Hierarchical Depth: Multi-Domain Insights

• Hierarchical depth is the multi-level organization of information processing, where systems decompose complex tasks into nested layers for efficiency.
• In neural networks and vision, it drives successive feature extraction and coarse-to-fine refinements, reducing sample complexity and enhancing performance.
• In directed networks and algebraic geometry, it quantifies flow order and stratification, offering robust insights into structure and invariant properties.

Hierarchical depth refers to both structural and algorithmic frameworks in which "depth"—either as a network architecture parameter, a flow-theoretic graph property, a geometric concept in computer vision, or an invariant in algebraic geometry—is defined, computed, and exploited in a multi-level or hierarchically-organized way. Because the concept appears in diverse mathematical and engineering domains, its rigorous meaning and computational significance vary. This article surveys rigorous formulations, measurement, and utilization of hierarchical depth across neural learning theory, computer vision (especially depth estimation), directed networks, and algebraic geometry.

1. Hierarchical Depth in Neural Network Theory

Recent theoretical advances formalize hierarchical depth as the organization of computation in neural networks into successive stages that progressively reduce effective dimension or complexity (Dandi et al., 19 Feb 2025). The central insight is that depth enables a compositional sequence of feature extraction and transformation, so that high-dimensional inputs are systematically coarse-grained into representations of decreasing dimension or nonlinearity.

Consider the "Single-Index Gaussian Hierarchical Target" (SIGHT), where input data $x\sim\mathcal{N}(0,I_d)$ passes through a sequence of mappings:

• First-layer: linear projection $W^*x$ to a low-dimensional subspace of dimension $d^{\epsilon_1}$.
• Second-layer: polynomial nonlinearity $P_k$ (degree $k$) and weighting $a^TA/\sqrt{d^{\epsilon_1}}$.
• Third-layer: final scalar nonlinearity $g^*$.

In deep networks trained by gradient descent, feature learning is hierarchical: each layer learns to approximate successive subspace projections and nonlinearities, converting a high-dimensional learning task into a sequence of lower-dimensional inductive steps. The "hierarchical depth" of a target is the number of such stages before arriving at the final output. Analysis shows that for a target function composed of $L$ such layers, each with decreasing subspace dimension, a depth-$L$ network learns at $L$ separate sample complexity thresholds, in contrast with the polynomial sample complexity required for shallow (kernel or two-layer) methods:

• First stage (learns $W^*$): $n_1=\Theta(d^{1+\epsilon_1})$ samples.
• Next stage (learns $k$-th order nonlinearity in reduced dimension): $n_2=\Theta(d^{k\epsilon_1})$ samples.
• Final stage (learns final scalar mapping): $O(1)$ samples.

Hierarchical depth therefore yields an exponential reduction in the required number of samples and computations with depth (Dandi et al., 19 Feb 2025).

2. Hierarchical Depth in Computer Vision Architectures

2.1. Monocular Depth Estimation: Hierarchical Plane Guidance

In the context of monocular depth estimation, hierarchical depth refers to architectures explicitly decomposing depth inference across multiple scales by propagating coarse-to-fine geometric information (Liu et al., 2024). In Plane2Depth, the architecture uses a set of learned "plane queries" as geometric prototypes, organized in a top-down hierarchy:

• Coarsest scale: initial set of plane queries $P_3$ models global planar structures.
• At each finer scale $l$, image features are modulated by plane queries from the previous level and further refined through masked cross-attention, producing new plane queries $P_l$.
• The final set of queries parameterizes plane normals and distances, with per-pixel depth computed via a closed-form derived from the pinhole camera model.

This hierarchical depth modeling produces greater spatial coherence and robustness, particularly in low-texture or repetitive regions. Ablation demonstrates that the top-down hierarchical approach outperforms non-hierarchical ("flat") alternatives in standard metrics (e.g., RMSE on NYU-Depth-v2 and KITTI) (Liu et al., 2024). The number of hierarchical plane prototypes (queries) is a key hyperparameter, with optimal values empirically found in the $32$–$64$ range.

2.2. Hierarchical Multi-scale Depth Estimation and Completion

Other models such as HMS-Net utilize hierarchical multi-scale encoder-decoder networks where features are progressively downsampled and then upsampled, with skip connections between matching layers. Sparsity-invariant convolutions and operations carry hierarchical masks across scales, allowing efficient fusion of global and local structure in both sparse depth completion and RGB-guided tasks (Huang et al., 2018).

Hierarchical fusion of features is also used in depth map super-resolution and saliency detection:

• Hierarchical Color Guidance in DSR learns to inject color details and semantic priors at multiple stages, achieving sharp boundaries and semantic consistency (Cong et al., 2024).
• HiDAnet for RGB-D saliency detection applies depth discretization at multiple granularities, aligning fine-grained and coarse granularity with CNN stages, and fuses multi-modal information via hierarchical attention (Wu et al., 2023).

2.3. Hierarchical Normalization

Hierarchical depth normalization (HDN) implements multiple normalizations at different spatial or depth-based contexts. Each pixel is normalized not only globally (over the entire image), but also relative to smaller spatial regions or quantile-based depth bins, permitting preservation of both large-scale scene structure and local detail (Zhang et al., 2022). This multi-context normalization significantly improves zero-shot transfer and boundary sharpness compared to global normalization.

3. Hierarchical Depth in Flow Hierarchies of Directed Networks

In network science, hierarchical depth quantifies one aspect of global order in a directed graph—how many "levels" of flow exist from roots to leaves.

Two non-equivalent formalizations are standard (Suchecki et al., 2013):

• Rooted depth: For each vertex reachable from a root (in-degree zero), the shortest-path length from any root. The average over all root–leaf pairs yields the network's rooted depth.
• Relative depth: After collapsing cycles into single nodes (strongly connected components), assign integer depths to nodes so every edge points from lower to higher depth and the depth difference along any edge is at least one. The network's depth is the (average) maximal path length in the resulting DAG.

These notions can diverge: rooted depth encodes "distance from origin" but does not enforce local consistency along all arcs; relative depth guarantees monotonicity but can obscure the global branching structure. Empirical studies show both depth measures peak near the percolation threshold in random directed graphs and then decay as loops dominate. Both require care with cycles, which must be collapsed for consistent depth assignment.

4. Hierarchical Depth in Stereo and Multiscale Matching

Hierarchical depth is a foundational strategy in stereo matching and other geometric vision algorithms. Multi-scale, pyramid-based methods build depth (or disparity) estimates from coarse to fine scales, propagating hypotheses down the Gaussian pyramid while refining only uncertain pixels at finer levels (Kaushik et al., 2019). At each scale, only a small search window is needed for most pixels due to the prior established by coarser levels, reducing both computation and memory relative to "flat" search or global energy minimization schemes.

This coarse-to-fine paradigm improves robustness and speed:

• At coarse scale, perform full-range block matching for all pixels.
• At each subsequent finer scale, upsample disparities and restrict searches for well-matched pixels to narrow windows.
• For ambiguous pixels, widen the search or apply additional regularization (median filtering, local costs).

Quantitative evaluation demonstrates substantial reductions in average disparity error and computational resource requirements compared to non-hierarchical methods, with only minor reduction in accuracy relative to more costly approaches (e.g., SGM) (Kaushik et al., 2019).

5. Hierarchical Depth in Algebraic Geometry

In algebraic geometry, hierarchical depth is formulated as a numerical invariant associated with filtrations of torsion-free sheaves—sequences of subsheaves of equal rank whose quotients are torsion sheaves supported in codimension one (Rahmati-asghar, 21 Dec 2025). Formally, for a sheaf $\mathcal{E}$ on a normal projective variety $X$, a hierarchical filtration is a sequence: $$\mathcal{E}_0 \subset \mathcal{E}_1 \subset \cdots \subset \mathcal{E}_h = \mathcal{E}$$ where each inclusion is saturated and $$\det(\mathcal{E}_i) \simeq \det(\mathcal{E}_{i-1})\otimes \mathcal{O}_X(D_i)$$ for some effective Cartier divisor $D_i$. The hierarchical depth $\delta_{\Lambda_0}(\mathcal{E})$ is the maximal length $h$ of such a filtration with specified normalization $\Lambda_0$.

Key properties:

• Determinantal bounds: On varieties of Picard rank one, $\delta_{\Lambda_0}(\mathcal{E}) \leq d - d_0$ for $\det(\mathcal{E}) \simeq \mathcal{O}_X(dA)$, $\Lambda_0 \simeq \mathcal{O}_X(d_0A)$.
• On smooth projective curves, depth equals the total degree difference: $$\delta_{\Lambda_0}(\mathcal{E}) = \sum_i d_i - \deg(\Lambda_0)$$.
• Commutativity: Elementary transforms along disjoint divisors commute, allowing reordering of hierarchical steps without changing total depth.

In dimension two, hierarchical depth transforms additively under birational contractions (minimal model program): each exceptional divisor increases depth by its multiplicity in the determinant difference. This has arithmetic significance in the theory of algebraic–geometric codes: hierarchical depth detects degeneracies, and birational simplification (contracting exceptional divisors) produces shorter codes with higher relative minimum distance (Rahmati-asghar, 21 Dec 2025).

6. Interpretative Remarks and Cross-Domain Significance

Hierarchical depth consistently emerges as a mechanism for representing structure across scales—be it in geometric reasoning, signal processing, learning, or algebraic invariants. A plausible implication is that algorithms or architectures that explicitly model, enforce, or exploit hierarchical depth achieve gains in robustness, generalization, computational efficiency, or structural interpretability.

Despite the differences in formalization and context, the essential role of hierarchical depth is the capacity to integrate global and local information, to decompose complex structures into nested strata, and to guide computation or inference by propagating information across these levels. In learning, it enables successive reduction of dimensionality and complexity; in geometry and vision, it ensures coarse-to-fine recovery of scene structure; in graph theory, it measures organizational order; and in algebraic geometry, it quantifies the stratification of vector bundles and connects to arithmetic applications.

7. Key Papers and Benchmarks

Domain	Hierarchical Depth Manifestation	Representative Papers
Deep learning theory	Successive reduction via depth	(Dandi et al., 19 Feb 2025)
Vision (depth est.)	Plane/GN/feature hierarchies, multiscale	(Liu et al., 2024, Zhang et al., 2022, Huang et al., 2018, Kaushik et al., 2019)
Directed networks	Rooted/relative depth as flow ordering	(Suchecki et al., 2013)
Algebraic geometry	Length of filtration of sheaves	(Rahmati-asghar, 21 Dec 2025)

• "The Computational Advantage of Depth: Learning High-Dimensional Hierarchical Functions with Gradient Descent" (Dandi et al., 19 Feb 2025)
• "Plane2Depth: Hierarchical Adaptive Plane Guidance for Monocular Depth Estimation" (Liu et al., 2024)
• "Hierarchical Normalization for Robust Monocular Depth Estimation" (Zhang et al., 2022)
• "HMS-Net: Hierarchical Multi-scale Sparsity-invariant Network for Sparse Depth Completion" (Huang et al., 2018)
• "Fast Hierarchical Depth Map Computation from Stereo" (Kaushik et al., 2019)
• "Hierarchy depth in directed networks" (Suchecki et al., 2013)
• "Hierarchical filtrations of torsion-free sheaves and birational geometry" (Rahmati-asghar, 21 Dec 2025)
• "HiDAnet: RGB-D Salient Object Detection via Hierarchical Depth Awareness" (Wu et al., 2023)
• "Learning Hierarchical Color Guidance for Depth Map Super-Resolution" (Cong et al., 2024)