Pyramid Stack: Concepts & Applications

• Pyramid stacks are layered, hierarchical structures that organize components recursively to achieve multiscale and systematic design across various disciplines.
• They enable precise methods in convex geometry, combinatorial tiling, and compressed stack algorithms, optimizing mesh generation and memory usage.
• Moreover, pyramid stacks underpin advanced deep learning architectures and multiscale data representations through effective integration of features across layers.

A pyramid stack refers to any systematically stacked or layered architecture, data structure, or geometric process exhibiting a pyramidal, multiscale, or hierarchical organization. The term arises in diverse scientific and engineering disciplines, including computational geometry, combinatorics, algebraic geometry, numerical analysis, machine learning, and computer vision. In all contexts, the fundamental concept involves constructing or analyzing a structure (physical, combinatorial, or algorithmic) by organizing components in a layered fashion—often with refinement or aggregation across scales—such that each layer builds upon or interacts with those below, producing a pyramidal or multiscale structure.

1. Geometric and Combinatorial Foundations

Pyramid stacks have rigorous definitions and applications in polytope theory and tiling models, playing a foundational role in computational and algebraic geometry.

1.1 Convex Geometry and Pyramidal Growth

In convex polytope theory, a pyramid over a polytope $P$ (apex $v$) is defined as the convex hull $Q = \mathrm{conv}(P, v)$ with $v$ not in the affine span of $P$. The iterative process of pyramidal stacking, where pyramids are recursively added onto facets of an existing polytope, is formalized as the pyramidal growth order. Gubeladze (Gubeladze, 2020) established that for 3-polytopes (and asymptotically, for all polytopes), any polytope can be constructed from any smaller included polytope by a sequence of pyramid stackings, preserving convexity in dimension 3 and, up to infinitesimal defect, in higher dimensions.

Table: Convexity and Pyramidal Stacking

Dimension	Convexity preserved?	Stack process
2	Yes	Triangular stacking
3	Yes	Facet-wise pyramids
≥4	Asympt. / Quasi	Limit/relaxation

This pyramidal extension framework is closely linked to the structure of posets in combinatorial geometry and the K-theory of monoid rings, where minimal pyramidal additions correspond to minimal object enlargements essential in algebraic invariants and homotopy theory.

1.2 Domino Tilings and Steep Tilings

Pyramid stacks also appear as combinatorial objects in domino tiling models. Bouttier, Chapuy, and Corteel (Bouttier et al., 2014) defined pyramid partitions (also called "pyramid stacks") as a limit case in the theory of steep tilings. These are infinite domino tilings of the plane with asymptotic data encoded by a binary word $w$, particularly $w = +^{\infty}-^{\infty}$ for stretches of north/east and south/west dominos, representing an infinite pyramid. The enumeration of these pyramid stacks is achieved via explicit product formulas arising from a vertex operator formalism, with connections to Schur processes, integrable systems, and limiting shape analysis.

2. Pyramid Stack in Algorithmic and Data Structures

2.1 Compressed Stack Technique and 1D Pyramid Approximation

In algorithm theory, the concept of pyramid stacks arises in the design of space-efficient algorithms for stack-based geometric problems. Barba et al. (Barba et al., 2012) introduced a compressed stack technique, permitting algorithms with stack bottlenecks to operate under severe memory constraints. The technique underpins the efficient resolution of the 1D pyramid approximation problem: Given a vector $f$, the objective is to find a unimodal (pyramidal) vector $\varphi$ minimizing $\|f-\varphi\|^2$. The solution involves constructing two monotonic hulls—left and right of the pyramid’s apex—using two stack-based geometric algorithms. When memory is limited, both stacks are compressed, and the procedures for stack manipulation and rollback reflect the underlying pyramid structure of the problem’s optimal solution. The term pyramid stack here refers to the logical organization of two monotonic stacks joined at the apex of the unimodal approximation.

2.2 Stack-Based Algorithms and Space-Time Trade-offs

The compressed stack framework yields general space-time trade-offs for stack-based algorithms, with time complexity scaling exponentially or polynomially in the allowed workspace, based on parameterization. For the 1D pyramid approximation, this permits solving the problem in $O(n \log_p n)$ time using $O(p \log_p n)$ space, with stack operations tailored to the pyramid structure induced by the unimodal constraint.

3. Pyramid Stack in Finite Element and Mesh Design

3.1 Pyramidal Elements in Finite Element Methods (FEM)

Pyramid stacks underlie the construction of lowest-order composite finite element spaces on pyramidal elements (Ainsworth et al., 2017). In hybrid mesh settings, such as those combining tetrahedra, hexahedra, and pyramids, pyramidal elements are subdivided into tetrahedra. The basis functions are defined piecewise on these sub-tetrahedra and patched to ensure conformity with standard elements (Lagrange, Nédélec, Raviart-Thomas) and exactness of the de Rham sequence. This approach, a composite (macro-element) pyramid stack, enables hybrid tetrahedral–hexahedral–pyramidal meshes that maintain optimal order of approximation and stability.

3.2 Schneiders' Hex-Meshing Pyramid Problem

The pyramid is also a key block in all-hex mesh conversion. The solution to Schneiders’ pyramid problem (Xiang et al., 2018) required generating a pyramid conforming to a subdivided 16-quadrilateral surface by stacking a minimal number of hexahedra (36 in the optimal solution). The systematic stacking of hexahedral elements until the pyramid surface matches the required pattern is an instantiation of the pyramid stack concept in computational meshing. The parity-changing template further exemplifies local stack reorganization to enable mesh transformation beyond parity constraints.

4. Pyramid Stack in Deep Learning and Computer Vision

4.1 Feature Pyramid Stacks and Grids

Feature pyramids are central in modern object detection and segmentation architectures. Classic Feature Pyramid Networks (FPN) explicitly stack cross-scale fusion blocks to aggregate multi-level features, but these are often limited by depth and simple pathwise stacking.

Recent architectures extend these ideas:

• Concatenated Feature Pyramid Networks (CFPN) (Sun et al., 2019): Use concatenation rather than addition to stack features from all pyramid levels, introducing an Inception module to explicitly model inter-level correlations.
• Feature Pyramid Grids (FPG) (Chen et al., 2020): Systematically extend pyramid structures into a deep, multi-pathway grid, allowing for multi-directional lateral connections across scales and pathways, thus stacking pathways and scales in a regular, extensible grid for richer feature aggregation.

4.2 Pyramid Convolution and Pyramid Stack

The pyramid convolution (PConv) module (Wang et al., 2020) performs 3D convolution over spatial and scale axes, with stacking of multiple such modules (pyramid stack) yielding a feature extractor with both spatial and scale-level receptive field growth. The SEPC module further integrates these convolutions for scale-equalization, demonstrating empirical gains over prior multi-scale feature fusion modules.

4.3 Implicit and Adaptive Pyramid Stacking

Implicit FPN (i-FPN) (Wang et al., 2020) models the feature pyramid as a fixed point of an implicit function, achieving an effect analogous to infinitely stacking FPN blocks, but with constant parameter cost.

In pyramid attention networks (Mei et al., 2020), pyramid stacks are instantiated in modules aggregating nonlocal information across scales, leveraging multi-scale self-similarity for image restoration tasks. This is done via building a pyramid of downsampled features and computing block-wise multi-scale attention.

Similarly, in semantic segmentation (e.g., DeepPyramid (Ghamsarian et al., 2022), DeepPyram (Ghamsarian et al., 2021)), pyramid stacks refer to the layered integration of context modules (pyramid view fusion), deformable receptive fields, and multi-level output supervision layers, each explicitly operating at different scales and stacked along the decoding path. These networks demonstrate improved performance in medical image segmentation tasks characterized by variable scale, deformability, and ambiguous boundaries.

4.4 Attention Pyramid Stack

The attention pyramid (Chen et al., 2021) in person re-identification stacks split-attend-merge modules across increasing granularity levels, each level building more fine-grained attention maps that are sequentially stacked and propagate multi-scale attention throughout the network.

5. Algebraic and Enumerative Aspects

5.1 BPS Pyramid Partitions and Cluster Recursions

In the enumeration of BPS invariants for toric Calabi–Yau threefolds, BPS pyramid partitions are encoded as stacks of stones (quiver paths), with finite or infinite types depending on the quiver framing (Eager et al., 2011). The partition functions associated to these stacks are computed via cluster algebra recursions—specifically, instances of the multidimensional octahedron recurrence. Cluster mutations at quiver nodes stack and rearrange pyramid configurations, and these transformations directly correspond to Seiberg duality in the field theory context. The generating functions for these pyramid stacks often yield celebrated integer sequences (Somos sequences), reflecting hidden algebraic structure.

5.2 Steep Tilings and Planar Stacks

Steep tilings of oblique strips, parameterized by a binary word $w$, realize domino tilings in the shape of a pyramid stack (Bouttier et al., 2014). Connections are forged between these tilings, sequences of interlaced partitions, and Schur processes, unifying the combinatorics of domino and plane partition models within the generalized pyramid stack framework.

6. Pyramid Stack in Multiscale Data Representations

6.1 Pyramid Transforms for Manifold Data

The multiscale analysis of manifold-valued data exploits pyramid stack constructions via subdivision and decimation operators (Mattar et al., 2021). The decomposition proceeds by recursive downsampling (decimation) and upsampling (non-interpolating subdivision) steps, producing a stack of detail coefficients across scales. For data sequences on a nonlinear manifold, this pyramid stack preserves analytical properties such as stability, coefficient decay, and reconstructability, enabling applications ranging from denoising to anomaly detection.

6.2 Convolutional Neural Pyramids

The convolutional neural pyramid (CNP) (Shen et al., 2017) organizes a deep regression network as a stack of increasingly coarser-resolution sub-networks, each processing downsampled versions of the input and feeding into a progressive upsampling and fusion path. Stacking these levels enables a dramatic increase in effective receptive field while preserving computation and memory efficiency.

7. Summary Table: Key Pyramid Stack Structures Across Domains

Domain	Stack Structure	Key Function
Convex Geometry	Facetwise pyramid addition	Polytope construction/extension (Gubeladze, 2020)
Tiling/Combinatorics	Domino/partition stacking	Partition function, integrability (Bouttier et al., 2014)
Algorithmics	Compressed stack pair	Unimodal/pyramid approximation (Barba et al., 2012)
FEM, Mesh Generation	Subdomain stacking, hex packing	Mesh compatibility, element minimization
Deep Learning	Layered multi-scale modules	Feature aggregation, receptive field (Chen et al., 2020, Wang et al., 2020, Wang et al., 2020)
Algebraic Geometry	Paths/stones in quiver stack	BPS counting via cluster recursion (Eager et al., 2011)
Multiscale Transforms	Pyramid of subdivision steps	Detail decomposition, denoising (Mattar et al., 2021)

8. Impact and Significance

The pyramid stack is a unifying abstraction with foundational impact across computational geometry, numerical analysis, combinatorics, representation theory, and deep learning. Its centrality lies in the systematic assembly, decomposition, or analysis of complex structures via recursive, multiscale, or hierarchical stacking. This enables efficient algorithms (e.g., space-time trade-offs), robust geometric and mesh designs (e.g., convex growth, mesh generation), and advanced neural network architectures (e.g., feature and attention pyramids), consistently yielding performance, stability, and analytical tractability across modalities.