Hierarchical Encoding Architectures

• Hierarchical Encoding Architectures are multi-scale neural frameworks that model structured data using manifold projections, hyperbolic spaces, and hierarchical positional encodings.
• They employ specialized modules and tailored loss functions to ensure consistency between coarse and fine representations, enhancing semantic accuracy and computational efficiency.
• Empirical benchmarks demonstrate state-of-the-art performance in language, vision, graphs, and time series by leveraging explicit hierarchical structures.

Hierarchical encoding architectures introduce explicit multi-scale structure into neural models, enabling representations that capture and leverage hierarchical relations across various data modalities and tasks. These architectures span a diverse design space, from manifold-based lexical projections in LLMs to hierarchical latent representations for image, video, graph, and multimodal data. Their unifying principle is the incorporation—either via architecture, positional encoding, loss design, or representational constraints—of inductive biases matching the intrinsic hierarchical structure of data domains or label ontologies. This article systematically delineates the foundational ideas, methods, algorithmic instantiations, and empirical evidence for hierarchical encoding architectures.

1. Theoretical Foundations of Hierarchical Encoding

A central characteristic of hierarchical encoding is the organization of learned representations to reflect structured relationships: tokens, nodes, or activations are not only embedded as points in $\mathbb{R}^d$ but mapped onto spaces such as Riemannian manifolds or hyperbolic balls that encode both local and global semantics. Importantly, hierarchical encoding seeks to ensure multi-scale semantic representation, providing stable interpolation between fine-grained and coarse-grained contexts. For instance, the Hierarchical Lexical Manifold Projection (HLMP) architecture represents each token as a position $p_i$ on a manifold $\mathcal{M}$ with a metric tensor $g_{\alpha\beta}$:

$$d_{\mathcal{M}}(x_i, x_j) = \inf_{\gamma} \int_0^1 \sqrt{g_{\alpha\beta}(\gamma(t))\,\dot{\gamma}^\alpha(t)\,\dot{\gamma}^\beta(t)}\,dt$$

The regularization loss aligns Euclidean distances with the manifold's geodesic structure:

$$\mathcal{L}_{\mathrm{hier}} = \mathbb{E}_{i,j} \left[ \big( \|e_i - e_j\|_2 - d_{\mathcal{M}}(x_i, x_j) \big)^2 \right]$$

This enforces that lexical relations at multiple abstraction levels are preserved by the embedding geometry (Martus et al., 8 Feb 2025).

Hierarchical encoding extends to temporal abstraction in predictive coding models, where multi-timescale dynamics are realized by level-specific leak constants $\tau_{\ell}$ within recurrent layers, leading higher network layers to develop slower, context-rich representations (Zhong et al., 2018).

Hierarchical organization of representations is also implemented in hyperbolic space for language and vision models. Here, entity or object embeddings reside in the Poincaré ball or Lorentz model, with deviceful loss functions (e.g., hyperbolic clustering and centripetal losses) that force radial and angular separation aligned with ancestor-descendant relations (He et al., 21 Jan 2024, Wang et al., 26 Nov 2024).

2. Architectural Patterns and Mechanisms

Hierarchical encoding is realized architecturally through the interplay of specialized modules, positional encoding innovations, and trainable attention or aggregation mechanisms.

Manifold-based Lexical Projections

HLMP integrates a hierarchical projection layer $P_h$ into transformer-based models. The projection of a base token embedding $e_i$ is given by a kernel-weighted sum over the vocabulary, decaying with hierarchical manifold distance and modulated by learned weights $\alpha_{ij}$:

$$P_h(e_i) = \sum_{j=1}^n \alpha_{ij} \exp(-\lambda d_{\mathcal{M}}(x_i, x_j)) e_j$$

Self-attention is biased to prefer hierarchically close tokens through an additive term $$f_{\mathrm{hier}}(h_i, h_j) = -\beta d_{\mathcal{M}}(x_i, x_j)$$ in the attention score computation (Martus et al., 8 Feb 2025).

Hierarchical Positional Encoding

HiNeRV and HIIF exemplify hierarchical encoding using multi-scale or hierarchical positional encoding. Instead of feeding positional information as fixed sinusoids or single-scale offsets, these methods inject learned or constructed multi-scale (level-wise) encodings at different network depths. For each upsampling or decoding block, hierarchical grids or multi-level modulus-folded coordinates are used to progressively refine the representation from coarse spatial/temporal structure to fine local detail (Kwan et al., 2023, Jiang et al., 4 Dec 2024).

Pooling and Latent Mean Encoding

In temporal hierarchies, latent mean encoding is used to achieve coherent multi-resolution forecasting: an encoder computes block-wise (e.g., weekly) means, a decoder predicts zero-sum deviations within each block, and the outputs compose via sum—guaranteeing consistency between coarse and fine forecasts (Salatiello et al., 24 Jun 2025).

Hierarchical Bracketing and Graph Hierarchies

Encoding structured trees or graphs often leverages minimal hierarchical bracketing. For projective dependency trees, the 12-symbol bracketing encoding represents the unique minimal rope cover structure, providing a minimal encoding for tag-based parsing (Ezquerro et al., 16 May 2025). In graph transformers, Hierarchical Distance Structural Encoding (HDSE) integrates multi-level shortest-path and coarsened-graph distances as trainable attention biases, allowing explicit modeling of hierarchical communities or neighborhoods (Luo et al., 2023).

3. Training Objectives and Optimization

Hierarchical encoding architectures frequently require tailored losses and optimizers to maintain the integrity of hierarchical structure during model training.

• HLMP employs a joint objective including the hierarchical regularization, language modeling loss, and an optional Laplace–Beltrami energy penalty to encourage smoothness on the manifold. Gradient computations must propagate through manifold distance terms, requiring automatic differentiation and Riemannian-aware optimizers that operate within tangent spaces of the embedding manifold (Martus et al., 8 Feb 2025).
• Hyperbolic encoding methods such as HiT utilize a composite loss combining a hyperbolic triplet loss (clustering) and a centripetal loss for ancestor-depth ordering. These are essential to jointly enforce both separation of sibling entities and radial ancestor-descendant alignment (He et al., 21 Jan 2024).
• In convolutional and autoencoding frameworks, hierarchical penalties are introduced to couple activations of parent and child latent units, e.g.,

$$R_{\mathrm{hier}}(z, \{z^{(i)}\}) = \sum_{i=1}^{m_\mathrm{top}} \sum_{j=1}^{m_\mathrm{low}} \max(0, z_j^{(i)} - z_i)^2$$

which restricts activation of child concepts unless the parent concept is active (Muchane et al., 1 Jun 2025).

• In graph hierarchical encodings, differentiable clustering or soft assignment matrices are learned to pool node features into cluster/subgraph representations for downstream transformer processing (Ngo et al., 2023).

4. Applications Across Modalities and Domains

Hierarchical encoding architectures are instantiated in a broad array of domains, reflecting the widespread presence of hierarchy in natural and artificial systems.

Natural Language

Hierarchical encoding methods have proved critical for lexical semantics, semantic parsing, and hierarchical text classification. HLMP achieves superior alignment to WordNet, robust semantic preservation, and improved cross-domain adaptability (Martus et al., 8 Feb 2025). HiT enables strong multi-hop transitive inference and hierarchy transfer in both WordNet and biomedical ontologies, outperforming fine-tuned baselines in F1 score (He et al., 21 Jan 2024). In code generation, tree-order (AST-based) positional encoding in transformers improves accuracy and prefix precision in program synthesis (Thellmann et al., 2022). Hierarchy-guided contrastive learning (HGCLR) for hierarchical text classification internalizes taxonomic constraints into transformer representations, surpassing static label-fusion recipes (Wang et al., 2022).

Vision and Multimodal

Vision-LLMs such as CLIP and METER exhibit layered hierarchical encoded patterns analogous to human neural processing, with layer-wise specialization and redundancy that mirrors biological representations in fMRI studies (Ren et al., 19 Oct 2025). Hyperbolic models for image retrieval (e.g., fine-tuned CLIP in Lorentzian space) directly capture part–object–scene hierarchies with substantial precision and recall improvements over Euclidean baselines for hierarchy-aware retrieval (Wang et al., 26 Nov 2024).

Graphs and Molecules

MGT and WavePE architectures allow learning at multiple scales in molecular graphs, with spectral graph wavelets supplying spatially and spectrally localized positional encoding. Hierarchical clustering pools learned atom representations into functional groups and repeats the process at higher scales, giving state-of-the-art regression performance on polymers and peptides (Ngo et al., 2023). Hierarchical mapping operator graphs systematically encode all symmetry-preserving coarse-graining operations for molecular systems, radically reducing the search space and enabling optimal mapping selection (Chakraborty et al., 2018). HDSE in graph transformers systematically integrates multi-scale topological distance information, outperforming methods based solely on shortest-path distance on graph-level and node-level benchmarks (Luo et al., 2023).

Images, Video, and 3D Data

Hierarchical positional encoding in HiNeRV enriches video INR models, yielding higher density multi-level detail at each upsampling stage, enabling state-of-the-art video compression at a fraction of the bitrate of non-hierarchical models (Kwan et al., 2023). HIIF analogously demonstrates that multi-scale hierarchical encoding (via modulus-folded offsets at each network depth) improves super-resolution in continuous image representation, achieving higher PSNR and better sample efficiency (Jiang et al., 4 Dec 2024). RALHE's octree-based hierarchical latent encoding, region-adaptively overfitted with joint rate–distortion objectives, achieves up to 2 dB higher PSNR for 3D Gaussian splatting compression compared to the best single-scale competitors (Sridhara et al., 26 Oct 2025).

Time Series

Latent mean encoding architectures for hierarchical time-series forecasting align encoder and decoder modules along aggregation levels (e.g., weekly and daily), with upsampling and deviation centering providing coherent multi-scale predictions without the need for explicit reconciliation post-processing (Salatiello et al., 24 Jun 2025).

5. Interpretability, Efficiency, and Generalization

Hierarchical encoding confers several desirable properties:

• Interpretability: Hierarchically organized embeddings make explicit the transitions between coarse and fine semantic/structural levels. Coordinates and distances on the embedding manifold correlate with external hierarchies (e.g., WordNet depth difference, part–object relations), and can be visualized directly (Martus et al., 8 Feb 2025).
• Computational Efficiency: Manifold projection and hierarchical pooling often reduce required memory, inference latency, and training time by promoting modularization and locality, as shown by the substantial savings in HLMP (e.g., 63.9 MB vs. 134.7 MB memory; 79.2 h vs. 174.2 h training) (Martus et al., 8 Feb 2025).
• Robustness and Generalization: Hierarchical encodings improve robustness under perturbations (e.g., adversarial noise) and maintain structure under domain shifts. Out-of-domain performance degrades more gracefully, due to the ability to reuse high-level abstractions for transferring to low-frequency or low-data scenarios (Martus et al., 8 Feb 2025).
• Scalability: High-level HDSE in graph transformers scales bias computation to billion-node graphs via coarsening and cross-level attention, with only linear overhead relative to graph size (Luo et al., 2023).
• Flexible Generalization: Hierarchical encoders can be extended across languages and modalities by initializing embeddings with external hierarchies or taxonomies and tuning hyperparameters (e.g., kernel sharpness $\lambda$, attention strength $\beta$) for the task (Martus et al., 8 Feb 2025).

6. Empirical Benchmarks and Quantitative Impact

Hierarchical encoding methods have achieved prominent empirical results:

Domain	Model/Method	Task/Metric	Hierarchical vs. Baseline	Reference
Language	HLMP	Alignment accuracy (WordNet, etc.)	0.89–0.94 vs. 0.65–0.75	(Martus et al., 8 Feb 2025)
Language	HiT	Multi-hop F₁ (WordNet subsumption)	0.90–0.92 vs. 0.21–0.34 (pre), 0.63 (ft)	(He et al., 21 Jan 2024)
NLP	HGCLR	Macro-F1 (WOS, NYT, RCV1-V2)	81.20 vs. ≤81.06	(Wang et al., 2022)
Molecular Graphs	MGT+WavePE	GAP (MAE, eV, polymers)	0.0387 (chem. accurate <0.043)	(Ngo et al., 2023)
Images	HIIF	PSNR (Super-Res, Urban100 x4)	26.51 vs. 26.38 (–hierarchy)	(Jiang et al., 4 Dec 2024)
Video	HiNeRV	BD-rate (UVG, PSNR)	–72.3% vs. HNeRV, –43.4% vs. DCVC	(Kwan et al., 2023)
3D Gaussian Data	RALHE	PSNR (low-bitrate rendering)	+2 dB vs. GPCC-GS; +0.8 dB vs. RDO-GS	(Sridhara et al., 26 Oct 2025)
Dependency Parsing	Hier. Bracketing	Complete-match, tokens/sec	12-label matching/↑speed vs. 16-/128-label	(Ezquerro et al., 16 May 2025)
Time Series	Latent Mean Enc.	WRMSSE (forecasting, M5)	0.620 vs. 0.640–1.085	(Salatiello et al., 24 Jun 2025)
Graphs	GT+HDSE	ZINC (MAE), MNIST (Acc)	0.159 vs. 0.226; 94.39% vs. 90.83%	(Luo et al., 2023)

7. Extensions, Open Problems, and Perspectives

Hierarchical encoding continues to evolve with research on:

• End-to-end curvature learning in hyperbolic or Riemannian embedding spaces for further adaptivity (He et al., 21 Jan 2024).
• Joint modeling of deeper or cross-modal hierarchies (e.g., cross-temporal and cross-sectional in time series; object-part-scene constructs in vision) (Wang et al., 26 Nov 2024).
• Dynamic, data-adaptive hierarchy construction (e.g., for arbitrary graphs, datasets without explicit ontologies) (Luo et al., 2023, Ngo et al., 2023).
• Hierarchy-aware attention mechanisms, hierarchical losses in masked-LM pretraining, and directly injecting hierarchical context via prompting or architectural innovations in transformers (He et al., 21 Jan 2024).

Hierarchical encoding architectures have established themselves as a critical ingredient for models that aim to reflect the multi-scale, structured nature of diverse real-world domains. Their growing empirical impact and theoretical foundations indicate a continued trajectory of broad adoption and further methodological sophistication.