Ananke: Contractive Torus Attractor Networks

Zhongping Ji

cs.CVSep 21, 2026

Ananke: Contractive Torus Attractor Networks

Authors: Zhongping Ji

Abstract

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 ( $\bigoplus_{k=1}^K \R^2$ ), 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.

Figures & tables

Figure 1: Geometric state decomposition and invariant toroidal attractor dynamics in 4D latent space. (a, b) A latent feature vector is orthogonally factored into independent 2D phase planes (

x=x_{1}\oplus x_{2}\in\mathbb{R}^{2}\oplus\mathbb{R}^{2}

). In each plane, signed gradient dissipation contracts perturbed states (

p_{i},q_{i}

) normally toward an invariant target circle (

\mathcal{C}_{1},\mathcal{C}_{2}

), while skew-symmetric Hamiltonian transport sustains continuous phase rotation. (c) The Cartesian product

\mathcal{C}_{1}\times\mathcal{C}_{2}

defines an invariant 2-torus

\mathbb{T}^{2}\subset\mathbb{R}^{4}

(depicted via its canonical embedding in

\mathbb{R}^{3}

). Composite feature trajectories

z_{i}=(p_{i},q_{i})

are asymptotically contracted from ambient space onto the compact toroidal surface, along which persistent phase orbits encode semantic invariants without representation collapse.

Figure 2: Topological transitions and multi-attractor dynamics of the Sum-of-Squares (SOS) algebraic potential. By modulating the Gram matrix and energy offset ( $R^{2}=1.0,0.60,0.50,0.50$ across panels (a)–(d)), the zero-level set $\mathcal{M}=\{x:H_{\rm SOS}(x)=0\}$ (black contours) undergoes Morse-type topological bifurcations, transitioning through configurations indexed by the 0-th Betti number $b_{0}(\mathcal{M})\in\{1,2,3,4\}$ . Feature trajectories (colored curves) originating from diverse initial states are contracted via transverse gradient dissipation onto locally attracting closed components and transported along tangential Hamiltonian orbits, enabling multimodal clustering within a single 2D phase subspace.

Figure 3: The three-stage compact CTAN hierarchy: a stride-two stem, stages of depths $(3,4,5)$ and channel widths $(32,64,96)$ , strided transitions, and a pooled classification head. Downsampling changes the state space and does not represent a diffeomorphism between the associated tori.

Architecture	Parameters	Top-1 accuracy
MobileNetV2 [ 19 ]	2.3M	70.90%
ShuffleNetV2 1.5 $\times$ [ 20 ]	2.6M	75.95%
ResNet-18 [ 1 ]	11.2M	76.75%
ResNet-50 [ 1 ]	23.7M	79.14%
CliffordNet-1 [ 8 ]	1.4M	77.82%
CliffordNet-2 [ 8 ]	2.6M	79.05%

Table 1: Top-1 classification accuracy on CIFAR-100 at

32\times 32

resolution without external pretraining. Best overall result is highlighted in bold; best sub-1M lightweight result is underlined.

Architecture	Parameters	Accuracy	MCC
ResNet-50 [ 1 ]	23.5M	88.50%	0.8689
DenseNet-161 [ 23 ]	26.5M	89.08%	0.8756
ViT-Small [ 2 ]	21.7M	75.50%	0.7209
ConvNeXt-Tiny [ 3 ]	27.8M	77.83%	0.7471
EfficientViT [ 24 ]	6.58M	82.83%	0.8041
CAN-Nano [ 8 ]	0.36M	90.33%	0.8902

Table 2: Diagnostic classification performance on Kvasir-v2 at

224\times 224

resolution trained from scratch. MCC denotes Matthews correlation coefficient.

Variant	Block Topology	Rotation Mode	Parameters	Accuracy
CTAN-Pure-1	Pure Flow Composition	Shared $\alpha_{0}$	0.47M	75.49%
CTAN-Pure-2	Pure Flow Composition	Dynamic $\alpha(\xi)$	0.81M	76.83%
CTAN-Res-1	Residual-Augmented	Shared $\alpha_{0}$	0.91M	78.41%

Table 3: CIFAR-100 block-topology comparison under fixed stage depths

(3,4,5)

and dimensions

(96,128,160)

ID	Update Mode	Radial Dissipation	Tangential Phase	Phase Velocity	Accuracy
A	No evolution ( $t=0$ )	None	None	—	73.39%
B	Hard radial projection ( $t\to\infty$ )	Hard	None	—	74.85%
C	Tangential only	None	Trigonometric	Shared $\alpha_{0}$	78.35%
D	Radial only	Soft log-decay	None	—	78.75%
E	Tangential only	None	Trigonometric	Dynamic $\alpha(\xi)$	79.35%
F	Full ELSF (Coupled)	Soft log-decay	Trigonometric	Dynamic $\alpha(\xi)$	79.92%

Table 4: Mechanics ablation on CTAN-Hier-1 on CIFAR-100. Isolating the contributions of normal logarithmic dissipation, tangential phase transport, and their coupled dynamics.

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