Hierarchical Concept Modeling

• Hierarchical concept modeling is a framework that organizes concepts across multiple abstraction levels, incorporating semantic, probabilistic, and geometric methods.
• Recent approaches leverage multi-layer architectures, rule-based reasoning, and sparse coding to enhance explainability and boost classification performance.
• The framework integrates external knowledge sources and structured supervision to align hierarchical representations with practical application needs.

Searching arXiv for papers on hierarchical concept modeling and related concept bottleneck/topic modeling approaches. Hierarchical concept modeling denotes a family of representational and inferential schemes in which concepts are organized across multiple levels of abstraction, specificity, or structural dependency rather than treated as a flat, independent set. Across recent work, the term covers several distinct but related settings: multilayer concept extraction and prompting in explainable medical image diagnosis (Dong et al., 4 Oct 2025); multi-level and label-aligned concept bottlenecks for image classification (Xie et al., 2 Apr 2026); supervised two-level concept hierarchies for leakage-resistant concept bottleneck models (Sun et al., 2024); multi-level concept discovery and recursive concept-embedding architectures (Hill et al., 10 Mar 2026); hierarchy-aware sparse recovery of concept embeddings in vision-language latent spaces (Nguyen et al., 11 Feb 2026); probabilistic concept layers in topic models (Tang et al., 2017, 0808.0973); incremental probabilistic concept formation for lifelong topic hierarchies (Singaravadivelan et al., 15 Apr 2026); directed-acyclic-graph reasoning over concepts via attention-selected rules (Debot et al., 26 Jun 2025); and geometric or linear representational analyses of hierarchical knowledge in embedding spaces and LLMs (Mishra et al., 30 Apr 2026, Sakata et al., 9 Apr 2026). Although these systems differ in modality, supervision, and formalism, they share a common objective: to encode concepts in a structured hierarchy so that prediction, retrieval, reasoning, or explanation can proceed from coarse structure to finer distinctions.

1. Formal perspectives and representational regimes

Hierarchical concept modeling appears in at least four formal regimes in the literature. In concept bottleneck and concept-embedding models, the hierarchy is usually defined over semantic abstraction levels, such as higher-level versus lower-level concepts, or top-level concepts versus discovered sub-concepts and sub-sub-concepts (Xie et al., 2 Apr 2026, Hill et al., 10 Mar 2026). In topic modeling, the hierarchy can be a probabilistic generative chain such as document $\rightarrow$ topic $\rightarrow$ concept $\rightarrow$ word, or a concept tree incrementally formed over document embeddings (Tang et al., 2017, Singaravadivelan et al., 15 Apr 2026). In symbolic and graph-based formulations, concepts are nodes in a directed acyclic graph or partial order, with edges representing logic rules, subsumption, or ordering relations (Debot et al., 26 Jun 2025, 0801.0131, Funk et al., 2023). In geometric formulations, hierarchy is encoded through structured subspaces, directional relations, radial potentials, or polar/angular decomposition (Mishra et al., 30 Apr 2026, Sakata et al., 9 Apr 2026).

A recurring distinction is between hierarchy in the concept space and hierarchy in the label space. HIL-CBM explicitly couples both: a higher-level concept vector $c^H$ supports prediction of basic-level labels $\hat{y}^H$, while a lower-level concept vector $c^L$ supports subordinate labels $\hat{y}^L$ (Xie et al., 2 Apr 2026). Deep-HiCEMs likewise represent a tree in which concepts have positive and negative descendants and can be intervened upon at multiple abstraction levels (Hill et al., 10 Mar 2026). By contrast, CoPA realizes hierarchy primarily through the visual feature hierarchy of a frozen encoder, extracting concept-aware embeddings from every layer and aggregating them into a multilayer concept representation aligned with textual concept definitions (Dong et al., 4 Oct 2025).

Several papers define hierarchy as an explicit structural constraint on admissible concept combinations. HCEP assumes that the correct concepts for an image form a rooted path in a hierarchy and performs hierarchical sparse pursuit under that support constraint (Nguyen et al., 11 Feb 2026). H-CMR constrains concept dependencies to a learned DAG, with acyclicity enforced by a learnable node-priority vector and rule-role masking (Debot et al., 26 Jun 2025). In the Concept-Oriented Model, inclusion induces a physical hierarchy while ordering induces a logical multi-dimensional structure; concepts become nodes in nested ordered sets, and the resulting partial order supports projection, de-projection, grouping, and constraint propagation (0801.0131).

2. Hierarchical concept bottlenecks and explainable vision models

Recent concept bottleneck research moves beyond flat concept sets by imposing semantic levels, structural constraints, or multilayer extraction. HIL-CBM is explicitly motivated by the claim that existing CBMs operate at a single semantic level in both concept and label space, whereas humans identify objects at different levels of abstraction using both general and specific features (Xie et al., 2 Apr 2026). Its architecture has two hierarchical concept bottleneck layers producing $c^H$ and $c^L$, and two hierarchical classification heads producing $\hat{y}^H$ and $\rightarrow$0. Concept supervision is label-free: GPT-4 generates concept sets, CLIP-Dissect provides concept similarity vectors, and a cubic cosine similarity loss aligns concept neurons with text concepts (Xie et al., 2 Apr 2026). Hierarchical consistency is enforced by a gradient-based visual consistency loss and a Tree-path KL Divergence loss, while sparse elastic-net classifiers preserve interpretability.

The empirical pattern reported for HIL-CBM is that hierarchy improves both interpretability and classification. On CIFAR-100 with a ResNet backbone and sparse final layer, HIL-CBM reaches $\rightarrow$1 for lower-level and higher-level accuracy, compared with $\rightarrow$2 for LF-CBM and $\rightarrow$3 for a standard sparse classifier (Xie et al., 2 Apr 2026). On ImageNet, HIL-CBM reaches $\rightarrow$4, compared with $\rightarrow$5 for LF-CBM and $\rightarrow$6 for SALF-CBM (Xie et al., 2 Apr 2026). Human evaluation on 35 ImageNet images with 50 Prolific raters reports mean scores around $\rightarrow$7–$\rightarrow$8 for LF-CBM and around $\rightarrow$9–$\rightarrow$0 for HIL-CBM on helpfulness and accuracy of explanations (Xie et al., 2 Apr 2026). This supports the paper’s claim that aligning explanations with prediction level improves interpretability.

SupCBM introduces a different two-level hierarchy: perceptual noun concepts and descriptive adjective concepts attached to those nouns (Sun et al., 2024). For each class, GPT-4 is queried for $\rightarrow$1 visual parts and $\rightarrow$2 descriptors per part, yielding up to $\rightarrow$3 second-level concepts per class; concept pooling then selects $\rightarrow$4 descriptors per part per image, so the ground-truth concept vector has $\rightarrow$5 nonzero entries (Sun et al., 2024). Label prediction is performed not by a trainable classifier but by a fixed intervention matrix $\rightarrow$6, with

$\rightarrow$7

The paper’s main claim is that this structure eliminates soft information leakage, because shared concepts contribute equally to competing labels and label differences arise only from unique concepts (Sun et al., 2024). On CIFAR-100, SupCBM reaches $\rightarrow$8, slightly above the feature baseline at $\rightarrow$9; on CUB-Bird it reaches $c^H$0, slightly below Feat at $c^H$1 but above all reported CBM baselines (Sun et al., 2024).

CoPA presents a hierarchical concept modeling framework for explainable diagnosis in which hierarchy is realized across the depth of a visual encoder rather than only across semantic labels (Dong et al., 4 Oct 2025). It operates on triplets $c^H$2 and extracts concept-aware embeddings $c^H$3 for each concept $c^H$4 at each encoder layer $c^H$5 via a Concept-aware Embedding Generator: $c^H$6

$c^H$7

These are aggregated into multilayer embeddings $c^H$8, aligned with textual concept states by contrastive loss, and fed into a gated disease predictor: $c^H$9 The framework further uses Concept Prompt Tuning, injecting layer-wise concept embeddings as prompts into frozen transformer layers: $\hat{y}^H$0 On PH$\hat{y}^H$1, CoPA reports AUC $\hat{y}^H$2, ACC $\hat{y}^H$3, F1 $\hat{y}^H$4, compared with MICA at AUC $\hat{y}^H$5, ACC $\hat{y}^H$6, F1 $\hat{y}^H$7; on Derm7pt it reports AUC $\hat{y}^H$8, ACC $\hat{y}^H$9, F1 $c^L$0, above the reported prior baselines (Dong et al., 4 Oct 2025). Concept intervention experiments on Derm7pt show accuracy increases by $c^L$1 and $c^L$2 when correcting one or two mispredicted concepts, and decreases by $c^L$3 and $c^L$4 when corrupting one or two correct concepts, which the paper interprets as evidence of faithful concept-based decision making (Dong et al., 4 Oct 2025).

HCEP provides a hierarchy-aware alternative to flat sparse concept recovery. It constructs concept atoms as child-parent differences in embedding space and assumes the correct concepts form a rooted path. Under subtree containment, sibling-cone disjointness, hierarchical orthogonality, and simplex conditions, child-parent difference vectors become usable concept directions (Nguyen et al., 11 Feb 2026). Hierarchical OMP restricts active atoms to children of the current deepest node and uses beam search to maintain multiple rooted-path hypotheses. The reported effect is improved support precision and recall relative to vanilla OMP, while maintaining competitive classification accuracy, especially in few-shot ImageNet settings (Nguyen et al., 11 Feb 2026).

3. Multi-level concept discovery and recursive concept embeddings

A separate line of work asks how hierarchies can be discovered rather than predefined. “Digging Deeper: Learning Multi-Level Concept Hierarchies” introduces Multi-Level Concept Splitting (MLCS) and Deep-HiCEMs (Hill et al., 10 Mar 2026). MLCS takes concept-aligned embeddings from a pretrained CEM and trains a Hierarchical Sparse AutoEncoder (HiSAE) with top-$c^L$5 sparsification at the first level and top-$c^L$6 sparsification in child-specific sub-encoders. The reconstruction is

$c^L$7

with sub-level activations gated by active top-level latents. This yields sub-concepts and sub-sub-concepts from only top-level supervision.

Deep-HiCEMs then represent the discovered hierarchy recursively. For a top-level concept $c^L$8, preliminary embeddings $c^L$9 and $\hat{y}^L$0 are produced; positive and negative sub-concept modules recursively generate child embeddings and compress them into parent-level positive and negative embeddings. The final concept embedding remains

$\hat{y}^L$1

The paper reports discovered-concept ROC-AUCs of $\hat{y}^L$2 on MNIST-ADD, $\hat{y}^L$3 on SHAPES, $\hat{y}^L$4 on CUB, $\hat{y}^L$5 on AwA2, and about $\hat{y}^L$6–$\hat{y}^L$7 on PseudoKitchens-2 (Hill et al., 10 Mar 2026). Task accuracy remains close to HiCEM: for example, Deep-HiCEM reaches $\hat{y}^L$8 on CUB and $\hat{y}^L$9 on AwA2 (Hill et al., 10 Mar 2026). The paper states that interventions on discovered concepts can improve performance, though on some datasets they may plateau or decrease, suggesting noise or imperfect alignment in discovered concept labels (Hill et al., 10 Mar 2026).

This discovery-oriented perspective differs from HIL-CBM and SupCBM. In HIL-CBM, hierarchy is given by coarse and fine labels; in SupCBM, it is generated per label from GPT-4 prompts and fixed by an intervention matrix (Xie et al., 2 Apr 2026, Sun et al., 2024). In MLCS and Deep-HiCEMs, hierarchy is induced from concept embeddings and then imposed on a recursive architecture (Hill et al., 10 Mar 2026). A plausible implication is that hierarchical concept modeling spans both ontology-guided and latent-discovery regimes, with corresponding trade-offs between semantic control and annotation efficiency.

4. Topic models and probabilistic concept formation

In topic modeling, hierarchical concept modeling often means inserting an explicit concept layer between topics and words. “Conceptualization Topic Modeling” replaces the standard document $c^H$0 topic $c^H$1 word assumption with document $c^H$2 topic $c^H$3 concept $c^H$4 word (Tang et al., 2017). In CLDA, each topic $c^H$5 has a multinomial over concepts plus atomic concepts: $c^H$6 while each document has a topic mixture

$c^H$7

If a word belongs to some concept in Probase, the model samples a concept $c^H$8 and then a word from $c^H$9, where $c^L$0 is the concept-word distribution from Probase (Tang et al., 2017). If not, the word is treated as an atomic concept. This grounds the hierarchy in an external knowledge base and yields lower perplexity than LDA and LLDA on Conf and AP datasets (Tang et al., 2017).

“Text Modeling using Unsupervised Topic Models and Concept Hierarchies” goes further by integrating a human-defined concept hierarchy into a topic model (0808.0973). In HCTM, each word is generated either by a topic route or by traversing a concept hierarchy from the root to an exit node and then sampling from the concept-word multinomial $c^L$1. The resulting document-level distribution is

$c^L$2

The model yields lower perplexity than pure topic models and flat concept-topic models, and supports visualization of document semantics as subtrees over named concepts (0808.0973). Here hierarchy is not learned from scratch; rather, a curated ontology is embedded inside a generative process and reweighted by corpus statistics.

CobwebTM represents a different probabilistic strategy: online concept formation over continuous embeddings (Singaravadivelan et al., 15 Apr 2026). Each concept node stores a diagonal Gaussian

$c^L$3

and hierarchy growth is driven by Category Utility: $c^L$4 where $c^L$5 is the differential entropy of the Gaussian at node $c^L$6. Structural operations are INSERT, NEW, MERGE, and SPLIT, chosen to maximize CU. The model is lifelong and nonparametric in topic count. On Spatiotemporal News, CobwebTM reports $c^L$7 versus about $c^L$8 for the best BERTopic variant; in lifelong experiments, ARI is about $c^L$9 on Stack Overflow and about $\hat{y}^H$0 on Spatiotemporal News, while Topic Centroid Drift is near zero (Singaravadivelan et al., 15 Apr 2026). This indicates that hierarchical concept formation can be made incremental and stable without backpropagation over a fixed-capacity latent space.

Older work on ontology construction from corpora likewise frames hierarchy induction as concept extraction plus partial-order learning. Formal Concept Analysis is used to transform a text-derived object-attribute incidence relation into a concept lattice and then into a compacted partial order over concepts and leaf terms (Cimiano et al., 2011). “Learning Concept Hierarchies through Probabilistic Topic Modeling” combines LDA-guided concept extraction with document-level subsumption tests: a concept $\hat{y}^H$1 subsumes $\hat{y}^H$2 if $\hat{y}^H$3 and $\hat{y}^H$4 (Anoop et al., 2016). “Towards Ontology Construction with LLMs” replaces corpus statistics with GPT-3.5 queries, producing a concept hierarchy as a preordered set $\hat{y}^H$5 over domain concepts (Funk et al., 2023). These lines suggest that hierarchical concept modeling in text predates current CBMs and that modern neural approaches inherit questions of subsumption, synonymy, multiple inheritance, and granularity from ontology learning.

5. Graph, rule, and order-based hierarchical reasoning

Hierarchical concept modeling is not limited to embeddings; it can also mean explicit rule-governed reasoning over concept graphs. H-CMR models concepts as nodes in a DAG and stores, for each concept $\hat{y}^H$6, a memory of $\hat{y}^H$7 rules over other concepts with roles $\hat{y}^H$8, $\hat{y}^H$9, or $\rightarrow$00 (Debot et al., 26 Jun 2025). A rule for concept $\rightarrow$01 is a conjunction over positive and negated parent concepts, and its evaluation under a binary concept assignment $\rightarrow$02 is

$\rightarrow$03

A neural selector attends over rules: $\rightarrow$04 and the concept probability is a mixture of rule outputs: $\rightarrow$05 Acyclicity is enforced by learnable node priorities $\rightarrow$06, which mask rule roles whenever $\rightarrow$07, and the paper states that the representable graph class is exactly the class of DAGs (Debot et al., 26 Jun 2025). Concept interventions propagate through the hierarchy because changing a parent affects both rule selection and rule evaluation, unlike in independent CBMs. This makes hierarchy itself part of the explanation.

The Concept-Oriented Model offers a more abstract structural account. A concept hierarchy is represented by nested ordered sets with inclusion defining hierarchical structure and ordering defining multi-dimensional structure (0801.0131). A core ontology is a structure $\rightarrow$08, and each element can be represented as a combination of super-elements along labeled dimensions. Syntactic constraints require that if a child element $\rightarrow$09 belongs to a parent concept $\rightarrow$10, then each coordinate of $\rightarrow$11 must belong to the corresponding super-concept $\rightarrow$12 (0801.0131). Projection and de-projection navigate concept hierarchies, and constraint propagation provides upward and downward reasoning. Although this work predates modern neural concept models, it formalizes hierarchy as both semantic generalization and multi-dimensional organization.

Sobolevsky’s model of hierarchical distance in networks gives yet another interpretation (Sobolevsky, 2017). A hierarchical distance $\rightarrow$13 satisfies an ultrametric-like inequality

$\rightarrow$14

and cuts at threshold $\rightarrow$15 yield community partitions. In the generic network model,

$\rightarrow$16

while in the spatial model geographical distance is factored via $\rightarrow$17 (Sobolevsky, 2017). This is not a concept bottleneck model, but it fits the broader notion of hierarchical concept modeling insofar as nested groupings are inferred from relational data and interpreted as concepts at multiple scales.

6. Geometric and linear representations of hierarchical knowledge

Several papers model hierarchy directly in embedding geometry. Polaris separates semantics from hierarchy using a polar hyperspherical representation (Mishra et al., 30 Apr 2026). Concepts are embedded on the unit sphere

$\rightarrow$18

with semanticity encoded by direction and hierarchy by an orbital potential $\rightarrow$19 derived from depth and descendant count: $\rightarrow$20 At inference, candidate parents are filtered by a parabolic gate in angle-radius space: $\rightarrow$21 Training combines a robust hyperspherical triplet loss, anisotropic spherical SVGD to prevent equator collapse, and an asymmetric vMF-KL containment loss in which broader parents have lower concentration than children (Mishra et al., 30 Apr 2026). On the Science dataset, Polaris reports R@1 $\rightarrow$22, R@5 $\rightarrow$23, Wu–Palmer $\rightarrow$24, MR $\rightarrow$25, compared with STEAM at R@1 $\rightarrow$26, R@5 $\rightarrow$27, Wu–Palmer $\rightarrow$28, MR $\rightarrow$29 (Mishra et al., 30 Apr 2026). The paper interprets these improvements as evidence that hierarchy and semantics should be decoupled geometrically.

“Linear Representations of Hierarchical Concepts in LLMs” analyzes hidden states rather than training a new model (Sakata et al., 9 Apr 2026). For a hierarchical relation $\rightarrow$30 specific to a domain and depth, it fits a linear map

$\rightarrow$31

from a child representation $\rightarrow$32 to a parent representation $\rightarrow$33, then uses a low-rank pseudo-inverse to define parent-specific concept vectors

$\rightarrow$34

These vectors classify parents by inner product and support causal interventions by adding $\rightarrow$35 to hidden states (Sakata et al., 9 Apr 2026). On Llama 3.1 8B, the reported LHE accuracy is $\rightarrow$36 for locations, $\rightarrow$37 for research topics, $\rightarrow$38 for persons, $\rightarrow$39 for organizations, and $\rightarrow$40 for organisms; causality scores are $\rightarrow$41, $\rightarrow$42, $\rightarrow$43, $\rightarrow$44, and $\rightarrow$45, respectively, all above the reported baselines (Sakata et al., 9 Apr 2026). Rank sweeps suggest hierarchy is encoded in a low-dimensional subspace of about 150–250 dimensions, and topological data analysis suggests structurally similar hierarchical geometry across domains despite domain-specific subspaces (Sakata et al., 9 Apr 2026).

HCEP can be read as a constructive counterpart to this representational analysis. It assumes that synset embeddings and child-parent differences obey cone and orthogonality conditions, then performs hierarchy-respecting sparse coding (Nguyen et al., 11 Feb 2026). A plausible synthesis is that hierarchy can be modeled either as a low-dimensional linear relation within a pretrained network (Sakata et al., 9 Apr 2026) or as a support constraint over structured concept atoms in a latent space (Nguyen et al., 11 Feb 2026). Polaris, in turn, suggests that if the underlying geometry is not Euclidean-linear but spherical-polar, then hierarchy should be separated from semantics in radial and angular components (Mishra et al., 30 Apr 2026).

7. Recurring technical themes, empirical patterns, and open issues

A first recurring theme is that flat concept modeling is systematically identified as insufficient. HIL-CBM criticizes single-level concept and label spaces (Xie et al., 2 Apr 2026). SupCBM criticizes holistic concept sets that are hard to observe and intervene on (Sun et al., 2024). CoPA criticizes reliance on final-layer features and lack of encoder guidance (Dong et al., 4 Oct 2025). HCEP criticizes vanilla sparse coding for producing hierarchy-inconsistent explanations (Nguyen et al., 11 Feb 2026). Across these papers, hierarchy is not merely an interpretive garnish; it is presented as a mechanism for improving concept quality, intervention faithfulness, and sometimes task accuracy.

A second theme is alignment between levels. In HIL-CBM, coarse labels are explained by coarse concepts and fine labels by fine concepts, with visual and semantic consistency losses coupling them (Xie et al., 2 Apr 2026). In Deep-HiCEMs, child modules are conditioned on parent embeddings and interventions propagate up the tree (Hill et al., 10 Mar 2026). In CoPA, shallow, intermediate, and deep features are aggregated into concept representations aligned with text semantics (Dong et al., 4 Oct 2025). In H-CMR, rule selection depends on parent concept predictions, making reasoning explicitly hierarchical (Debot et al., 26 Jun 2025). This suggests that successful hierarchical concept modeling requires not only multi-level representations but also constraints linking those levels.

A third theme is the use of external knowledge or LLMs to specify concepts. Probase defines concept-word distributions in CLDA/CLLDA (Tang et al., 2017). CALD and ODP define explicit concept hierarchies in HCTM (0808.0973). GPT-4 generates label-free concepts in HIL-CBM (Xie et al., 2 Apr 2026) and part-attribute concept hierarchies in SupCBM (Sun et al., 2024). GPT-3.5 is used directly to construct concept hierarchies in ontology learning (Funk et al., 2023). This suggests that hierarchical concept modeling often depends on priors external to the downstream dataset, whether ontologies, taxonomies, or LLM-generated semantic lists.

A fourth theme is tension between interpretability and model flexibility. H-CMR shows that explicit rule memories and DAGs can remain universal binary classifiers while preserving concept- and task-level interpretability (Debot et al., 26 Jun 2025). HIL-CBM shows only a small gap between sparse and dense classifiers (Xie et al., 2 Apr 2026). SupCBM uses a fixed intervention matrix rather than a learned label head to reduce leakage (Sun et al., 2024). CobwebTM avoids fixed latent capacity by restructuring the hierarchy online (Singaravadivelan et al., 15 Apr 2026). These results collectively suggest that hierarchical structure can sometimes increase both transparency and performance, but this is not automatic; it depends on the specific representational bottleneck and loss design.

Several open issues recur. One is hierarchy depth. HIL-CBM uses two levels and explicitly notes challenges in extending to deeper hierarchies because very abstract concepts may be non-visual and intermediate taxonomic names may not help users (Xie et al., 2 Apr 2026). MLCS and Deep-HiCEMs are architecturally recursive but evaluated mainly at two discovered levels (Hill et al., 10 Mar 2026). SupCBM is limited to a two-level part-plus-descriptor hierarchy (Sun et al., 2024). Another issue is evaluation. Some papers report human studies or intervention curves (Xie et al., 2 Apr 2026, Dong et al., 4 Oct 2025), while others rely on structural metrics such as Parent–Child Coherence, Sibling Diversity, or concept-bank ROC-AUC (Singaravadivelan et al., 15 Apr 2026, Hill et al., 10 Mar 2026). A plausible implication is that hierarchical concept modeling still lacks a shared evaluation protocol spanning faithfulness, usability, structural quality, and downstream utility.

Another open problem is robustness to noisy or emergent concepts. CobwebTM notes sensitivity to document order and dependence on pretrained encoders (Singaravadivelan et al., 15 Apr 2026). HIL-CBM notes dependence on GPT-4 and CLIP (Xie et al., 2 Apr 2026). Deep-HiCEMs notes that discovered concept interventions can sometimes harm performance (Hill et al., 10 Mar 2026). Ontology construction with GPT-3.5 reports sloppiness, attribute inflation, hallucinations, and wrong relation type, mitigated but not eliminated by verification and KRIS insertion (Funk et al., 2023). These observations indicate that hierarchical concept modeling often trades annotation cost for uncertainty in concept validity.

Finally, cross-modal generalization remains an active direction. CoPA is designed for dermoscopic and skin images but identifies design principles—concept anchors, multi-layer aggregation, concept-guided prompting, textual alignment, and gated aggregation—that could transfer to other domains (Dong et al., 4 Oct 2025). CobwebTM explicitly suggests multimodal topic modeling via continuous representations (Singaravadivelan et al., 15 Apr 2026). Polaris demonstrates taxonomy expansion in trees, DAGs, and multimodal hierarchies, including CUB-200-2011 (Mishra et al., 30 Apr 2026). Linear analyses of hierarchical concepts in LLMs imply that pretrained models already encode interpretable hierarchy subspaces (Sakata et al., 9 Apr 2026). Taken together, these results suggest that hierarchical concept modeling is evolving toward a general methodology for structured interpretability across text, images, multimodal data, and knowledge graphs, rather than remaining a niche extension of flat concept bottlenecks.