High-Dimensional Convolution

• High-dimensional convolution is an extension of classical convolution that generalizes the operator to D dimensions, enabling efficient function approximation and structured kernel designs.
• It mitigates exponential growth in computation and memory by leveraging sparse strategies and hybrid kernel patterns, reducing FLOPs and enhancing scalability.
• This framework underpins advanced neural network architectures and non-Euclidean models, improving performance in image processing, 3D recognition, and discrete optimization.

High-dimensional convolution generalizes classical convolution operations beyond the usual one, two, or three spatial dimensions, thereby encompassing a range of mathematical, computational, and algorithmic frameworks with extensive applications in machine learning, numerical analysis, optimization, and geometry. The following sections detail major paradigms, theoretical properties, practical instantiations, and computational implications as developed in recent arXiv literature.

1. Mathematical Foundations and Operator Generalizations

High-dimensional convolution on $\mathbb{Z}^D$ is defined as an extension of the classical linear operator. Let $D$ denote ambient dimension, $$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$ the input, $$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$ a collection of kernel weights indexed by offset $i \in \mathcal{N}^D \subset \mathbb{Z}^D$ (kernel support), and $$\mathcal{C}^{\mathrm{in}}, \mathcal{C}^{\mathrm{out}}$$ sparse (or dense) sets of input and output integer coordinates. The generalized convolution is

$$x^{\mathrm{out}}_u = \sum_{i \in \mathcal{N}^D: u+i \in \mathcal{C}^{\mathrm{in}}} W_i\, x^{\mathrm{in}}_{u+i}, \quad \forall u \in \mathcal{C}^{\mathrm{out}}$$

This structure admits dense and sparse, strided or dilated, hyper-cubic or arbitrary kernel supports. In high-dimensional function approximation, a convolution kernel $\Psi_{\mathbf{c},\mathbf{h}}(\mathbf{x})$ is often factorized as a tensor product of univariate kernels over each axis: $$\Psi_{\mathbf{c},\mathbf{h}}(\mathbf{x}) = \prod_{k=1}^d \Psi_{c_k, h_k}(x_k)$$ with smoothness and localization ensured through parameter choice, enabling efficient regularization and approximation in high-dimensional domains (Choy et al., 2019, Gao et al., 2024).

2. Computational Complexity and Sparse Implementation Strategies

A principal computational challenge for high-dimensional convolution is the exponential growth of tensor sizes, kernel supports, and memory requirements:

• Dense computation: For a side length $S$, kernel size $D$0, and $D$1 dimensions, the cost is $D$2 FLOPs, $D$3 memory.
• Sparse strategy: If only $D$4 input voxels are populated (e.g., 3D/4D LIDAR), coordinate-feature pairs are quantized and stored as $D$5, with $D$6 the list of $D$7 coordinates and $D$8 the $D$9 features. The per-voxel cost is then $$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$0 (Choy et al., 2019).

Kernel designs such as the "hybrid" pattern in 4D, employing a small 3D spatial block and a 1D temporal cross,

$$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$1

achieve substantial parameter and FLOP reductions, empirical speedups (1.3x over full 4D convolutions for $$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$2), and improved generalization (Choy et al., 2019).

3. High-Dimensional and Higher-Order Convolutional NN Architectures

Standard high-dimensional convolutions are used in convolutional neural network (CNN) backbones for spatial-temporal and volumetric data:

• The Minkowski Engine supports 1D to 4D convnets with full autograd via sparse-tensor BLAS, batched hash lookups, and efficient kernel-mapping (Choy et al., 2019).
• In density functional neural approximations, fully convolutional architectures implement mappings from densities $$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$3 to correlation functions $$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$4 in 2D and 3D, with convolutions as weighted densities and learnable back-projections (Glitsch et al., 19 Feb 2025).

Higher-order convolutions generalize linear convolution to Volterra–like expansions, incorporating polynomials of local patch features: $$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$5 For each order $$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$6, $$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$7 is an order-$$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$8 tensor and $$x: \mathbb{Z}^D \rightarrow \mathbb{R}^{C_{\mathrm{in}}}$$9 the $$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$0-fold patch outer product. This architecture efficiently captures higher-order image statistics—e.g., 3rd- and 4th-order correlations important in natural image processing—while achieving optimal performance at modest parameter counts (Azeglio et al., 2024).

4. Algebraic and Non-Euclidean High-Dimensional Convolutions

Vector-map convolutions (V-MapConv):

• Generalize complex (2D) and quaternion (4D) convolutions to arbitrary D by weight-sharing and learnable inter-subspace mixing. The parameterization reduces from $$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$1 to $$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$2, maintains entanglement via a $$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$3 mixing matrix, and is empirically superior in parameter economy and test performance for large-D data (e.g., satellite bands) (Gaudet et al., 2020).

Volumetric convolution in non-Euclidean domains:

• On the unit ball $$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$4, convolution must respect $$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$5 symmetry. This is accomplished by projecting functions onto a Zernike polynomial basis, performing convolution via coefficient multiplication and rotation-invariant re-projection: $$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$6 This construction is differentiable, highly structured, efficiently implemented, and enables explicit symmetry measurement, offering advantages for shape analysis and 3D recognition (Ramasinghe et al., 2019).

5. High-Dimensional Convolution in Approximation Theory and Numerical Analysis

In high-dimensional function approximation, convolution serves as a regularization primitive for quasi-interpolants:

• A two-step procedure convolves the function with a localized, smooth kernel (controlling "convolution error"), then discretizes with a sparse-grid quadrature ("discretization error"): $$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$7 For a Smolyak sparse grid, the number of nodes grows as $$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$8, mitigating the curse of dimensionality compared to tensor grids ($$W_i \in \mathbb{R}^{C_{\mathrm{out}} \times C_{\mathrm{in}}}$$9). Achieved rates are $i \in \mathcal{N}^D \subset \mathbb{Z}^D$0, and empirical results demonstrate near-predicted convergence even for $i \in \mathcal{N}^D \subset \mathbb{Z}^D$1 (Gao et al., 2024).

6. High-Dimensional Convolution for Discrete Optimization and Parameterized Algorithms

The $i \in \mathcal{N}^D \subset \mathbb{Z}^D$2 convolution extends naturally to $i \in \mathcal{N}^D \subset \mathbb{Z}^D$3 dimensions for applications in multi-constraint discrete optimization: $i \in \mathcal{N}^D \subset \mathbb{Z}^D$4 This high-dimensional formulation is polynomially equivalent (for fixed $i \in \mathcal{N}^D \subset \mathbb{Z}^D$5) to multi-constraint $i \in \mathcal{N}^D \subset \mathbb{Z}^D$6-$i \in \mathcal{N}^D \subset \mathbb{Z}^D$7 and unbounded Knapsack and related decision problems. For fixed dimension, parameterized algorithms achieve

$i \in \mathcal{N}^D \subset \mathbb{Z}^D$8

where $i \in \mathcal{N}^D \subset \mathbb{Z}^D$9 is the number of distinct weights and $$\mathcal{C}^{\mathrm{in}}, \mathcal{C}^{\mathrm{out}}$$0. The equivalence cycle between convolution, Knapsack, and associated decision problems also yields conditional quadratic lower bounds (assuming the 1D $$\mathcal{C}^{\mathrm{in}}, \mathcal{C}^{\mathrm{out}}$$1-convolution hypothesis) for high-dimensional convolution algorithms (Grage et al., 2024).

7. Empirical Results and Practical Considerations

Empirical evidence underscores the benefits and limitations of high-dimensional and higher-order convolution:

• On large-scale sparse 3D and 4D datasets (e.g., ScanNet, Synthia), high-dimensional sparse convnets (e.g., MinkowskiNet42, MinkNet20/32) outperform 2D–3D hybrid or pointwise alternatives, with significantly better mIoU and runtime (Choy et al., 2019).
• Higher-order convolutions (HoCNN, HoResNet-18) provide consistent gains (e.g., $$\mathcal{C}^{\mathrm{in}}, \mathcal{C}^{\mathrm{out}}$$2 accuracy in image classification) and exhibit qualitatively richer representations as validated via representational similarity analysis (Azeglio et al., 2024).
• In function approximation, sparse-grid convolution-based quasi-interpolants yield rapid algebraic convergence for $$\mathcal{C}^{\mathrm{in}}, \mathcal{C}^{\mathrm{out}}$$3 up to $$\mathcal{C}^{\mathrm{in}}, \mathcal{C}^{\mathrm{out}}$$4, with strong robustness and efficiency (Gao et al., 2024).
• Neural DFT models using pure convolutional architectures achieve near-simulation accuracy for density predictions in 2D and generalize efficiently to 3D, speedup factors up to $$\mathcal{C}^{\mathrm{in}}, \mathcal{C}^{\mathrm{out}}$$5 over naive local-ML computations, with the main bottleneck being consistent training data generation in high $$\mathcal{C}^{\mathrm{in}}, \mathcal{C}^{\mathrm{out}}$$6 (Glitsch et al., 19 Feb 2025).

8. Outlook and Ongoing Challenges

Ongoing challenges in high-dimensional convolution include:

• Managing exponential scaling in kernel size, memory, and computation as dimension increases.
• Designing kernel parameterizations (e.g., hybrid patterns, multi-scale or attention-based kernels) to efficiently capture relevant geometric or statistical dependencies without incurring full $$\mathcal{C}^{\mathrm{in}}, \mathcal{C}^{\mathrm{out}}$$7 costs.
• Developing symmetry-aware architectures for non-Cartesian domains (e.g., balls, spheres, SE(3)), including further efficient implementations of steerable or equivariant kernels.
• Addressing complexities in optimization problems (e.g., multi-constraint Knapsack, ILPs) where hardness equivalence with high-dimensional convolution persists.
• Enhancing training and data preparation pipeline scalability for scientific and engineering applications requiring high-dimensional structured data processing.

High-dimensional convolution unifies several active research frontiers and is central to contemporary advances in geometry-aware deep learning, large-scale function approximation, discrete optimization, and physics-informed machine learning (Choy et al., 2019, Gao et al., 2024, Azeglio et al., 2024, Glitsch et al., 19 Feb 2025, Grage et al., 2024, Gaudet et al., 2020, Ramasinghe et al., 2019).