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Learning High-Frequency Functions Made Easy with Sinusoidal Positional Encoding
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Proposes sinusoidal positional encoding (SPE), which places a trainable sinusoidal activation after a fixed Fourier positional encoding so the frequencies become adaptive, and shows gains in few-view NeRF, FastSpeech TTS, and NTK 1D regression.
Problem
Fourier-feature positional encodings require manual, task-specific tuning of the frequency hyperparameters and struggle to learn high-frequency functions, especially with limited data.
Approach
Keep the fixed PE as an initial basis and apply a trainable sinusoidal activation sin(omega * PE(x)) as the first layer, so frequencies are learned adaptively within a bounded search space; it is a drop-in replacement for PE requiring a single-line change.
Key results
- On few-view NeRF (Blender, 8 views), FreeNeRF+SPE+GAN reaches PSNR 25.202 and SSIM 0.910 versus 24.259 and 0.883 for FreeNeRF.
- SPE gives PSNR gains over FastSpeech and PE in all five speech cases, e.g., Case 4 improves about 4.6% over vanilla FastSpeech.
- In NTK 1D regression, SPE converges about 2x faster than PE within 50,000 iterations and reduces training loss by two orders of magnitude at p=0.5.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace FreeNeRF's PE with SPE | FreeNeRF · Few-view NeRF (Blender) | PSNR: 24.259 -> 24.951 | Helped |
| Add SPE to FreeNeRF | FreeNeRF · Few-view NeRF (Blender) | SSIM: 0.883 -> 0.898 | Helped |
| Add GAN training to FreeNeRF with PE (ablation) | FreeNeRF · Few-view NeRF (Blender) | PSNR: 24.259 -> 24.531 | Helped |
| Add GAN training to FreeNeRF with SPE (ablation) | FreeNeRF + SPE · Few-view NeRF (Blender) | PSNR: 24.951 -> 25.202 | Helped |
| Add GAN training to FreeNeRF with SPE (ablation) | FreeNeRF + SPE · Few-view NeRF (Blender) | SSIM: 0.898 -> 0.910 | Helped |
| Replace PE with GRFF (σ=1995) | FreeNeRF · Few-view NeRF (Blender) | PSNR: 24.259 -> 24.382 | Helped |
| Replace sinusoidal activation with sawtooth activation (ablation) | FreeNeRF + SPE · Few-view NeRF (Blender) | PSNR: 24.951 -> 23.975 | Hurt |
| Replace sinusoidal activation with PReLU (ablation) | FreeNeRF + SPE · Few-view NeRF (Blender) | PSNR: 24.951 -> 24.812 | Hurt |
| Remove PE from SPE (ablation) | FreeNeRF + SPE · Few-view NeRF (Blender) | PSNR: 24.951 -> 15.183 | Hurt |
| Replace FastSpeech's PE with SPE | FastSpeech · Speech synthesis, Cases 1–5 | PSNR: 103.4 -> 104.4; 96.3 -> 97.7; 101.6 -> 103.2; 102.2 -> 103.9; 96.4 -> 97.6 | Helped |
| Replace FastSpeech's PE with SPE | FastSpeech · Speech synthesis, Cases 1–5 | SSIM: 0.210 -> 0.223; 0.209 -> 0.223; 0.228 -> 0.239; 0.229 -> 0.240; 0.236 -> 0.262 | Helped |
| Add PE to FastSpeech | FastSpeech · Speech synthesis, Cases 1–5 | PSNR: 100.7 -> 103.4; 94.4 -> 96.3; 100.2 -> 101.6; 99.3 -> 102.2; 94.7 -> 96.4 | Helped |
Surprising findings
- Hash-encoding NeRFs (InstantNeRF, NeRFfacto) perform far worse than PE methods in the few-view setting (PSNR about 14.7-14.9).
- Adding PE to FastSpeech with the same L showed little change in the speech experiments, and GRFF gave only negligible gains over PE.
Limitations
- Only the first layer's sinusoidal activation is replaced; the choice of L must still be tuned up to network size, and optimal L is found by increasing until no improvement.
- Few-view NeRF results rely on GAN adversarial training for the best numbers, and the GAN component is separate from SPE.
- Speech experiments use an unofficial FastSpeech implementation and PSNR/SSIM on spectrograms, which may not reflect perceptual quality.
Experimental setup
Datasets: Blender synthetic dataset speech synthesis data (FastSpeech setup) 1D regression data
Benchmarks: Blender synthetic dataset (few-view NeRF) speech spectrogram PSNR/SSIM comparison 1D regression with NTK
Metrics: PSNR SSIM WDPR RWDE training loss test loss
Baselines: NeRF DietNeRF MipNeRF FreeNeRF InstantNeRF NeRFfacto APENeRF FastSpeech PE (NTK)
Builds on: NeRF positional encoding FreeNeRF FastSpeech Fourier features for MLPs (Tancik et al.) SIREN
Closest papers by meaning
- Efficiently Learning Significant Fourier Feature Pairs for Statistical Independence Testing (NeurIPS 2024 · similarity 0.84)
- FreSh: Frequency Shifting for Accelerated Neural Representation Learning (ICLR 2025 · similarity 0.84)
- Adaptive Wavelet-Positional Encoding for High-Frequency Information Learning in Implicit Neural Representation (AAAI 2025 · similarity 0.84)
- Representing Sounds as Neural Amplitude Fields: A Benchmark of Coordinate-MLPs and a Fourier Kolmogorov-Arnold Framework (AAAI 2025 · similarity 0.84)
- Visual Fourier Prompt Tuning (NeurIPS 2024 · similarity 0.84)
- Memory-Efficient Training with In-Place FFT Implementation (NeurIPS 2025 · similarity 0.84)
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