# Ananke: Contractive Torus Attractor Networks

arXiv: https://arxiv.org/abs/2609.24737 · HTML: https://mlsanity.com/paper/2609.24737/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 25 references that are in the map.
- First posted on arXiv 21 Sep 2026.

## Builds on (most cited first, 18 of 18)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [ShuffleNet V2: Practical Guidelines for Efficient CNN Architecture Design](https://mlsanity.com/paper/1807.11164/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [EfficientViT: Memory Efficient Vision Transformer with Cascaded Group Attention](https://mlsanity.com/paper/2305.07027/) (2023)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation](https://mlsanity.com/paper/1701.04247/) (2017)
- [CliffordNet: All You Need is Geometric Algebra](https://mlsanity.com/paper/2601.06793/) (2026)

## Built on by (most cited first, 2 of 2)

- [Minkowski Attractor Networks: Closed-Form Hyperbolic Flows for Visual Representations](https://mlsanity.com/paper/2609.37817/) (2026)
- [HySPE: Positional Encoding via Symplectic Dual Shears](https://mlsanity.com/paper/2610.10154/) (2026)

## Abstract (arXiv)

We introduce Ananke, a representation-learning framework that scaffolds latent representations onto a structured product-torus prior, and its flagship visual backbone realization, Contractive Torus Attractor Networks (CTAN). By factorizing high-dimensional latent spaces into an orthogonal direct sum of two-dimensional phase planes (bigoplusk=1K R2), Ananke coordinates feature updates via a decoupled dual-phase continuous flow: skew-symmetric Hamiltonian transport moves features tangentially along energy level sets to preserve semantic phase invariants, while signed gradient dissipation contracts transverse perturbations normally toward target invariant manifolds. For circular potential families with frozen parameters, logarithmic radial feedback yields the Exact Log-Symplectic Flow (ELSF), an analytical closed-form mapping with exact exponential decay of log-radius error that evaluates in a single forward pass without numerical integration. We establish local input-to-state bounds for level-set deviations and log-radius errors, and characterize the normal hyperbolicity and persistence of the ideal product torus under bounded perturbations. We further formulate the architecture through Lie--Trotter operator splitting, unifying spatial depthwise diffusion with local manifold contraction, and analyze both exact trigonometric flows and hardware-friendly symplectic dual-shear variants. Across natural image benchmarks (CIFAR-100) and clinically challenging endoscopy datasets (Kvasir-v2), CTAN demonstrates exceptional parameter efficiency: an ultra-compact hierarchical model with merely 0.27M parameters achieves 90.52% accuracy on Kvasir-v2, outperforming 25M+ baselines (ResNet-50, DenseNet-161) by nearly two orders of magnitude in capacity, while scaled variants attain 80.32% top-1 accuracy on CIFAR-100.