GFUM: Decoupling Features in SCI

• GFUM is a network design principle that decouples the physical data-fidelity channel from the neural feature space in spectral compressive imaging.
• It enhances model capacity and interpretability by separating physics-driven gradients from auxiliary feature channels in deep unfolding networks.
• GFUM achieves improved reconstruction quality, evidenced by a PSNR gain of ~1 dB, while reducing computational costs compared to conventional methods.

The Generalized Feature Unfolding Mechanism (GFUM) is a network design principle introduced in the context of low-rank deep unfolding networks (LRDUN) for spectral compressive imaging (SCI). GFUM decouples the physical dimensionality of the data-fidelity term from the neural feature dimensionality in proximal network priors, providing both interpretability and enhanced model capacity. This mechanism systematically addresses the redundancy, ill-posedness, and computational inefficiency found in conventional deep unfolding networks that operate directly on high-dimensional hyperspectral image (HSI) tensors from 2D measurements (Huang et al., 23 Nov 2025).

1. Conventional Deep Unfolding Limitations and GFUM Motivation

Conventional deep unfolding for SCI reconstructs the full 3D HSI tensor $\mathbf X \in \mathbb{R}^{H \times W \times B}$ from a 2D measurement $\mathbf y$, causing a significant dimensionality gap (recovering $HWB$ unknowns from $HW'$ measurements) and severe stage-wise ill-posedness. Each stage of such networks refines an entire high-dimensional HSI cube, resulting in high computational costs and memory usage due to full-cube convolutions.

GFUM is motivated by the need to break the rigid correspondence between the physical rank $r$ (number of latent basis vectors in the rank-$r$ subspace) and the network feature dimension $d$ in the prior model. By projecting rank-$r$ physical variables into an augmented $d$-dimensional feature space ($d>r$), GFUM preserves a low-dimensional physical data-fidelity channel while allowing auxiliary channels to support the learning of richer feature representations. This design enables the network to efficiently utilize both physics-driven constraints and learned high-level priors, without conflating their respective dimensionalities.

2. Mathematical Formulation

The foundational SCI reconstruction problem is typically cast as minimizing a data-fidelity loss: $$\min_{\mathbf X} \frac{1}{2} \|\mathbf y - \mathcal H(\mathbf X)\|_2^2,$$ solved via proximal gradient descent (PGD): $$\mathbf X^{k+1} = \operatorname{prox}_{\lambda_k \mathcal{R}}\big( \mathbf X^k - \eta_k \nabla f(\mathbf X^k) \big),$$ where $$f(\mathbf X) = \frac{1}{2} \|\mathbf y - \mathcal H(\mathbf X)\|_2^2$$ and $\mathcal R$ is a learned prior.

In LRDUN, the reconstruction problem is reformulated in terms of low-rank factors: $\mathbf E \in \mathbb{R}^{B \times r}$ (spectral basis) and $\mathbf A \in \mathbb{R}^{HW \times r}$ (spatial subspace). The alternating PGD updates are: $$\begin{aligned} \mathbf e^{i+1/2} &= \mathbf e^i - \rho_e\,\Phi_{A^i}^T\left(\Phi_{A^i}\,\mathbf e^i-\mathbf y\right), \quad \mathbf e^{i+1} = \operatorname{prox}_{\lambda_e\rho_e, \mathcal R_e}(\mathbf e^{i+1/2}), \ \mathbf a^{i+1/2} &= \mathbf a^i - \rho_a\,\Phi_{E^{i+1}}^T\left(\Phi_{E^{i+1}}\mathbf a^i-\mathbf y\right), \quad \mathbf a^{i+1} = \operatorname{prox}_{\lambda_a\rho_a, \mathcal R_a}(\mathbf a^{i+1/2}). \end{aligned}$$

GFUM augments $\mathbf e^i$ and $\mathbf a^i$ into feature vectors $\mathbf e^i_{\mathrm{feat}} \in \mathbb{R}^d$ and $\mathbf a^i_{\mathrm{feat}} \in \mathbb{R}^d$ respectively, partitioned as: $$\mathbf e^i_{\mathrm{feat}} = \big[\, \underbrace{\mathbf e^i_{\mathrm{feat}}[1:r]}_{\text{physical part}},\;\underbrace{\mathbf e^i_{\mathrm{feat}}[r+1:d]}_{\text{auxiliary}}\,\big]$$ with analogous partitioning for $\mathbf a^i_{\mathrm{feat}}$.

The GFUM update for the E-branch comprises four steps: (i) Slice out the physical subspace, (ii) Apply the physics-driven gradient, (iii) Carry forward the auxiliary part unchanged, (iv) Concatenate and process through a learned proximal prior network $\mathrm{ProxyNet}_E$.

The same logic applies to the A-branch with its dedicated prior module $\mathrm{ProxyNet}_A$.

3. Stepwise GFUM Update Procedure

The following pseudocode represents one unfolding stage with GFUM, detailing the separation of physical and auxiliary features and their propagation:

e_phys = E^i_feat[:, 1:r] # physical part e_aux = E^i_feat[:, (r+1):d] # auxiliary part e_half = e_phys - rho_e * (Φ_{A^i}^T (Φ_{A^i} e_phys - y)) e_aux_half = e_aux E^{i+1}_feat = ProxyNet_E(concat(e_half, e_aux_half)) a_phys = A^i_feat[:, 1:r] a_aux = A^i_feat[:, (r+1):d] a_half = a_phys - rho_a * (Φ_{E^{i+1}}^T (Φ_{E^{i+1}} a_phys - y)) a_aux_half = a_aux A^{i+1}_feat = ProxyNet_A(concat(a_half, a_aux_half))

This stepwise separation preserves interpretability by ensuring the data-fidelity gradient is applied only to the physical rank-$r$ subspace, while the auxiliary $(d-r)$-dimensional channel is propagated for enhanced capacity in the learned prior.

4. Enhanced Representational Benefits and Empirical Impact

GFUM increases the expressive capacity of the unfolding network by permitting $d > r$. This enables the ProxyNet modules to access and refine not only the physical rank-$r$ subspace (directly tied to the physics of the SCI problem) but also an auxiliary $(d-r)$-dimensional feature channel. This auxiliary space carries complementary information such as high-frequency textures, spectral mask cues, and proximal parameters.

Empirical studies demonstrate that, for fixed $r=11$ (matching the effective intrinsic HSI rank), increasing $d$ from $r$ to $16$ increases PSNR from $38.3$ dB to $39.4$ dB on the KAIST dataset, saturating for larger $d$. Disabling GFUM (i.e., enforcing $d=r$) reduces PSNR by $\sim1$ dB. Conversely, excessively large $d$ values increase computational cost (FLOPs) with diminishing returns in fidelity. This illustrates that GFUM achieves a tradeoff, permitting flexible design of networks that manifest both interpretability (via physical consistency) and capacity (via auxiliary space) (Huang et al., 23 Nov 2025).

5. Hyperparameters, Computational Efficiency, and Practical Guidance

Key hyperparameters under GFUM include the physical rank $r$ and the feature dimension $d$:

• Physical rank $r$: Should align with the intrinsic spectral subspace dimension (e.g., $r=11$ for typical HSI). Too small $r$ leads to underfitting, whereas too large $r$ reduces the share available for auxiliary features.
• Feature dimension $d$: Governs the tradeoff between prior capacity and computational cost; empirical results indicate $d=16$ for $r=11$ provides an optimal balance.

LRDUN configured with GFUM (9 unfolding stages) achieves state-of-the-art reconstruction—PSNR $\approx 41$ dB—for a computational cost of $\sim 30$ G FLOPs, compared to full-cube deep unfolding networks that exceed $70$ G FLOPs. Even a compact 3-stage variant reaches $39.4$ dB PSNR at only $\sim 10$ G FLOPs (Huang et al., 23 Nov 2025).

Parameter	Typical Value	Impact
Physical rank $r$	11	Spectral fidelity
Feature dim $d$	16	Capacity/cost trade
Unfolding stages	3–9	Performance/cost

A plausible implication is that similar decoupling strategies could benefit other inverse problems where physics-driven dimensionality is intrinsically lower than potential network prior capacities.

6. Broader Context and Interpretability

GFUM provides a principled way to decouple the structure imposed by the physical measurement process from the representational needs of deep neural network priors. This separation underwrites both the interpretability (the physical rank-$r$ factors retain explicit correspondence with the sensing model) and the practical efficiency (auxiliary features enrich priors without incurring unnecessary computational burden of full-cube convolutions).

In summary, GFUM constitutes a central advancement in the architecture of deep unfolding networks for spectral compressive imaging by disentangling physical modeling constraints from network prior flexibility, yielding both high reconstruction quality and computational efficiency (Huang et al., 23 Nov 2025).