TT-net: Quantum Inspired Tensor Network Denoising in Conditional GANs

Technical Summary: TT-net: Quantum Inspired Tensor Network Denoising in Conditional GANs

Problem Statement
Image denoising, the task of recovering clean images from degraded versions, remains a critical challenge in fields ranging from medical imaging to consumer photography. While Generative Adversarial Networks (GANs) have become a standard for this task due to their ability to preserve fine details, specific architectural choices regarding feature map processing vary. A notable approach, exemplified by SVD-Net, utilizes Singular Value Decomposition (SVD) as a per-channel filtering mechanism within the generator. However, SVD applied as a single-cut decomposition treats each channel of a feature map (a 3D tensor of channels, height, and width) independently. By unfolding a feature map into a single matrix per channel, this method inherently prevents the model from accessing or exploiting relationships between different channels, potentially limiting the denoising capability.

Methodology
The authors introduce TT-Net, a conditional GAN architecture that replaces the per-channel SVD block of SVD-Net with a two-cut Tensor Train (TT) decomposition, a method inspired by Matrix Product States (MPS) from quantum physics.

• Architecture: TT-Net inserts the denoising block at the same three locations in the generator as SVD-Net (after each encoder downsampling stage).
• Decomposition Mechanism:
  1. First Cut: The input feature map $X \in \mathbb{R}^{C \times H \times W}$ is reshaped into $M_1 \in \mathbb{R}^{C \times (HW)}$. SVD is applied to this matrix, where rows represent channels. This allows the decomposition to identify dominant patterns of agreement across channels, a capability absent in SVD-Net.
  2. Second Cut: The retained components are reshaped to merge the channel axis with the height axis, creating $M_2 \in \mathbb{R}^{(\chi_1 H) \times W}$. A second SVD is applied to capture spatial regularities (e.g., edges, textures) shared across the retained channel directions and height positions.
  3. Reconstruction: The filtered feature map is reconstructed by contracting the three resulting cores.
• Adaptive Thresholding: To ensure a fair comparison, TT-Net employs an adaptive per-cut thresholding scheme. Since the two cuts operate multiplicatively on the retained energy, the thresholds are set to $\theta_1 = \theta_2 = \sqrt{0.9} \approx 0.9487$ to match SVD-Net's target of retaining 90% of the energy ($\theta = 0.9$).
• Experimental Control: The study isolates the decomposition mechanism as the sole variable. TT-Net and SVD-Net share identical generator/discriminator structures, loss functions (combining MSE, MS-SSIM, and adversarial loss), optimizers, and hyperparameters. They are evaluated against each other, as well as against EigenGAN and a non-decomposition Pix2pix baseline, on the CIFAR-10 dataset with three noise types: Gaussian, motion blur, and salt-and-pepper.

Key Contributions

1. Cross-Channel Access: The introduction of a tensor-train denoising block that allows the decomposition to directly access and utilize cross-channel structural information, unlike the isolated per-channel processing of SVD-Net.
2. Fair Comparison Framework: The development of an adaptive thresholding scheme that equalizes the "retained energy" target between the single-cut SVD and the two-cut TT decomposition, enabling a rigorous, like-for-like ablation study.
3. Empirical Validation: A controlled experimental comparison demonstrating that the multi-cut tensor decomposition outperforms the single-cut SVD approach across all tested noise conditions.

Results

• Quantitative Performance: TT-Net consistently outperformed SVD-Net across all three noise types (Gaussian, motion blur, salt-and-pepper) in both Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM).
  • Gaussian Noise: TT-Net achieved the best results among all four models tested (including Pix2pix and EigenGAN), surpassing SVD-Net by 2.68 dB PSNR and 0.0513 SSIM.
  • Motion Blur: TT-Net showed a significant improvement (8.32 dB PSNR) over SVD-Net. However, the authors note this gap is partially confounded by a discriminator collapse in the SVD-Net training run, which did not occur in TT-Net.
  • Salt-and-Pepper: TT-Net outperformed SVD-Net by 0.99 dB PSNR and 0.0165 SSIM.
• Training Dynamics: Analysis revealed that TT-Net's adversarial loss term ($G_{adv}$) consistently saturates to a stagnant state early in training across all noise types, whereas SVD-Net's adversarial loss continues to fluctuate. Despite this saturation, TT-Net's reconstruction quality (driven by $L_2$ and MS-SSIM losses) continues to improve throughout training. This suggests that once the discriminator reaches a state where it cannot distinguish real from generated images (outputting ~0.5), further improvements are driven by the reconstruction loss and the denoising mechanism itself, rather than adversarial feedback.

Significance and Claims
The paper claims that the primary significance of TT-Net lies in demonstrating that quantum-inspired tools (specifically Tensor Trains/MPS) can serve as practical, real-world feature filters for deep learning applications.

The authors support the hypothesis that allowing a denoising mechanism to compare channels against one another, rather than treating them in isolation, leads to measurable improvements in reconstruction quality. The work identifies a specific open question regarding the contribution of the adversarial component in TT-Net—specifically, whether the model's performance relies on the adversarial loss at all once the reconstruction loss is optimized—but explicitly states that this work identifies the phenomenon without resolving it. The authors conclude that while the motion-blur results are confounded by training instability in the baseline, the consistent gains on Gaussian and salt-and-pepper noise provide direct evidence of the efficacy of the cross-channel decomposition mechanism.