Topological Hierarchical Decompositions (THD)

• THD is a family of multi-scale frameworks that decompose spaces by tracking the evolution of connected components across varying scales.
• The methodology leverages generalized merge trees, recursive MAPPER analysis, and cosheaf constructs to extend and refine classical clustering approaches.
• Applications of THD range from unsupervised data explanation to EEG representation learning, offering a unified strategy for multi-scale decomposition and prediction.

Searching arXiv for the cited THD-related papers to ground the article in current preprints. Topological Hierarchical Decompositions (THDs) are hierarchical structures that organize data, spaces, or structured domains by tracking connected components or nested regions across multiple scales. In one explicit formalization, a THD is the generalized merge tree of a filtration of spaces $F:P\to\mathbf{Top}$, namely $T_F=(\pi_0 F)$, so that branches record how connected components evolve as the parameter in the poset $P$ varies (Joyce et al., 2023). In recursive topological data analysis, THD also denotes a tree of MAPPER-derived topological networks obtained by repeatedly decomposing connected components of a simplicial summary of the data (Brown et al., 2018). In topology-aware representation learning for EEG, the same phrase is instantiated by a fixed multi-scale decomposition of channels, a hierarchy-aligned tokenizer, and an autoregressive model that predicts coarse-to-fine spatial structure over time, thereby turning EEG into a multi-scale, topology-aware sequence (Yang et al., 5 Nov 2025). The available formulations indicate that the term designates a family of closely related multi-scale constructions rather than a single universally fixed object.

1. Definitional scope and principal usages

Across the cited literature, THD appears in three technically distinct but structurally related forms.

Setting	THD object	Basic mechanism
Filtration of spaces	Generalized merge tree $T_F=(\pi_0 F)$	Tracks connected components across a poset of scales
MAPPER-based unsupervised analysis	THD tree of topological networks	Recursively splits connected components of a MAPPER 1-skeleton
EEG generic representation learning	Brain Topology Hierarchy with THVQ-VAE and BAR	Uses a multi-scale partition of channels and coarse-to-fine autoregression

In the merge-tree formulation, the central object is categorical. A filtration is a functor $F:P\to\mathbf{Top}$, and the THD is the category of elements of $\pi_0 F$. Its objects are pairs $\beta_p=(p,\beta)$ with $\beta\in\pi_0(F(p))$, and arrows encode how components merge as the filtration parameter increases (Joyce et al., 2023). This makes THD a topological generalization of a dendrogram.

In the MAPPER-based formulation, THD is an unsupervised or semi-supervised method that starts with a full dataset $X$, builds a topological summary of $X$ using the MAPPER algorithm, identifies connected components in the resulting graph, recursively applies MAPPER to the data points in each sufficiently large connected component, and organizes the resulting complexes into a tree of topological networks called the THD tree (Brown et al., 2018). Here the hierarchy is explicit and recursive.

In THD-BAR for EEG, topological hierarchical decompositions are instantiated by a carefully designed Brain Topology Hierarchy (BTH) and a matching hierarchical VQ-VAE plus autoregressive model. The BTH is a multi-scale, nested partition of EEG channels that reflects spatial proximity of electrodes, coarse anatomical or functional regions, and nested inclusions. The hierarchy is not merely a prior or a precomputed graph; it redefines the sequential structure on which the autoregressive model operates (Yang et al., 5 Nov 2025).

A plausible implication is that these usages share a common pattern: each introduces a hierarchy derived from topological or quasi-topological structure, and each uses that hierarchy to control either decomposition, representation, or prediction.

2. Topological and categorical foundations

A basic topological foundation for THD-like constructions is decomposition theory. A decomposition of a space $T_F=(\pi_0 F)$0 is a partition $T_F=(\pi_0 F)$1 into pairwise disjoint sets whose union is $T_F=(\pi_0 F)$2. The associated decomposition space $T_F=(\pi_0 F)$3 is the quotient space whose underlying set is $T_F=(\pi_0 F)$4, equipped with the quotient topology induced by the decomposition map $T_F=(\pi_0 F)$5. The notes on decomposition space theory state that this is the fundamental mechanism in any THD-like construction: each level of a hierarchy corresponds to a quotient of the original space by some decomposition (Kalmar, 2021).

The same notes emphasize conditions under which such quotient levels remain topologically well behaved. A closed upper semi-continuous decomposition can be characterized by the existence of saturated neighborhoods, by openness and closedness properties of unions of decomposition elements, or equivalently by the decomposition map being closed. In metric settings, a usc decomposition yields a metrizable decomposition space, and in complete metric spaces Bing’s shrinkability criterion identifies shrinkability with the decomposition map being a near-homeomorphism. In the language of hierarchical decompositions, near-homeomorphism functions as a fidelity criterion: quotienting changes the representation while preserving the topological type when shrinkability holds (Kalmar, 2021).

The categorical formulation of THD given in the topological data analysis literature is more specific. For a filtration $T_F=(\pi_0 F)$6, the generalized merge tree is

$T_F=(\pi_0 F)$7

This construction makes connected components the primitive invariant of the hierarchy. The paper further frames clustering as an instance of the connected components functor

$T_F=(\pi_0 F)$8

and then extends clustering from a single space to a filtration of spaces, so that hierarchical structure arises from the way connected components merge across scales (Joyce et al., 2023).

Cosheaf language supplies an additional layer of structure. For a locally connected space $T_F=(\pi_0 F)$9, the assignment $P$0 is a cosheaf on open sets. Given a map $P$1, the Reeb functor produces the Reeb cosheaf

$P$2

The display space of this cosheaf recovers the associated Reeb graph, and the paper uses this framework to reinterpret Mapper as a pixelization of a cosheaf. In this setting, a THD is not only a hierarchy of components but also an object compatible with functorial and cosheaf-theoretic operations (Joyce et al., 2023).

3. MAPPER, clustering, and recursive topological decomposition

The recursive THD procedure in applied topological data analysis is built from MAPPER. Let $P$3 be a finite dataset in a metric space, $P$4 a filter space, $P$5 a filter function, and $P$6 an open cover of $P$7. The MAPPER simplicial complex is

$P$8

where the pullback cover is obtained by taking inverse images of cover elements and splitting them into connected components. In the HELOC formulation, only the 1-skeleton of this simplicial complex is used; vertices correspond to clusters induced from cover bins, and edges record overlapping membership (Brown et al., 2018).

A THD tree is then constructed recursively. The process initializes with a MAPPER network on the full dataset. Its connected components $P$9 define subpopulations $T_F=(\pi_0 F)$0 consisting of the data points in the nodes of component $T_F=(\pi_0 F)$1. Each sufficiently large component becomes a child node, and MAPPER is recursively re-applied on that subset. If the network is connected, the resolution parameter is increased while the overlap parameter is kept fixed until a split occurs. Recursion stops when a component is below the split threshold or further refinement is no longer informative (Brown et al., 2018).

This formulation differs sharply from classical decision trees and also from standard hierarchical clustering. The splits are unsupervised in construction, because labels are not used to build the hierarchy. Labels are overlaid only after the THD is built, to characterize groups and provide explanations. The paper also emphasizes that THD does not require all points to appear in leaf clusters, since small components may be treated as outliers or may stop splitting early (Brown et al., 2018).

The more abstract THD framework broadens this picture by topologizing several standard clustering paradigms. For a metric filtration, one can define $T_F=(\pi_0 F)$2 as a union of metric balls and then form the THD $T_F=(\pi_0 F)$3, which plays the role of a continuous dendrogram. For density-based clustering, the paper defines covers around core points and represents DBSCAN-type behavior as a join of canonical clusterings induced by those cover elements. For linkage methods, single linkage and complete linkage are cast as THDs of offset filtrations indexed by recursively defined scales. Mapper itself is subsumed through the identification of the Mapper functor with a pixelization

$T_F=(\pi_0 F)$4

and multiscale Mapper is described via a filtration of covers and the corresponding merge tree of Mapper nerves (Joyce et al., 2023).

In this family of constructions, THD functions as a unifying formalism for turning clustering into a topological hierarchy of connected components.

4. Topology-hierarchical decompositions in EEG representation learning

In THD-BAR, the hierarchical object is the Brain Topology Hierarchy. Let $T_F=(\pi_0 F)$5 be the set of EEG channels and let $T_F=(\pi_0 F)$6 be the number of scales, with $T_F=(\pi_0 F)$7 in the experiments. At scale $T_F=(\pi_0 F)$8, the hierarchy defines a partition

$T_F=(\pi_0 F)$9

whose subsets are non-overlapping and whose union is $F:P\to\mathbf{Top}$0. The nesting property requires that each coarse group at scale $F:P\to\mathbf{Top}$1 is a union of finer groups at scale $F:P\to\mathbf{Top}$2. The resulting structure is a tree-like topological decomposition of the EEG sensor layout (Yang et al., 5 Nov 2025).

The hierarchy is constructed using standard EEG montages, spatial proximity, and coarse anatomical alignment. The five-scale example is: $F:P\to\mathbf{Top}$3 whole brain, $F:P\to\mathbf{Top}$4 major regions, $F:P\to\mathbf{Top}$5 sub-regions, $F:P\to\mathbf{Top}$6 channel clusters, and $F:P\to\mathbf{Top}$7 channels. The construction is largely rule-based and anatomy-informed rather than learned, but it is described as flexible with respect to other montages or sensor geometries (Yang et al., 5 Nov 2025).

The Topology-Hierarchical VQ-VAE (THVQ-VAE) converts continuous EEG into discrete multi-scale tokens aligned with the hierarchy. Starting from an encoder output $F:P\to\mathbf{Top}$8, the model repeatedly downsamples from the finest channel layout $F:P\to\mathbf{Top}$9 to a coarser partition $\pi_0 F$0, quantizes the downscaled features, dequantizes them through a shared codebook, upscales them back to the finest layout, and subtracts a residual through a scale-specific convolution. In compact form, $\beta_p=(p,\beta)$7 The shared codebook and residual structure mean that coarse tokens explain large-scale structure first and finer scales model residual detail (Yang et al., 5 Nov 2025).

The Brain Autoregressive module redefines autoregression as next-scale-time prediction. If $\pi_0 F$1 denotes the multi-scale tokens at time $\pi_0 F$2, the factorization is

$\pi_0 F$3

Within each time step, prediction proceeds coarse to fine; across time, prediction proceeds past to future. The transformer input is flattened in the order

$\pi_0 F$4

and the scale-time-wise mask allows each token $\pi_0 F$5 to attend to all earlier times at all scales and to coarser scales at the same time, while forbidding attention to later times and to equal or finer scales at the current time (Yang et al., 5 Nov 2025).

This construction treats EEG channels as a topological object, with electrodes functioning as nodes in an implicit graph or topological space and the BTH functioning as a multi-scale partition or clustering of that graph. The paper explicitly compares this to multiresolution partitions such as wavelet trees or quadtree decompositions, but over scalp or head topology rather than image pixels (Yang et al., 5 Nov 2025).

5. Related hierarchical topological frameworks

A closely related two-level framework appears in work on hierarchical datasets of graph-structured samples. There the hierarchy is observations within samples within datasets. Each sample is represented by a diffusion operator $\pi_0 F$6 derived from a kernel graph on its observations. At the dataset level, samples become vertices of a weighted simplicial complex, where edge and triangle weights are defined through alternating diffusion operators such as

$\pi_0 F$7

The filtration

$\pi_0 F$8

then yields persistent homology and persistence diagrams that summarize the global topological structure of each dataset (Aloni et al., 2021).

This framework does not produce an explicit THD tree in the sense of MAPPER recursion or generalized merge trees, but it is hierarchical and topological in a closely related sense. The fine level is geometric, encoded by diffusion operators on sample graphs, and the coarse level is topological, encoded by filtrations of weighted simplicial complexes and persistent homology. The paper describes this as a method for building an informative representation of hierarchical datasets, and the construction is explicitly presented as close in spirit to what one might call a Topological Hierarchical Decomposition (Aloni et al., 2021).

An even more structural analogue arises in infinite graph theory. Every connected graph admits a rooted tree-decomposition of finite adhesion into connected parts with upwards disjoint separators that displays all topological ends. In this setting, the hierarchy comes from a refining sequence of induced connected subgraphs

$\pi_0 F$9

and from a decomposition tree built from the components of the complements $\beta_p=(p,\beta)$0. Ends of the decomposition tree are in bijection with the undominated, or topological, ends of the graph (Pitz, 2021).

This graph-theoretic construction can be read as a rigorous THD of a graph at infinity. Each branch represents a nested sequence of regions separated by finite boundaries, and each infinite branch corresponds to a unique topological end. The relation to THD is therefore structural rather than terminological: hierarchy is provided by the tree-decomposition, and topology is encoded by the end space that the decomposition displays (Pitz, 2021).

6. Interpretability, empirical evidence, and limitations

THDs are repeatedly presented as an explanatory device. The topological data analysis formulation states that hierarchical data structures referred to as Topological Hierarchical Decompositions can be used as a basis for explainability in unsupervised data analysis, because they encode how clusters form or split as parameters vary (Joyce et al., 2023). In the HELOC application, THD is used to explain loan risk without using labels to construct the hierarchy. The dataset contains 10,459 anonymized HELOC applications, with 5,000 “Good” and 5,459 “Bad” outcomes. Two THDs are built using different filters, and labels are overlaid afterward to characterize subgroups. Feature differences between sibling groups are identified with the KS-score for continuous variables and the hypergeometric distribution for categorical variables. One cited subgroup, split 1.1.2, contains 88.5% “Bad” outcomes and is characterized by high credit card utilization (Brown et al., 2018).

In EEG representation learning, the empirical case for topology-hierarchical design is based on large-scale pre-training on 17 datasets followed by validation on 10 downstream datasets spanning 5 distinct tasks. The authors report that single-scale configurations perform significantly worse than the full five-scale configuration in THVQ-VAE reconstruction metrics and BAR pretraining metrics, and that the full scale-time-wise mask outperforms scale-wise-only and time-wise-only variants. Across downstream evaluation, THD-BAR variants consistently outperform prior general EEG models on 9/10 datasets; the paper highlights gains over NeuroLM on DEAP by $\beta_p=(p,\beta)$1, on SEED by $\beta_p=(p,\beta)$2, and on TUEV from $\beta_p=(p,\beta)$3 to $\beta_p=(p,\beta)$4. Visualization also shows that THVQ-VAE improves PCC by $\beta_p=(p,\beta)$5 over a standard VQ-VAE from NeuroLM (Yang et al., 5 Nov 2025).

The limitations are likewise specific to each formulation. The recursive MAPPER approach is sensitive to the metric, filter function, resolution, overlap or gain, clustering choices within bins, and split threshold; the paper explicitly notes that different THD settings generate different explanations, even though recurring themes may persist (Brown et al., 2018). The categorical and cosheaf-based formulation assumes local connectivity and, for several results, locally finite good covers and finite indexing posets. It also notes that computational cost and noise sensitivity remain practical concerns, and that a full theory beyond $\beta_p=(p,\beta)$6 is still open (Joyce et al., 2023). In THD-BAR, the hierarchy is fixed and rule-based, changing electrode layouts requires redefining the BTH manually, and alternative topological decompositions such as data-driven clustering of functional connectivity graphs are not explored (Yang et al., 5 Nov 2025).

These limitations clarify the present status of THD. In one line of work, THD is a precise topological object—the merge tree of a filtration. In another, it is a recursive MAPPER-based decomposition used for unsupervised explanation. In a third, it is a domain-specific multi-scale topological ordering that shapes tokenization and autoregressive prediction. Taken together, the cited works suggest that the common invariant is not a single algorithmic template, but the use of topology-derived hierarchy to organize decomposition, representation, and interpretation across scales.