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Quantum-Inspired Trainable and Parameter-Efficient Tensor Networks for Image Inpainting

Shiwen An, Konstantinos Slavakis

arXiv:2609.17298v1eess.IVcs.CVcs.LG

TL;DR

Image inpainting must reconstruct missing pixels, and transform-based recovery depends on both sparsity and coherence. This work trains quantum-inspired tensor-network transforms, including a diagonal QFT relaxation that preserves minimum coherence by construction. The learned models outperform fixed transforms and per-image optimization while matching much larger unitary architectures with far fewer parameters.

  • Problem

    Transform-based inpainting performance depends on coherence as well as sparsity, while the task itself requires reconstructing images from partial pixel observations.

    Source: §1 · p. 1, §2 · p. 1

  • Method

    The paper trains quantum-inspired tensor-network transforms for inpainting using gradient-based phase optimization, with the diagonal QFT relaxation preserving minimum coherence through its circuit structure.

    Source: §1 · p. 1

  • Results

    The learned diagonal model outperforms fixed transforms and per-image optimization schemes while matching much larger unitary architectures with far fewer parameters.

    Source: §1 · p. 1

  • Takeaways & Limitations

    The diagonal QFT model achieves minimum coherence throughout training without explicit penalties, and the conclusion reports efficient training with plain Adam and competitive performance using far fewer parameters.

    Source: §1 · p. 1, §5 · p. 4

Abstract

from arXiv · show

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 $O(N^2 \log N)$ computational cost for $N\times N$ 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.

1. INTRODUCTION

Image inpainting has several established approaches, but transform-based recovery depends on more than sparsity: coherence also affects recovery. The paper introduces trainable quantum-inspired tensor-network transforms whose circuit structure preserves minimum coherence, and reports strong performance with far fewer parameters than larger learned architectures.

1. INTRODUCTION

  • Transform-based recovery depends on coherence as well as sparsity, motivating trainable low-coherence transforms for inpainting.The introduction notes that wavelets can yield sparser approximations than the DFT yet perform worse under the same solver.Source: §1 · p. 1
  • The proposed quantum-inspired tensor-network transforms adapt QFT circuits by relaxing diagonal gates while retaining one Hadamard per bit.The basis retains FFT-like fast application at O(N^2 log N) for N × N images and invertibility via its adjoint.Source: §1 · p. 1
  • Gradient-based phase optimization applies the tensor-network circuits to image inpainting without Riemannian optimization.The work extends QFT-gate relaxations previously trained via Riemannian optimization for compression [21].Source: §1 · p. 1
  • The diagonal QFT relaxation and its phase-only subfamily guarantee minimum coherence throughout training without an explicit coherence penalty.The guarantee follows from circuit topology and distinguishes these models from unconstrained learned dictionaries and QFT circuits with free Hadamards.Source: §1 · p. 1
  • The learned diagonal model outperforms fixed transforms and per-image optimization schemes while matching much larger unitary architectures with far fewer parameters.The comparison is reported for Sec. 4.Source: §1 · p. 1

2. THE IMAGE-INPAINTING PROBLEM

The section defines inpainting as recovering an image from partial pixel observations and sets up learned transform-based recovery. Its training loss averages over random sampling masks, while the tensor-network parameters factorize across image rows and columns.

2. THE IMAGE-INPAINTING PROBLEM

  • An image is observed through a binary mask, and inpainting recovers the full image from the observed entries.The mask is modeled with independent Bernoulli entries of success probability p, so pN^2 pixels are observed on average.Source: §2 · p. 1
  • The transform parameters are learned from training data by minimizing a recovery loss averaged over random sampling masks.The expectation over masks is part of the loss, and the recovered image is obtained after K recovery steps.Source: §2 · p. 1
  • Fig. 1 traces the construction from a separable 2D DFT through a radix-2 level and trainable circuit to the analysis map.The circuit diagram marks trainable diagonals in red and fixed Hadamards in blue.Source: Figure 1 · p. 2
  • The tensor-network parameters split into row and column components to respect the image’s two-dimensional structure.The section defines 2D coherence as the product of coherences along the two dimensions.Source: §2 · p. 2

3. THE TENSOR NETWORKS

The section builds an FFT-factorized unitary transform, relaxes its gates into trainable models, and trains them through an unrolled recovery map. It motivates QFT (diagonals) through isometry and minimum coherence.

3.1. An FFT-factorized isometric family

  • The FFT circuit recursively replaces dense Fourier blocks with Hadamard and controlled-phase layers; each parameterized gate remains unitary, so its adjoint is the inverse.The construction starts from the DFT and frees controlled-phase angles to form the trainable network.Source: §3.1 · p. 2

3.2. A ladder of relaxations

  • The relaxation ladder frees pair phases and then Hadamards; tying diagonal phases by gate distance yields 4(n −1) parameters per axis.Adjacent model comparisons isolate one additional freedom at a time.Source: §3.2 · pp. 2–3

3.3. Recovery as an unrolled map

  • Recovery alternates transform-domain hard thresholding with consistency on observed pixels, and differentiates the fixed-iteration computation graph to train gate angles.The output matches observed entries but is not constrained to remain k-sparse.Source: §3.3 · p. 3

3.4. Isometry without manifold optimization

  • Adam updates on periodic diagonal phases preserve gate unitarity without Riemannian projections; applying the diagonal model or its adjoint costs O(N^2 log N) per image.The implementation combines the Hadamards and commuting diagonal gates without forming dense T.Source: §3.4 · p. 3

3.5. Coherence is pinned

  • QFT (diagonals) and its phase-only subfamily retain minimum coherence, µ = 1, throughout training by circuit structure, without an explicit penalty.The guarantee does not extend to QFT (rotations), whose free Hadamards can allow coherence above the minimum bound.Source: §3.5 · p. 3

4. NUMERICAL TESTS AND DISCUSSION

Across the numerical tests, the diagonal QFT model outperforms fixed transforms and per-image methods while using far fewer parameters than larger learned architectures. The experiments also compare coherence, sparsity, and the performance trade-offs of freeing QFT circuit components.

4.1. Setup

  • Tests use 750 DIV2K crops for training, 50 for validation, and 100 for testing; each test image has one randomly drawn mask fixed across methods and budgets.Methods are evaluated at per-image budget optima over six sparsity levels; transforms run for K = 300 iterations, while QFT models train for K = 100 with T = 200 steps.Source: §4.1 · p. 3

4.2. Comparison with fixed transforms and per-image methods

  • The tensor train trails QFT (diagonals) by 0.60dB in PSNR but leads by 0.005 in MS-SSIM; the constrained butterfly leads by 0.17dB at 64× the parameters.The unconstrained butterfly loses 8dB under a single-budget protocol, while transform learning achieves 0.1dB below the DCT.Source: §4.2 · pp. 3–4
  • QFT (diagonals) beats the DFT by 2.02dB and the best fixed transform by 1.59dB in PSNR, leading both PSNR and MS-SSIM on all 100 test images.Table 1 covers fixed bases, per-image fits, trained transforms, and proposed QFT networks on 100 DIV2K test images at 512×512 and p = 10%.Source: Table 1 · p. 3, §4.2 · p. 3
  • Across sampling rates, QFT (diagonals) leads the DFT on both metrics; at 512² it uses 288 trainable reals and costs two-fifths of the tensor train per step.Transforms trained at p = 10% are applied unchanged; QFT (diagonals) scales as N^1.7, and training costs 741s, equivalent to approximately 570 tensor-train fits.Source: §4.2 · p. 4

4.3. The coherence-sparsity trade-off

  • Wavelets’ superior sparsity does not prevent worse inpainting PSNR than the less-sparse, lower-coherence DFT; soft thresholding improves all bases without reversing the family ordering.The comparison uses fixed-k hard-thresholding; at 60% sampling, soft thresholding narrows Symlet-8’s high-rate deficit from 3.0 to 2.3dB.Source: §4.3 · p. 4

4.4. Freeing the QFT circuit: performance vs. structure

  • Freeing diagonal gates adds 0.11dB over phase-only updates, while freeing Hadamards adds 0.24dB and 0.013 MS-SSIM but raises coherence and requires retractions.At the largest attempted Cayley step, training diverges (µ = 697, 12.5dB); the diagonal subfamily keeps µ = 1 with only 0.24dB loss and plain Adam.Source: §4.4 · p. 4

5. CONCLUSION

The conclusion presents quantum-inspired tensor-network circuits as trainable transforms for image inpainting. The diagonal QFT model obtains minimum coherence through circuit topology and achieves competitive performance with far fewer parameters than alternative learned transforms.

  • The work introduces quantum-inspired tensor-network circuits as trainable transforms for image inpainting.Source: §5 · p. 4
  • The diagonal QFT model guarantees minimum coherence at every parameter value through circuit topology, without explicit penalties.Source: §5 · p. 4
  • The model trains with plain Adam and achieves competitive performance with far fewer parameters than alternative learned transforms.Source: §5 · p. 4

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