Hierarchical Topological Clustering (HTC)

• Hierarchical Topological Clustering (HTC) is a framework that uses multiscale connectivity from topological filtrations to extract nested clusters and outliers.
• It employs techniques like Vietoris–Rips filtration and PMFG-based DBHT to track merging of connected components across scales.
• HTC offers practical insights by revealing complex, non-convex structures without preset cluster counts or extensive parameter tuning.

Searching arXiv for recent and foundational papers on hierarchical topological clustering and closely related topological hierarchical clustering frameworks. Hierarchical Topological Clustering (HTC) denotes a family of clustering frameworks in which cluster structure is extracted from topological or topologically constrained objects across scales, rather than from a single cut of a metric space or a pre-specified number of groups. In the most direct recent formulation, HTC constructs a Vietoris–Rips filtration for a finite point cloud under an arbitrary chosen distance and interprets persistent $H_0$ connected components as clusters and outliers (Carpio et al., 31 Dec 2025). In an earlier graph-theoretic usage, HTC is realized through the DBHT technique, where clusters and both intra-cluster and inter-cluster hierarchies are extracted from a Planar Maximally Filtered Graph (PMFG) and its separating $3$-cliques (Song et al., 2011). Related work formalizes such multiscale cluster structures as merge trees of filtrations, or Topological Hierarchical Decompositions (THDs), thereby placing HTC within a broader topological and categorical theory of hierarchical clustering (Joyce et al., 2023).

1. Conceptual scope and defining principles

At its core, HTC treats clustering as the study of how connected components appear, persist, and merge under a scale parameter or under a topologically constrained representation. In the persistent-homology formulation, the operative invariant is $H_0$, so the hierarchy is the evolution of connected components in a filtration. In the graph-theoretic formulation, the operative structure is a sparse topologically embedded graph whose separators induce a hierarchy. In the THD formulation, the hierarchy is the merge tree of the connected-components functor $\pi_0$ applied to a filtration of spaces (Carpio et al., 31 Dec 2025, Song et al., 2011, Joyce et al., 2023).

Formulation	Core object	Hierarchy mechanism
Persistent-homology HTC	Vietoris–Rips filtration	Merging of $H_0$ components
DBHT-style HTC	PMFG with separating $3$-cliques	Bubble tree and converging bubbles
THD	Filtration $F$ of spaces	Merge tree $T_F=(\pi_0F)$

This common emphasis on connectedness distinguishes HTC from methods that optimize a fixed objective over a flat partition. The 2025 HTC paper motivates this distinction by noting that K-means requires a fixed number of clusters and works poorly for non-convex shapes, standard hierarchical clustering depends strongly on the linkage rule, and DBSCAN requires parameter choices such as $\varepsilon$ and a minimum number of points (Carpio et al., 31 Dec 2025). The topological viewpoint instead begins from nested structure: either metric neighborhoods, open covers, or sparse embeddings, and then reads cluster organization from how components are related across scales.

A recurrent misconception is that “topological” necessarily implies the use of higher-order holes or full persistent homology. In the explicit HTC algorithm, higher-order homology is not needed; the entire clustering procedure uses only $H_0$, that is, connected components (Carpio et al., 31 Dec 2025). This places HTC close to classical clustering in intent, but topological in representation and in its treatment of scale.

2. Persistent $3$0 construction and the filtration-based algorithm

For a finite point cloud $3$1 with distance $3$2, the HTC algorithm in the 2025 formulation constructs a Vietoris–Rips filtration $3$3. At threshold $3$4, points are vertices, an edge $3$5 is included when $3$6, and higher-dimensional simplices are included whenever all their edges are present. The filtration satisfies

$3$7

At each $3$8, the number of connected components is the Betti number $3$9, and HTC interprets each connected component as a cluster (Carpio et al., 31 Dec 2025).

The paper discretizes the filtration by setting

$H_0$0

then choosing a grid resolution

$H_0$1

and filtration values

$H_0$2

The algorithm initializes at $H_0$3, where each point is its own cluster, and then recursively merges linked clusters as $H_0$4 increases. Connectivity can be encoded by a cluster-link matrix $H_0$5, with $H_0$6 when there exist $H_0$7 and $H_0$8 such that $H_0$9. Equivalently, one may first compute the point-link matrix

$\pi_0$0

The output is not only a dendrogram height, but the hierarchy of clusters and their members at each filtration value (Carpio et al., 31 Dec 2025).

This construction is intentionally metric-agnostic. The 2025 paper states that HTC can be implemented with any distance choice, and then instantiates Euclidean distance, Wasserstein distance, and Fermat distance in different applications (Carpio et al., 31 Dec 2025). A plausible implication is that the filtration mechanism is fixed while the scientific meaning of a cluster is delegated to the metric.

3. Persistence, merge structure, and the interpretation of outliers

The hierarchical component of HTC arises because connected components form a nested family as the threshold grows. At $\pi_0$1, every point is isolated; as $\pi_0$2 increases, points linked by distances smaller than $\pi_0$3 form components; those components merge whenever a new edge bridges them; and eventually a single component contains all points. The paper describes this as a “topological hierarchy of clusters,” and interprets the order of mergers as meaningful structure: components that merge only at large $\pi_0$4 are more isolated and are therefore natural candidates for outliers (Carpio et al., 31 Dec 2025).

Persistence is the central interpretive device in this framework. A cluster that exists across a wide interval of $\pi_0$5 is persistent, and a point or cluster that remains isolated until very late is a persistent outlier. The paper emphasizes that dominant clusters usually correspond to dense or central regions of the data, while outliers can be single points or entire groups of points, termed collective outliers. The last-to-merge components are singled out as especially significant (Carpio et al., 31 Dec 2025).

The THD framework provides a general topological formalization of this same idea. For a filtration of spaces $\pi_0$6, the THD is defined as the generalized merge tree

$\pi_0$7

Vertices correspond to connected components $\pi_0$8, and edges record how components map under the filtration. For metric clustering, the filtration is given by

$\pi_0$9

where $H_0$0. The associated merge tree $H_0$1 is then the THD (Joyce et al., 2023). This suggests that HTC can be understood not merely as an algorithmic recipe, but as a special case of a broader theory in which clustering is the multiscale behavior of $H_0$2.

The same section of the 2025 HTC paper also clarifies the relation between barcodes and cluster identity. The persistence barcode indicates the range of filtration values over which clusters remain alive, but HTC improves on a bare barcode by also identifying which data points belong to each persistent component (Carpio et al., 31 Dec 2025). In other words, persistence provides significance, while the hierarchy retains membership.

4. Graph-theoretic, categorical, and continual-learning variants

A distinct but historically important usage of HTC appears in the DBHT technique. There the starting point is a similarity-weighted graph

$H_0$3

where $H_0$4 is the vertex set, $H_0$5 the edge set, $H_0$6 the edge-weight set, and $H_0$7 the edge-distance set. The graph is filtered into a PMFG by adding edges in decreasing similarity order subject to the planarity constraint. Because the PMFG is maximal planar, it contains $H_0$8 edges and supports a decomposition by separating cycles, especially separating $H_0$9-cliques (Song et al., 2011).

Each separating $3$0-clique $3$1 divides the graph into an interior $3$2 and exterior $3$3. Cutting along all such cliques yields “bubbles,” and the resulting bubble tree $3$4 is oriented by comparing the aggregate weights

$3$5

The direction is assigned toward the side with larger weight. Converging bubbles, whose incident edges all point inward, are treated as centers of clusters. Vertex assignment is then resolved by an attachment strength

$3$6

followed, for remaining vertices, by average shortest-path proximity

$3$7

The resulting discrete clusters support an intra-cluster and inter-cluster hierarchy with complete-linkage distances

$3$8

and

$3$9

The paper characterizes DBHT as deterministic, unsupervised, parameter-free, and non-iterative (Song et al., 2011).

A different extension of hierarchical topological clustering arises in ART-based continual learning. The HCAEA framework uses the Correntropy-Induced Metric,

$F$0

together with automatic threshold estimation, edge aging, and recursive divisive partitioning. The hierarchical version trains CAEA at the top level, partitions data according to learned nodes, and then recursively trains child CAEA models on those subsets until no additional layer is created (Masuyama et al., 2022). This is topological in the paper’s terminology because the method maintains a growing graph of nodes and edges, and hierarchical because refinement proceeds by successive divisions of the learned topological structure.

At a more abstract level, multiparameter hierarchical clusterings can be flattened into optimization problems. The 2021 paper on flattening multiparameter hierarchical clustering treats the flattening procedure as a functor

$F$1

from a category of multiparameter hierarchical partitions to a category of binary integer programs, and further introduces a Bayesian update algorithm whose composition with flattening satisfies a consistency property (Shiebler, 2021). This does not define HTC in the narrow algorithmic sense, but it places hierarchical topological clustering objects within applied category theory and learning pipelines.

5. Empirical domains and comparative behavior

The explicit 2025 HTC algorithm is demonstrated on a fragmented front in epithelial tissue, image quality assessment, Spanish trade data, and breast cancer gene-expression data. In the epithelial case, Euclidean distance is used on $F$2D points extracted from an image of an interface between healthy and malignant cells. HTC finds a main cluster representing the interface, detached islands of malignant cells, and last-joining clusters corresponding to deeper invasive malignant islands. The paper contrasts this with K-means, average-linkage hierarchical clustering, and DBSCAN, arguing that HTC yields a more interpretable interface/island separation without trial-and-error tuning of $F$3 and $F$4 (Carpio et al., 31 Dec 2025).

The same paper uses Wasserstein distance $F$5 for image comparison and reports that HTC separates highly compressed blurry images, uncompressed images, and images with line defects. In Spanish trade data, after normalization

$F$6

Euclidean distance yields a large cluster of countries with low interaction while dominant trade partners remain persistent outliers. In breast cancer gene-expression data, both Euclidean and Fermat distances are applied to normalized expression values, again using

$F$7

with $F$8 and $F$9 taken from healthy samples. HTC identifies persistent outliers including CCNE1, SMC1B, CDKN2A, CDC6, PKMYT1, and CDK1, which the authors note are known to be relevant to breast cancer prognosis or therapy (Carpio et al., 31 Dec 2025).

The DBHT paper provides a different empirical profile. On the Iris dataset, with similarity

$T_F=(\pi_0F)$0

DBHT reports an adjusted Rand index of $T_F=(\pi_0F)$1, compared with $T_F=(\pi_0F)$2 for Q-cut, $T_F=(\pi_0F)$3 for kNN-Spectral, $T_F=(\pi_0F)$4 for k-means++, and $T_F=(\pi_0F)$5 for SOM. On synthetic hierarchical data with $T_F=(\pi_0F)$6 large clusters, $T_F=(\pi_0F)$7 medium clusters, and $T_F=(\pi_0F)$8 small clusters, DBHT exactly recovers the $T_F=(\pi_0F)$9 large clusters and reports approximately $\varepsilon$0 ARI at the medium level and approximately $\varepsilon$1 ARI at the small level. In lymphoma gene-expression profiling, DBHT retrieves $\varepsilon$2 sample clusters; all FL samples gather in one cluster, almost all CLL samples gather in another, and DLBCL samples split into four clusters with reported survival rates of $\varepsilon$3, $\varepsilon$4, $\varepsilon$5-$\varepsilon$6, and $\varepsilon$7 (Song et al., 2011).

Across these examples, the empirical claim is not that HTC optimizes a single universal criterion, but that it retains shape sensitivity, outlier visibility, and cluster membership across scales. This suggests that HTC is especially suited to problems where the distinction between dominant structure and informative exceptions matters as much as partition accuracy.

6. Limitations, ambiguities, and adjacent meanings

The 2025 HTC paper is explicit about several caveats. Its complexity is stated to be at most

$\varepsilon$8

with a lower bound of order $\varepsilon$9, where $H_0$0 is the number of clusters at scale $H_0$1. The method is described as efficient for small and moderate datasets, while larger datasets may call for density-based algorithms. The same paper also notes that the chosen metric strongly shapes the interpretation, that the filtration grid is selected heuristically through $H_0$2 and $H_0$3, and that substantive meaning of persistent outliers remains domain-dependent (Carpio et al., 31 Dec 2025).

DBHT has a different structural limitation: planarity is essential. The decomposition into separating cycles, bubbles, and the bubble tree depends on the PMFG’s topological embedding, so the method’s hierarchy is tied to the properties of planar filtered graphs (Song et al., 2011). HCAEA, in turn, inherits the strengths and weaknesses of ART-based continual learning: automatic threshold estimation and stability-plasticity management are central advantages, but the paper notes weaker performance on datasets with many classes and identifies concept drift as future work (Masuyama et al., 2022).

A further source of ambiguity is terminological rather than methodological. In a separate arXiv literature, HTC denotes Hierarchical Text Classification rather than Hierarchical Topological Clustering. That usage includes sequence-to-tree generation with constrained decoding (Yu et al., 2022), zero-shot classification via knowledge graphs and LLMs (Zang et al., 8 May 2025), and few-shot prompt-based classification with sibling contrastive learning (Xiong et al., 17 Apr 2026). Those works study hierarchical taxonomies of labels, not unsupervised clustering of data clouds. The acronym therefore spans distinct research programs.

Within topological data analysis more broadly, HTC sits beside THDs, mapper, Reeb graphs, and categorical approaches to hierarchical clustering. The THD paper treats clustering as a cosheaf-theoretic problem over open covers and nerves, and describes mapper as a pixelization $H_0$4 (Joyce et al., 2023). The multiparameter flattening paper shows that hierarchical clustering objects can also be transported into binary integer programs and Bayesian learning procedures (Shiebler, 2021). Taken together, these directions indicate that HTC is best understood not as a single closed algorithmic family, but as a topological approach to multiscale clustering in which connected components, graph embeddings, or hierarchical functors are made explicit and computationally actionable.