows arranged as a tree

def pretopo_hierarchy(rows, graphs, thresholds):
    # One weighted directed graph and threshold per family of tests
    # (the families are described in words, not as formulas)
    seeds = [{x} for x in rows]

    # Grow each seed to a fixed point under the pseudoclosure, where a row
    # joins set A when a positive DNF over the graph tests holds
    closures = [fixed_point(pseudoclosure, A, graphs, thresholds)
                for A in seeds]

    # Arrange sets into a tree by attraction between them
    # (attraction has no formula in the paper)
    return arrange_as_tree(closures, attraction)

Contradictions

• Summary versus tables. The conclusion says the PaCMAP variant performs best "consistently" 2512.03070 §9. The tables do not support this on every dataset: Pretopo-UMAP is best on the 10-cluster synthetic case 2512.03070 tab. 10, and Pretopo-FAMD is best on Penguins by Calinski-Harabasz 2512.03070 tab. 6. This page uses the tables.
• Paper and linked code. The paper's linked code is not used as a source here. Its main repository returned "Repository not found" at compile time, and the card records further differences between the paper's description and the code. Implementations that follow only the paper's text may not reproduce its tables. See the card's "Paper vs code" section 2512.03070 §4.8.

Papers

• Mixed Data Clustering Survey and Challenges: survey and benchmark of clustering for mixed numeric and categorical tables, with the PretopoMD tree-producing method and a pretopology-based tandem family.