TT-net: Quantum Inspired Tensor Network Denoising in Conditional GANs
Authors: Michal A. Sterzel, Marko J. Rančić
Organizations: University of Luxembourg, FSTM, Department of Computer Science Campus Belval, 6 Av. de la Fonte, L-4364 Esch-Belval, Esch-sur-Alzette, Luxembourg
Developed as a workhorse for classical simulations of quantum algorithms and quantum many-body systems, Tensor Network methods have entered the scientific mainstream in quantum physics. Among various types of tensor networks, Tensor Trains (commonly know as Matrix Product States in the quantum computing community) have already found applications in machine learning. These methods often rely on a powerful linear algebra tool called the Singular Value Decomposition (SVD). Several conditional GAN architectures for image denoising incorporate SVD as a single-cut decomposition step applied to generator feature maps. In this work we introduce TT-Net, which replaces the per-channel SVD denoising block with a two-cut tensor-train decomposition capable of accessing cross-channel information directly, a capability absent from contemporary alternatives. In a controlled comparison differing only in this decomposition mechanism, TT-Net outperforms SVD-Net on PSNR and SSIM across all three noise types tested (Gaussian, motion blur, and salt-and-pepper), supporting the hypothesis that cross-channel access improves denoising quality. Training-dynamics analysis further shows that TT-Net's adversarial loss term consistently saturates to a stagnant state across all three noise types, more so than SVD-Net's, while reconstruction quality continues to improve regardless, raising an open question about the adversarial component's contribution that this work identifies but does not resolve. Furthermore, for Gaussian noise our method outperforms both the EigenGAN and the state of the art Pix2pix method which does not assume any linear algebra decompositions and does not retain any linear algebra information. Our manuscript shows how quantum inspired tools can be used as practical real world feature filters for deep learning applications.
Figures & tables
Insertion point
Channels
Height
Width
After 1st encoder stage
64
32
32
After 2nd encoder stage
128
16
16
After 3rd encoder stage
256
8
8
Table 1: Feature map shape at each TT-Net / SVD-Net insertion point.
Model
lr (G, D)
β1
β2
Optimizer
Pix2pix
2×10−4
0.5
0.999
Adam
EigenGAN
2×10−3
0.5
0.999
Adam
SVD-Net
1×10−4
0.9
0.999
Adam
TT-Net
1×10−4
0.9
0.999
Adam
Table 2: Optimized hyperparameters used for each model, with a uniform batch size of 32 applied across all four.
Gaussian
Motion Blur
Salt & Pepper
Model
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
Pix2pix
25.52
0.7466
36.18
0.9643
41.79
0.9881
EigenGAN
25.18
0.7545
26.71
0.8398
34.24
0.9513
SVD-Net
23.60
0.7455
18.02 †
0.4260 †
28.98
0.8761
TT-Net
26.28
0.7968
26.34
0.8058
29.97
0.8926
Table 3: Test-set PSNR and SSIM (standard windowed formulation) across noise types, using each model’s optimized hyperparameters (Section 4.1 ) with a uniform batch size of 32. Bold indicates the best value across all four models. TT-Net achieves the best PSNR and SSIM of any model on Gaussian noise. † SVD-Net’s motion-blur result reflects a late-training discriminator collapse rather than a converged model (Section 5.3 ).
Figure 1: Representative test-set outputs across all four models and three noise types. Ground truth is shown once per row since it is identical across models. Each model’s own noisy input and output are shown alongside it.
Figure 3: Total generator loss ( LG=LGadv+Lrecon , Section 4.2 ) over training, SVD-Net (top) vs. TT-Net (bottom), by noise type, linear scale.
Figure 4: Generator loss components ( Gadv , L2, MS-SSIM, recon) over training, SVD-Net (top) vs. TT-Net (bottom), by noise type. See Section 4.2 for the loss definitions, Lrecon=δ⋅LMS-SSIM+(1−δ)⋅LL2,δ=0.84 .
Figure 5: Validation PSNR over training, SVD-Net vs. TT-Net, by noise type.
Figure 6: Validation SSIM (windowed) over training, SVD-Net vs. TT-Net, by noise type.
Figure 7: MS-SSIM loss component over training, SVD-Net vs. TT-Net, by noise type.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 2: Discriminator output probabilities D(real) / D(fake) over training, SVD-Net vs. TT-Net, by noise type.
We introduce the tubal tensor train (TTT) decomposition, a tensor-network model that combines the t-product algebra of the tensor singular value decomposition (T-SVD) with the low-order core structure of the tensor train (TT) format. For an order-(N+1) tensor with a distinguished tube mode, the proposed representation consists of two third-order boundary cores and N−2 fourth-order interior cores linked through the t-product. As a result, for bounded tubal ranks, the storage scales linearly with the number of modes, in contrast to direct high-order extensions of T-SVD. We present two computational strategies: a sequential fixed-rank construction, called TTT-SVD, and a Fourier-slice alternating scheme based on the alternating two-cores update (ATCU). We also state a TT-SVD-type error bound for TTT-SVD and illustrate the practical performance of the proposed model on image compression, video compression, tensor completion, and hyperspectral imaging.
Lab of Machine Learning and Knowledge Representation, Innopolis University, Innopolis, Russia · Innopolis University, Innopolis, Russia · Skolkovo Institute of Science and Technology, Center for Artificial Intelligence Technology, Moscow, Russia +1
This work introduces quantum-inspired tensor-network circuits as trainable transforms for image inpainting. Among the proposed architectures, the diagonal quantum Fourier transform (QFT) relaxation is invertible with O(N2logN) computational cost for N×N images, inherently preserving minimum coherence throughout training via its circuit structure and eliminating the need for explicit coherence penalties. Unconstrained gradient-based phase optimization (Riemannian-optimization free) enables efficient learning from randomly sampled training data, allowing the learned transform to generalize to test images observed through fixed sampling masks. Numerical tests show that the learned models outperform fixed transforms and per-image optimization while matching the performance of much larger unitary architectures, yet with far fewer parameters.
Shiwen An, Konstantinos Slavakis
Institute of Science Tokyo, Department of Information and Communications Engineering, Yokohama, Japan
Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.
Gustav J L Jäger, Martin B Plenio, Hans-Martin Rieser
Deutsches Zentrum für Luft- und Raumfahrt e.V. – Institut für KI-Sicherheit Wilhelm-Runge-Straße 10, 89081 Ulm, Germany · Universität Ulm – Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, 89081 Ulm, Germany