PaperlayerView exact revision on arXiv

Source-linked AI summary

Minkowski Attractor Networks: Closed-Form Hyperbolic Flows for Visual Representations

Zhongping Ji

arXiv:2609.37817v1cs.CV

TL;DR

Flat representations can distort multi-scale tree-like visual hierarchies, while existing hyperbolic architectures face computational and numerical challenges. MAN uses Minkowski spacetime for closed-form radial dynamics and Lorentz transport. On CIFAR-100, MAN reaches up to 81.82% accuracy with 2.13M parameters.

  • Problem

    Flat spaces’ polynomial volume growth can distort multi-scale tree-like visual hierarchies, while hyperbolic models face computationally cumbersome operations, optimization, and numerical challenges.

    Source: §1 · pp. 1–2

  • Method

    MAN embeds representations in Minkowski spacetime, combining cone lifting, closed-form radial relaxation, and Lorentz transport within an operator-splitting framework.

    Source: §1 · p. 2, §7 · p. 13

  • Results

    81.82% accuracy on CIFAR-100 with 2.13M parameters is the reported peak for MAN-2D.

    Source: §7 · p. 13

  • Takeaways & Limitations

    MAN-2D’s D/2 channel blocks maximize filter granularity, while MAN-4D evaluates closed-form spacetime isometries through commuting Cartan subalgebras.

    Source: §1 · p. 2, §7 · p. 13

    MAN-2D’s individual H1 factors and their product are intrinsically flat, so its advantage is not intrinsic negative curvature.

    Source: §4.5 · p. 6

Abstract

from arXiv · show

Geometric representation learning predominantly scaffolds representations onto flat Euclidean subspaces or compact product tori ($\mathbb{T}^K$). However, flat manifolds possess vanishing curvature and polynomial volume growth, inherently suffering from metric distortion when embedding multi-scale, tree-like visual hierarchies. While hyperbolic spaces ($\mathbb{H}^m$) circumvent this via constant negative curvature ($K<0$) and exponential volume expansion, prior hyperbolic deep architectures are hindered by computationally cumbersome Riemannian optimization, non-linear gyrovector calculus, and floating-point instabilities. In this work, we introduce \textbf{Minkowski Attractor Networks (MAN)}, an operator-splitting-inspired framework that embeds representations within pseudo-Riemannian Minkowski spacetime ($\mathbb{R}^{1,m}$). By framing hyperbolic manifolds as quadric level sets, MAN resolves hyperbolic geometry by combining linear Lorentz group transport with non-linear cone lifting and closed-form radial rescaling, evaluating in a single forward pass without numerical ODE solvers or iterative retractions. We establish \textbf{MAN-2D} ($\mathbb{R}^{1,1} \to \mathbb{H}^1$) as our primary, high-throughput visual backbone, which maximizes channel factorization granularity into $D/2$ independent two-dimensional Minkowski blocks. We further formulate \textbf{MAN-4D} ($\mathbb{R}^{1,3} \to \mathbb{H}^3$) as a spacetime extension, leveraging a commuting Cartan-subalgebra parameterization of $\mathrm{SO}^+(1,3)$ to evaluate 4D Lorentz isometries via two commuting 2D planar maps without matrix-exponential overhead.

1 Introduction

The introduction motivates hyperbolic representations for tree-like visual hierarchies but identifies computational, optimization, numerical, and hardware obstacles. It presents MAN as a Minkowski-spacetime framework with closed-form local dynamics and reports CIFAR-100 results for MAN-2D and MAN-4D.

Motivation

  • Flat spaces’ polynomial volume growth can distort embeddings of multi-scale tree-like visual hierarchies, whereas negative-curvature hyperbolic space has exponential volume expansion.The introduction frames hyperbolic geometry as a way around the dimensional bottleneck of embedding branching trees in fixed low-dimensional flat spaces.Source: §1 · p. 1
  • Prior hyperbolic methods face undefined standard operations, costly gyrovector calculations or Fréchet means, manifold-constrained optimization, finite-precision instability, and GPU-unfriendly projections.The cited numerical risks include loss of accuracy or NaN failures in Poincaré and Lorentz models.Source: §1 · pp. 1–2

Minkowski formulation

  • MAN represents hyperbolic manifolds as quadric level sets in Minkowski spacetime, where Lorentz isometries become linear matrix multiplications in SO+(1,m).Lie–Trotter splitting pairs standard depthwise convolutions for feature mixing with an exact single-pass closed-form local reaction mapping.Source: §1 · p. 2
  • MAN separates transverse radial relaxation from tangential Lorentz transport; with shared frozen parameters, radial error decays exponentially while normalized-state pairwise hyperbolic distances are preserved.These are conditional guarantees for the exact MAN reaction flow, not an unconditional statement about the learned architecture.Source: §1 · p. 2

Contributions

  • MAN-2D reaches 81.03% and 81.82% accuracy, while MAN-4D reaches 80.80% and 81.75%, outperforming flat tori and a 23.7M ResNet-50.MAN-2D uses 1.02M and 2.13M parameters; MAN-4D uses 0.97M and 2.01M.Source: §1 · p. 2
  • MAN-2D uses D/2 independent two-dimensional blocks for maximal filter granularity, while MAN-4D uses commuting boost and rotation generators to reduce parameter overhead from 2.0D to 1.75D.MAN-2D’s H1 factors are intrinsically flat, but their non-compact coordinates avoid the periodic phase-locking of compact tori.Source: §1 · p. 2

2 Related Work

The related work situates MAN among continuous-depth and geometric models, then distinguishes it from prior hyperbolic deep-learning approaches by its closed-form reaction dynamics and structured Lorentz transport.

Continuous-Depth Representations

  • Neural ODEs, Hamiltonian and symplectic networks, and operator splitting connect continuous dynamics to neural architectures; CTAN applies operator splitting on flat tori.MAN extends this continuous paradigm to non-Euclidean pseudo-Riemannian spacetimes.Source: §2 · p. 3

Hyperbolic Deep Learning

  • Prior hyperbolic work includes Poincaré embeddings, gyrovector methods, hyperbolic graph networks, continuous Riemannian flows, and Lorentz-model networks.MAN instead uses closed-form logarithmic radial relaxation with low-dimensional structured Lorentz transport.Source: §2 · p. 3
  • Poincaré embeddings introduced hyperbolic geometry to deep learning for hierarchical structures, while gyrovector calculus generalized the approach [7].The related-work passage also names hyperbolic graph networks [12] and continuous Riemannian normalizing flows [13].Source: §2 · p. 3

3 Foundations: Minkowski Spacetime and the Hyperboloid Model

The foundations define Minkowski spacetime and its causal cones, then realize hyperbolic space as a hyperboloid whose induced tangent metric is positive definite. They describe its curvature, volume growth, and geodesic distance.

3.1 Pseudo-Riemannian Metric and Causal Cones

  • A Minkowski state separates a timelike scalar X0, interpreted as hierarchy depth, from spacelike coordinates X representing branching directions.The ambient space is R1,m and uses the diagonal metric eta = diag(1, -1, ..., -1).Source: §3.1 · p. 3
  • The future timelike cone consists of states with positive time coordinate and positive Minkowski norm.The text distinguishes timelike, lightlike, and spacelike states by the sign of their Minkowski norm.Source: §3.1 · p. 3

3.2 The Lorentz Hyperboloid H𝑚

  • Hyperbolic space H^m of radius R is realized as the forward sheet of a two-sheeted hyperboloid in Minkowski spacetime; its tangent metric is positive definite.The tangent-space argument relies on tangent vectors being orthogonal to a strictly timelike point under the Minkowski metric.Source: §3.2 · p. 3
  • For m ≥ 2, H^m has constant negative sectional curvature and exponentially growing geodesic-ball volume, while H1 is intrinsically flat with volume 2r.The passage states that H1 has vanishing intrinsic curvature tensor.Source: §3.2 · p. 3
  • The foundations define intrinsic geodesic distance between hyperboloid points and use the reversed Cauchy–Schwarz inequality to ensure the arccosh argument is at least 1.This guarantees the distance expression’s arccosh input is in its stated domain.Source: §3.2 · p. 4

4 Methodology: Minkowski Attractor Networks

MAN replaces compact torus targets with hyperbolic quadric level sets in Minkowski blocks, retaining decoupled local flows. Operator splitting combines spatial depthwise convolution with a closed-form reaction flow whose radial correction and Lorentz transport have distinct roles.

4.1 Recap: The Product-Manifold Scaffolding Prior

  • MAN replaces CTAN’s compact circular product-torus targets with invariant hyperbolic quadrics in Minkowski blocks, preserving decoupled solvability.The scaffold factorizes representations into low-dimensional subspaces whose fibers are conditioned on visual context.Source: §4.1 · p. 4

4.2 Operator-Split Reaction-Diffusion Flow

  • The operator-split layer applies multi-scale spatial depthwise convolution for neighbor context, then a closed-form Minkowski reaction flow within each block.Reaction-step parameters are held fixed across the local integration interval.Source: §4.2 · p. 4, §4.3 · p. 4

4.3 Continuous Dynamics and Orthogonal Decomposition

  • Tangential Lorentz flows contribute nothing to the Minkowski norm, while logarithmic radial dissipation contracts radial error exponentially toward the target level set.For fixed shared flow parameters, the dynamics are non-expansive under the stated product metric.Source: §4.3 · p. 5

4.4 Cone Lifting and End-to-End Layer Definition

  • Cone lifting handles activations outside the forward cone, with a Softplus-based timelike coordinate ensuring a valid state before the conditional flow.The full layer orders cone lifting before the exact conditional flow.Source: §4.4 · p. 6

4.5 Primary Workhorse: MAN-2D Architecture

  • MAN-2D partitions channels into D/2 independent Minkowski planes, maximizing adaptive-filter granularity without intrinsic negative curvature.Each H1 factor and their product are intrinsically flat, but non-compact coordinates avoid compact-torus phase-locking.Source: §4.5 · p. 6

4.6 Spacetime Generalization: MAN-4D Architecture

  • MAN-4D restricts Lorentz dynamics to commuting boost and rotation generators, yielding two independent planar maps that preserve the Lorentz quadratic form exactly.The full operator evaluates in a single pass; unrestricted 6-DOF dynamics would require six context channels per block and increase register pressure and intermediate memory traffic.Source: §4.6 · p. 7

4.7 Intermediate Formulation: MAN-3D (H2)

  • MAN-3D targets true negative curvature, but its three-channel divisor breaks power-of-two Tensor Core alignment and causes memory-slicing overhead in unfused implementations.Source: §4.7 · p. 8

5 Theoretical Analysis and Systemic Trade-Offs

The symplectic connection clarifies that these conservative maps preserve different quadratic forms, so symplecticity alone does not make a dual-shear an exact Lorentz boost. Across geometric scaffolds, MAN-2D prioritizes channel-filter granularity, while MAN-4D prioritizes per-factor negative curvature and parameter compactness.

5.1 Elliptic and Hyperbolic Transport: A Symplectic Perspective

  • The algebraic boost B(a) remains an exact Lorentz boost while avoiding explicit cosh and sinh evaluation.For real a, it corresponds to rapidity arsinh(a) and requires elementary arithmetic and one square root.Source: §5.1 · p. 9
  • The hyperbolic dual-shear preserves a parameter-dependent quadratic form, not the Minkowski metric preserved by exact Lorentz transport.Both are symplectic, but the dual-shear generally changes the Minkowski radius.Source: §5.1 · pp. 9–10

5.2 Comparison of Geometric Scaffolds

  • MAN-2D maximizes independent filtering granularity with D/2 blocks, whereas MAN-4D offers per-factor negative curvature with 1.75D rather than 2.0D context-generator width.The 4D design couples longitudinal depth boosts with continuous transverse category rotations.Source: §5.2 · p. 11

6 Experiments

On CIFAR-100, Minkowski models improve over flat tori across both scale tiers, and Tier-2 MAN-2D also outperforms ResNet-50 with substantially fewer parameters. The dimensional comparison highlights a trade-off between MAN-2D's fine-grained channel factorization, MAN-4D's compactness, and MAN-3D's computational cost.

6.1 Experimental Setup

  • The models were trained from scratch on CIFAR-100 for 200 epochs using AdamW and cosine annealing.CIFAR-100 has 50,000 training and 10,000 testing images across 100 fine-grained categories at 32 × 32 resolution.Source: §6.1 · p. 11

6.2 Main Benchmark Results

  • 81.82% accuracy in Tier-2 lets MAN-2D-2 outperform 23.7M ResNet-50 by +2.68% with 11× fewer parameters.Across Tier-1, the reported flat-torus to Minkowski gain is from ∼78.41% to ∼81.03%.Source: §6.2 · p. 11, Table 2 · p. 12

6.3 Dimensional Scaling

  • +2.39% to +2.68% accuracy separates the flat compact torus from Minkowski formulations H1, H2, and H3 in the Tier-1 comparison.The passage attributes the contrast to compact periodic orbits versus non-compact Minkowski flows.Source: §6.3 · pp. 11–12, Table 3 · p. 12
  • 81.03% makes MAN-2D-1 the highest-accuracy Tier-1 model, while MAN-4D-1 reaches 80.80% with 0.97M parameters.MAN-2D uses D/2 independent planes and faster throughput; MAN-4D uses fewer blocks and a narrower context generator.Source: §6.3 · p. 12
  • 81.01% accuracy for MAN-3D does not surpass MAN-2D-1, while its 1.42M parameters and slowest training throughput make it less efficient in the reported comparison.The passage attributes the added parameter cost to non-standard channel widths and the slowest throughput to intermediate memory traffic.Source: §6.3 · p. 13

7 Conclusion

The conclusion presents MAN as a Minkowski-spacetime framework combining closed-form dynamics with visual architectures, reporting up to 81.82% CIFAR-100 accuracy with 2.13M parameters.

7 Conclusion

  • MAN combines cone lifting, conditional Lorentz transport, and closed-form radial relaxation without iterative ODE integration, reaching up to 81.82% CIFAR-100 accuracy with 2.13M parameters.MAN-2D maximizes channel-filter granularity; MAN-4D evaluates closed-form spacetime isometries through commuting Cartan subalgebras.Source: §7 · p. 13

A.1 Closed-Form Formulation of Full 6-DOF Lorentz Dynamics

The unconstrained 6-DOF Lorentz transformation is evaluated analytically through a spatial rotation followed by a hyperbolic boost. It is metric-preserving but has higher latency in deep cascades than the factorized Cartan flow.

A.1 Closed-Form Formulation of Full 6-DOF Lorentz Dynamics

  • The unconstrained 6-DOF Lorentz transformation evaluates analytically without series truncation.It combines a spatial rotation with a hyperbolic boost.Source: §A.1 · p. 15
  • The rotation uses Rodrigues’ formula with axis k and angle 𝜗, while the boost acts along direction n with magnitude 𝜑.The parameter definitions are k = 𝝑/∥𝝑∥, n = 𝝋/∥𝝋∥, 𝜗 = ∥𝝑∥, and 𝜑 = ∥𝝋∥.Source: §A.1 · p. 15
  • Although mathematically exact and metric-preserving, the operator has higher latency than the factorized Cartan flow in deep cascades.Source: §A.1 · p. 15

A.2 Cayley–Hamilton Expansion for 𝔰𝔬(1, 2)

For Ω in 𝔰𝔬(1, 2), the Cayley–Hamilton relation reduces its exact exponential to a closed-form evaluation. The coefficient functions are defined piecewise by the sign of κ, with a stable evaluation of Bκ(t) to prevent cancellation near zero.

A.2 Cayley–Hamilton Expansion for 𝔰𝔬(1, 2)

  • For Ω ∈ 𝔰𝔬(1, 2), the Cayley–Hamilton theorem gives Ω^3 = κΩ, enabling exact exponential evaluation without infinite-series truncation.κ = b1^2 + b2^2 − ω^2 for spatial rotation ω and boosts b1, b2.Source: §A.2 · p. 15
  • Aκ(t) is sinh(t√κ)/√κ for κ > 0, t for κ = 0, and sin(t√−κ)/√−κ for κ < 0.Source: §A.2 · p. 15
  • Bκ(t) is evaluated stably to prevent numerical cancellation as κ approaches zero.The supplied expression distinguishes κ = 0 and κ < 0 cases.Source: §A.2 · p. 15

Related papers

Loading 2609.37817v1…