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Implicit Representations via Operator Learning
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Proposes Operator INR (O-INR), which reformulates implicit neural representations as integral operators mapping positional-encoding function spaces to signal spaces, parameterized by convolutions instead of MLPs.
Problem
Implicit neural representations (INRs) fit each signal in isolation as an MLP mapping coordinates to values, making downstream processing (derivatives, interpolation, classification in weight space) difficult and often requiring large MLPs.
Approach
Treat INR fitting as learning an integral operator between a function space of sinusoidal positional encodings and a signal function space; parameterize the kernel as a convolution (optionally continuous convolution) with sinusoidal nonlinearities, which removes the need for MLPs and yields closed-form derivatives via convolution properties.
Key results
- On 2D image representation, O-INR reaches 34.04 dB PSNR on Train and 32.91 dB on Rocket, comparable to WIRE and iNGP, outperforming SIREN, MFN and NFFB on several images.
- On MNIST and Fashion-MNIST weight-space classification, O-INR achieves 96.57% and 80.18% vs DWSNets 85.71% and 67.06%.
- On image denoising, O-INR achieves 24.28 dB on Astronaut, comparable to WIRE (24.48) and above SIREN (20.94).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use O-INR rather than the strongest reported baseline | 2D image representation · Fern, Coffee, Walnut, Rocket, and Train | PSNR: 34.04 vs 34.13; 36.4 vs 37.16; 32.04 vs 37.99; 32.12 vs 34.11; 32.91 vs 39.55 | Mixed |
| Use O-INR rather than the strongest reported baseline | Image denoising · Astronaut, Cat, Kodak05, Kodak19, and Rocket | PSNR: 24.28 vs 25.22; 25.1 vs 27.4; 21.9 vs 23.6; 22.7 vs 24.6; 26.11 vs 26.27 | Mixed |
| Use O-INR weights instead of DWSNets weights for nearest-neighbor classification | MNIST and Fashion-MNIST · MNIST and Fashion-MNIST | test accuracy: 85.71 -> 96.57; 67.06 -> 80.18 | Helped |
| Use continuous rather than discrete convolution (ablation) | O-INR · Coffee-mug image representation | PSNR: 32.04 -> 31.18 | Hurt |
| Use continuous rather than discrete convolution (ablation) | O-INR · Astronaut image denoising | PSNR: 24.48 -> 23.5 | Hurt |
| Train O-INR with 82% of slices missing (ablation) | O-INR brain-volume reconstruction · 3D brain images | test MSE: 7.66e-4 -> 1.08e-3 | Hurt |
| Use O-INR rather than SIREN for surface reconstruction | 3D surface reconstruction · Surface reconstruction | IoU: +0.0334 | Helped |
Surprising findings
- Interpolating convolutional weights between independently trained O-INRs yields coherent images even without weight matching, whereas interpolating SIREN layers does not.
Limitations
- Convolutional parameterization in 3D and higher dimensions can be computationally expensive.
- Performance on arbitrary non-grid inputs (relevant to 3D applications) is not maintained.
- Does not consistently outperform baselines; iNGP is better on 2D image PSNR.
Experimental setup
Datasets: Kodak scikit-image images Stanford 3D Scanning Repository UCF-101 TGIF ImageNet-1K ADNI CelebA MNIST Fashion-MNIST
Benchmarks: Kodak UCF-101 TGIF ImageNet-1K ADNI MNIST Fashion-MNIST
Metrics: PSNR IoU MSE training time classification accuracy
Baselines: SIREN WIRE MFN iNGP NFFB DWSNets
Builds on: SIREN WIRE Multiplicative filter networks Continuous convolutions (Romero et al.) Functa
Closest papers by meaning
- Grids Often Outperform Implicit Neural Representation at Compressing Dense Signals (NeurIPS 2025 · similarity 0.90)
- VI^3NR: Variance Informed Initialization for Implicit Neural Representations (CVPR 2025 · similarity 0.90)
- Learning Transferable Features for Implicit Neural Representations (NeurIPS 2024 · similarity 0.89)
- MIRE: Matched Implicit Neural Representations (CVPR 2025 · similarity 0.89)
- Implicit Neural Representations and the Algebra of Complex Wavelets (ICLR 2024 · similarity 0.89)
- SINR: Sparsity Driven Compressed Implicit Neural Representations (CVPR 2025 · similarity 0.89)
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