High-dimensional Equivalent (HiE)

• High-dimensional Equivalent (HiE) is a concept that rigorously defines correspondences between complex high-dimensional structures and their lower-complexity, equivalent forms.
• It employs methods from graph theory, random matrix theory, and differential geometry to achieve combinatorial, spectral, and geometric equivalences in various domains.
• HiE facilitates reduced computational complexity and interpretability in applications spanning geometric topology, neural network design, and theoretical gravity.

The term “High-dimensional Equivalent” (HiE) encompasses rigorous notions and constructions wherein high-dimensional mathematical, physical, or computational structures are shown to possess equivalents—either exact or combinatorial—in alternative domains, typically of lower formal or computational complexity. This equivalence may be geometric, combinatorial, spectral, or dynamical, depending on context. Recent results establish HiE relationships in fields ranging from geometric topology and group theory to implicit neural networks and higher-dimensional gravity.

1. Formal Definitions of High-dimensional Equivalence

High-dimensional equivalence is instantiated by precise identification of structures—actions, spectral behaviors, subdivision rules, or kernel matrices—in high dimensions that admit either:

• Combinatorial reduction (structural preservation under complexity minimization),
• Spectral congruence (equality or deterministic correspondence of spectral data across different architectures),
• Geometric/dynamical reformulation (faithful reproduction of field equations or invariants under alternate geometric variables).

Examples:

• In finite subdivision rules, any $d$-dimensional subdivision rule $R$ (with complex $S_R$) can be reduced in the combinatorial sense to a 3-dimensional rule $(R_3, X_3)$ whose history graph reconstructs that of the original $R$ (Rushton, 2015).
• In deep equilibrium model (DEQ) theory, the spectral behavior (CK/NTK matrices) for high-dimensional input (Gaussian mixture regime) generated by implicit DEQ architectures is shown to be captured by appropriately designed shallow explicit networks, up to deterministic equivalence in the kernel eigenspectra (Ling et al., 5 Feb 2024).
• In higher-dimensional gravity, the Lovelock action in arbitrary $d$ dimensions admits a teleparallel equivalent, where the dynamical content and symmetries persist under reformulation from curvature-dominated (Levi-Civita) to torsion-dominated (Weitzenböck) geometries (Astudillo-Neira et al., 2017).

2. Construction Methods and Characterizations

HiE constructions depend on: (a) structural maps (e.g., history graphs, spectral kernel relations), (b) nonlinear algebraic equations yielding parameter congruence, and (c) explicit reduction algorithms or continuation techniques.

Subdivision Rules – Combinatorial Equivalence:

• The history graph $\Gamma(R, X)$ encodes how $d$-cells subdivide across iterates; combinatorial equivalence is asserted if two rules yield isomorphic labeled graphs.
• Every labeled graph decomposition (with levels and predecessor maps) can be realized by a finite subdivision rule in dimension 3, by explicit construction using 3-balls and colored boundary disks to encode vertex and edge labels, modulo ideal set augmentation (Rushton, 2015).

Spectral Equivalence in Neural Architectures:

• For Gaussian mixture inputs $x_i \in \mathbb{R}^p$, the DEQ model yields CK/NTK matrices whose spectra are determined by four scalar nonlinear equations involving the activation function $\phi$ and initial weight variances $\sigma_a, \sigma_b$ (Ling et al., 5 Feb 2024). There exist shallow explicit networks whose depth-$L$ activation parameters can be solved via a system (M) to exactly match these kernels in operator norm.

Gravity Theories – Geometric Equivalence:

• The teleparallel equivalent of Lovelock gravity replaces the action built from curvature two-forms $R^{ab}$ (Lanczos–Lovelock action) with one built from torsion/contorsion forms $K^{ab}$ in a Weitzenböck connection. Dimensional continuation of Euler densities generates closed, gauge-invariant forms of identical dynamical content (Astudillo-Neira et al., 2017).

3. Mathematical Frameworks Enabling HiE

The realization of HiE relies on several mathematical frameworks:

• Graph-theoretic combinatorics: For subdivision rules, structures are encoded via labeled graphs and combinatorial subdivision rules.
• Random matrix theory and implicit function analysis: In DEQ/NN equivalence, random matrix operator convergence and fixed-point equations drive the equivalence.
• Differential geometry and exterior calculus: For gravity theories, equivalence is formulated using differential forms, connections, and geometric invariants.

Context	High-dimensional Structure	Equivalent (Reduced) Structure
Finite subdivision	$d$-complex & subdivision rule $R$	3D CW complex, history graph
Neural architectures	DEQ, infinite depth kernels	Shallow explicit NN, parameter match
Gravity theories	Lovelock action in $d$ dimensions	Teleparallel action via contorsion

4. Examples, Applications, and Explicit Reductions

Subdivision Rules:

• The $4$-dimensional barycentric subdivision rule on the simplex is shown to be combinatorially equivalent to a concrete 3D rule on a ball with five boundary disks, with all combinatorial data preserved under the reduction (Rushton, 2015).

DEQ/NN Spectral Equivalence:

• Tanh-DEQ in the GMM regime is matched by 1-layer explicit NNs with “hard-Tanh” activations; likewise, ReLU-DEQ is reduced to a 2-layer leaky-ReLU explicit model. Both cases achieve CK/NTK spectral identity and nearly identical performance in classification accuracy and computational efficiency (Ling et al., 5 Feb 2024).

Teleparallel Gravity:

• The teleparallel Lovelock action, constructed by dimensional continuation of lower even-dimensional Euler densities, yields closed forms manifestly invariant under the Poincaré group and diffeomorphisms. This reformulation retains dynamical equivalence: the field equations for the contorsion reduce to those for the Levi–Civita connection (Astudillo-Neira et al., 2017).

5. Theoretical Implications and Limitations

• The equivalence of high-dimensional constructs to lower-dimensional or more tractable representations provides insight into both complexity collapse and structural invariance. For subdivision rules, it formalizes the notion that high-dimensional combinatorial data admit visualization and manipulation in low dimensions.
• In machine learning, HiE theory shows that the infinite-depth implicit model class (DEQ) brings no “spectral” advantage over well-constructed shallow explicit networks in specified high-dimensional regimes, with all kernel spectra governed by a handful of scalar parameters (Ling et al., 5 Feb 2024).
• In field theory, teleparallel equivalents allow gauge and covariance properties (closedness, topological invariance) to persist under geometric reformulation.

This suggests that, at least for certain regimes and structural constraints, complexity or dimensional proliferation does not entail fundamentally new mathematical or computational content—subject always to the preservation of critical invariants (spectral, topological, dynamical).

6. Consequences and Applications

• Every finite subdivision rule in arbitrary dimension may be replaced (for combinatorial/tessellation purposes) by a 3D rule, with consequences for understanding boundaries of hyperbolic groups and compactifications in geometric group theory (Rushton, 2015).
• In neural network theory, efficiency and interpretability of shallow explicit models are justified by rigorous spectral equivalence to their DEQ counterparts, enabling computational savings and guided architecture design (Ling et al., 5 Feb 2024).
• In gravity, the geometric equivalence informs both action quantization approaches and the investigation of gauge and topological aspects of fundamental interactions (Astudillo-Neira et al., 2017).

A plausible implication is that future research across mathematics, physics, and machine learning will increasingly rely on high-dimensional equivalence techniques to reduce, analyze, and harness complex phenomena through their more tractable lower-dimensional or explicit representations.