Hierarchical Label Assignment (HLA)

• Hierarchical Label Assignment (HLA) is a structured multi-label classification framework that mandates ancestry closure, ensuring parent labels accompany every child label.
• Modern methodologies incorporate decision-theoretic approaches, deep architectures, and constraint-aware decoding to optimize global ranking and enforce hierarchy consistency.
• HLA applies to diverse domains such as text, audio, and vision, leveraging techniques like label propagation and semi-supervised learning to address label scarcity and imbalance.

Hierarchical Label Assignment (HLA) refers to the process of assigning structured, multi-label outputs consistent with an explicit or implicit class hierarchy—typically a tree or directed acyclic graph (DAG). HLA is central to hierarchical multi-label classification tasks in domains including text, audio, and vision. Key challenges include enforcing hierarchy constraints, correctly modeling dependencies between labels, and ensuring accuracy and interpretability, especially as hierarchies and label sets grow large. Modern HLA research encompasses decision-making frameworks, explicit loss formulations, end-to-end deep architectures, and post-processing algorithms that all aim to exploit the relational structure among labels.

1. Formal Definitions and Hierarchical Constraints

In HLA, every object $x$ is associated with a subset of labels $Y \subseteq \mathcal L$, with $\mathcal L$ structured by a hierarchy (tree or DAG) $\mathcal G=(\mathcal L, E)$. The fundamental consistency constraint is ancestry closure: if a child label is assigned, all its ancestors must be assigned as well. This is sometimes encoded via upward-propagation postprocessing of label matrices, as in the Hierarchical Label Propagation (HLP) paradigm (Tuncay et al., 26 Mar 2025), or directly as domain constraints in training and inference algorithms (Chen et al., 2022, Ye et al., 2022).

A general principle is as follows:

• Tree/DAG constraint: For all $x$, if label $\ell$ is assigned, then every ancestor $a$ of $\ell$ (i.e., $a \in \mathrm{Ancestors}(\ell)$) must also be assigned: $$Y_\ell = 1 \implies \forall a \in \mathrm{Ancestors}(\ell), ~ Y_a=1$$.

State spaces then comprise all indicator vectors $y \in \{0,1\}^{|\mathcal L|}$ respecting these constraints.

2. Global Ranking and Decision-Theoretic Approaches

A prominent line of work views HLA as an optimal decision-making problem under hierarchical constraints. The key contributions are:

• Multidimensional Local Precision Rate (mLPR): For each sample-label pair (event), $mLPR$ is the eventwise probability of correctness given all scores and the hierarchy; $mLPR_i = \mathbb P(Y_i=1|S_1, ..., S_n)$ (Ye et al., 2022).
• CATCH Objective: The Conditional Area under the Curve of Hit (CATCH) is defined as $$CATCH(\pi) = \sum_{i=1}^{n} (n - i + 1) \, mLPR_{\pi_i}$$ for a hierarchical ordering $\pi$, which is maximized when events are sorted by decreasing true $mLPR$, subject to topological consistency.
• HierRank Algorithm: Empirically estimated $mLPR$ enables the HierRank algorithm to merge chains in the hierarchy, producing the CATCH-optimal assignment (Ye et al., 2022). This approach automatically ensures hierarchy-respecting solutions while optimizing a global accuracy criterion.

Similarly, the HierLPR framework uses local precision rates (LPR) and block-merge untangling to optimize an early-hit area-under-curve (eAUC) criterion, providing a computationally efficient and statistically grounded assignment procedure (Ho et al., 2018).

3. Deep Learning Architectures Incorporating Hierarchy

Modern deep HLA models encode hierarchy both in the architecture and in loss/cost functions:

• Residual and Granularity-specific Feature Extraction: In image tasks, the hierarchical residual network (HRN) explicitly injects parent-level information into child-level representations via additive fusion, with combinatorial loss functions that marginalize over all legal hierarchy-consistent assignments (Chen et al., 2022).
• Attention and Label-Interaction: LA-HCN uses level-specific, label-based attention modules to hierarchically extract features from text, propagating label-conditional embeddings and masking across levels (Zhang et al., 2020).
• Contrastive and Alignment Losses: HTLA employs a text-label alignment loss, leveraging BERT and a Transformer-based label encoder (GPTrans) to jointly learn document and label embeddings such that correct label assignments are reinforced in the shared space via contrastive objectives (Kumar et al., 1 Sep 2024).
• Hyperbolic Embedding Approaches: Models embedding both input and label spaces into the Poincaré ball (e.g., HyperIM (Chen et al., 2019), Joint Hyperbolic Label Embedding (Chatterjee et al., 2021)) allow HLA to arise from the intrinsic geometry, facilitating structure-aware classification even when the explicit hierarchy is only latent.

4. Generation and Constraint-Aware Decoding

The generative sequence modeling paradigm defines HLA as a sequence of level-wise predictions with explicit constraint integration:

• Hierarchical Multi-Label Generation with Probabilistic Level-Constraint (HMG-PLC): The label sequence is generated level by level, with a masked decoder ensuring level- (and ancestry-) consistency. A probabilistic penalty term further guides the output count to match corpus-level or user-specified per-level label numbers. Inference proceeds with hierarchical beam search under constraint masks, guaranteeing no cross-level violations (Chen et al., 30 Apr 2025).
• Constraint penalties are blended with sequence-level negative log-likelihood to control output composition tightly, outperforming non-constrained generative models and rigid path-based methods such as HECTOR on complex taxonomies.

5. Hierarchical Label Propagation and Postprocessing

Practical HLA sometimes augments or postprocesses base classifier outputs to enforce the hierarchy via label propagation:

• Label Propagation: For a known ontology, upward closure is enforced by propagating each positive label to all its ancestors, yielding a denser, hierarchy-consistent label matrix that can be used both for training and inference postprocessing (Tuncay et al., 26 Mar 2025). This approach is computationally efficient ($O(NL)$) and model-agnostic.
• Impact: On benchmarks such as AudioSet, HLP boosts mean average precision especially for smaller models and increases label density per sample, highlighting its importance as a baseline for hierarchy-aware training and evaluation.

6. Active and Semi-supervised HLA; Label-efficient Annotation

Label scarcity is more extreme in hierarchical contexts due to data fragmentation over deep nodes. Hierarchical approaches to annotation and label inference include:

• Bounded Expectation of Label Assignment (BELA): An adaptive, tree-structured split-label strategy, BELA selects queries or splits to maximize a rigorous lower bound on the expected correctly labeled examples (Herbst et al., 2019). Bias-reduction tactics, such as separating train/test label pools and “post-split forgetting,” enforce valid statistical guarantees during adaptive tree growth.
• Semi-supervised HLA: Semi-supervised architectures can leverage both labeled and unlabeled data by propagating pseudo-labels via structure- or similarity-aware neighbor aggregation, allowing robust assignment even in data-poor subtrees (details in SSHMC-BLI abstract, but full exposition not available in data) (Serrano-Pérez et al., 30 Apr 2024).

7. Application-Specific and Specialized HLA Variants

HLA principles are adapted to meet application-specific requirements:

• Tiny Object Detection: The RFLA framework introduces Hierarchical Label Assignment based on Receptive Field Distance (RFD), assigning positive samples using a two-stage scoring and rescue mechanism that rectifies the severe imbalance of anchor-based or anchor-free assigners for tiny objects (Xu et al., 2022).
• Fine-grained and Partial Supervision: Models such as HRN can handle labels at variable tree-deepness, integrating mutual-exclusion and subsumption constraints across partially observed or noisy levels (Chen et al., 2022).

Summary Table: Selected HLA Methodologies

Method (Paper)	Core Mechanism	Hierarchy Enforcement
HierRank (Ye et al., 2022)	mLPR-based ranking	Algorithmic, global sorting
HRN (Chen et al., 2022)	Residual fusion	Loss-based, marginalization
LA-HCN (Zhang et al., 2020)	Label-specific attn	Architecture and masking
HTLA (Kumar et al., 1 Sep 2024)	Contrastive loss	Implicit via embedding
HMG-PLC (Chen et al., 30 Apr 2025)	Constrained gen	Masked output, PLC penalty
BELA (Herbst et al., 2019)	Adaptive splitting	Tree structure, bounds
HLP (Tuncay et al., 26 Mar 2025)	Label propagation	Upward transitive closure
RFLA/HLA (Xu et al., 2022)	RFD scoring, rescue	Top-k ranking per GT

All methods seek to guarantee hierarchical consistency—either explicitly (via tree- or DAG-based constraints), via loss regularization, or post-hoc label set closure—while simultaneously maximizing classification or ranking performance under the compositional and statistical constraints induced by the label structure. Hierarchical label assignment is thus a unifying framework for structured prediction across a broad spectrum of pattern recognition tasks.