Hyperbolic State Space Hallucination

• Hyperbolic State Space Hallucination is a framework for fine-grained domain generalization that uses state space hallucination and hyperbolic manifold consistency to achieve style-invariant representations.
• The SSH module quantitatively perturbs channel-wise statistics to simulate diverse style variations, while the HMC module employs hyperbolic geometry to enforce embedding consistency.
• Empirical evaluations demonstrate significant accuracy improvements over benchmarks, validating the effectiveness of multi-scale style enrichment in maintaining fine-grained separability.

Hyperbolic State Space Hallucination (HSSH) is a framework for fine-grained domain generalization (FGDG) designed to produce representations that are robust to unseen domain-specific style variations, particularly those that threaten the discernment of subtle category-defining patterns. HSSH advances the state-of-the-art by integrating two modules—State Space Hallucination (SSH) and Hyperbolic Manifold Consistency (HMC)—within a Vision Mamba backbone, achieving style-invariant fine-grained separability via style extrapolation and hyperbolic embedding consistency. This approach addresses the fragility of fine-grained recognition tasks (e.g., subtle plumage differences in bird species) under strong cross-domain style shifts such as illumination and color, achieving superior accuracy on multiple FGDG benchmarks (Bi et al., 10 Apr 2025).

1. Architectural Overview

HSSH builds on a four-stage Vision Mamba encoder, introducing SSH and HMC in each block to handle style perturbations and exploit hyperbolic geometry for discriminative embedding. At each encoder block $i$, the batch state tensor $$s^i = \mathbf{F}^i \in \mathbb{R}^{B\times C_i\times H_i\times W_i}$$ undergoes style hallucination via SSH to yield $\hat s^i = \widetilde{\mathbf{F}^i}$. Both $s^i$ and $\hat s^i$ are mapped to hyperbolic embeddings $z^i = \phi(s^i),\; \hat z^i = \phi(\hat s^i)$ via exponential map, and HMC enforces style invariance by minimizing their geodesic distance $\sum_{i=1}^4 d_\mathbb{H}(z^i,\hat z^i)$. The model’s classifier is shared between final-layer original and hallucinated states to enforce fine-grained categorization consistent across styles.

2. State Space Hallucination (SSH)

SSH operates blockwise, quantifying channel-wise "style" for each sample via mean and standard deviation over spatial locations:

$$\mu^i_{j,k} = \frac{1}{H_iW_i}\sum_{h,w}F^i_{j,k,h,w},\qquad \sigma^i_{j,k} = \sqrt{\frac{1}{H_iW_i}\sum_{h,w}(F^i_{j,k,h,w}-\mu^i_{j,k})^2}$$

for $j\in[1..B]$, $k\in[1..C_i]$. A least-squares fit of $\sigma = \gamma \mu + b$ extracts a slope $\gamma^i$; the range $[\min\gamma^i,\max\gamma^i]$ is extrapolated to $$[2\min\gamma^i-\max\gamma^i,\,2\max\gamma^i-\min\gamma^i]$$, from which a randomized $\widetilde{\gamma}^i$ is sampled. This defines a hallucinated style pair $(\widetilde{\mu}^i, \widetilde{\sigma}^i)$ along the extrapolated line. The feature tensor is re-normalized:

$$\hat F^i = \widetilde{\sigma}^i\frac{F^i-\mu^i}{\sigma^i}+\widetilde{\mu}^i$$

producing $\hat s^i$, the style-hallucinated state. Performing SSH at all four encoder stages enhances multi-scale style enrichment, as verified by empirical ablation.

3. Hyperbolic Manifold Consistency (HMC)

To ensure style perturbations via SSH do not alter fine-grained identity, HMC maps both $s^i$ and $\hat s^i$ into a Poincaré ball $$\mathcal{B}_c^n = \{x\in\mathbb{R}^n : c\|x\|^2 < 1\}$$ of constant negative curvature $-c$ using the exponential map at the origin:

$$\phi(x) = \exp_0^c(x) = \tanh\!\left(\sqrt{c}\|x\|\right)\frac{x}{\sqrt{c}\|x\|}$$

The hyperbolic distance is given by

$$d_{\mathbb{H}}(u,v) = \frac{2}{\sqrt{c}}\tanh^{-1}\left(\sqrt{c}\|-\!u\oplus_c v\|\right)$$

where Möbius addition is

$$u\oplus_c v = \frac{(1+2c\langle u,v\rangle +c\|v\|^2)u + (1-c\|u\|^2)v}{1+2c\langle u,v\rangle +c^2\|u\|^2\|v\|^2}$$

Minimizing the hyperbolic geodesic between $z^i$ and $\hat z^i$ encourages style-invariant fine-grained separation, leveraging the ball’s exponential distance scaling for semantic amplification.

4. Training Objectives and Optimization

At the final block ($i=4$), $s^4$ and $\hat s^4$ are input to a shared linear classifier $\phi_\mathrm{cls}$ of dimension $C_\mathrm{fine}$. Cross-entropy classification losses are incurred for both states:

$$\mathcal{L}_\mathrm{cls} = -\frac{1}{B}\sum_{j=1}^B \sum_{k=1}^{C_\mathrm{fine}} y_{j,k}\log[\phi_\mathrm{cls}(s^4)_{j,k}]$$

$$\widetilde{\mathcal{L}}_\mathrm{cls} = -\frac{1}{B}\sum_{j=1}^B \sum_{k=1}^{C_\mathrm{fine}} y_{j,k}\log[\phi_\mathrm{cls}(\hat s^4)_{j,k}]$$

HMC imposes the style-invariance constraint:

$$\mathcal{L}_\mathrm{HMC} = \sum_{i=1}^4 \sum_{j=1}^B d_\mathbb{H}(z^i_j, \hat z^i_j)$$

The overall loss is:

$$\mathcal{L} = \mathcal{L}_\mathrm{cls} + \widetilde{\mathcal{L}}_\mathrm{cls} + \lambda\mathcal{L}_\mathrm{HMC},\qquad \lambda=0.5$$

Optimization is performed using Adam for 100 epochs, following standard deep learning practices.

5. Role of Hyperbolic Geometry in Fine-grained Discrimination

Hyperbolic space encodes hierarchical and high-order statistical relations, enabling exponential expansion of distance near the ball boundary. This property allows the model to represent many closely related fine-grained classes without crowding, facilitating the discernment of minute semantic differences—such as subtle texture or shape cues—between categories. Projecting state embeddings to a negatively curved manifold amplifies these subtle cues, while suppressing linear style variations arising from SSH, thereby ensuring fine-grained separability even under substantial appearance shifts.

6. Empirical Evaluation in Fine-grained Domain Generalization

HSSH achieves state-of-the-art results on three FGDG benchmarks:

• CUB ↔ Paintings (birds): VMamba baseline: 63.47% avg.; FSDG (prior best): 55.10% avg.; HSSH: 66.03% avg. (↑2.56% over VMamba, ↑16.14% over FSDG)
• RS-FGDG (remote-sensing scenes): VMamba: 66.85% avg.; HSSH: 69.65% avg. (↑2.80%)
• Birds-31 (natural-image domains): VMamba: 88.24% avg.; FSDG: 82.31% avg.; HSSH: 90.69% avg. (↑2.45% over VMamba, ↑8.38% over FSDG)

Ablation studies on CUB–Paintings show SSH alone raises accuracy from 63.47% to 64.86%, with further gains to 66.03% when HMC is added. Hallucinating all four stages of the encoder produces greater performance than any subset, highlighting the importance of multi-scale style enrichment and hyperbolic alignment. These results support the effectiveness of HSSH in maintaining discriminative fine-grained recognition across substantial domain-induced style variation.

7. Contextual Significance and Implications

The introduction of Hyperbolic State Space Hallucination represents a shift in FGDG methodology, combining linear-time style extrapolation with hyperbolic manifold consistency to safeguard fine-grained feature separability under diverse unseen conditions. This design effectively mitigates the collapse of subtle semantic cues under cross-domain style shifts, as evidenced by substantial improvements over prior methods. A plausible implication is that negative curvature embedding may find broader applicability in other problem domains requiring robust hierarchical or fine-grained structure under distributional drift (Bi et al., 10 Apr 2025).