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Neural Fourier Transform: A General Approach to Equivariant Representation Learning
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Proposes Neural Fourier Transform (NFT), a framework that learns encoder/decoder pairs making a nonlinear group action linear in latent space, with theory linking linear equivariant features to group-invariant kernels.
Problem
Equivariance learning usually requires known, linear group actions on data; many real symmetries (fisheye warps, 3D rendering viewpoint changes) act nonlinearly and implicitly, so standard Fourier analysis and group convolution do not apply.
Approach
Learn an encoder and decoder from short observed tuples (x, g∘x) so that the group acts linearly on the latent space, decomposed into irreducible representations (frequencies); prove existence and uniqueness via G-invariant kernels and show the autoencoder loss acts as a data-dependent nonlinear Fourier filter.
Key results
- On time-warped signals, NFT recovers the true frequencies with (FN, FP) = (0.035, 0.022) over 100 random frequency sets, and 0.04/0.028 with 5000 samples.
- On time-warped signal compression, g-NFT achieves reconstruction MSE 3.59e-5 (noiseless) versus 1.98e-2 for G-NFT and 8.10 for DFT with 16 frequencies.
- On rotated MNIST out-of-domain linear probing, g-NFT norm consistently outperforms competitors on OOD datasets, while supervised SteerableCNN wins only in-domain.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use known group element identities in g-NFT instead of group-only knowledge in G-NFT | NFT · Time-warped 1D signals, noiseless | reconstruction error: $1.98$ ($\pm 1.89$) $\times 10^{-2}$ -> $3.59$ ($\pm 1.29$) $\times 10^{-5}$ | Helped |
| Use known group element identities in g-NFT instead of group-only knowledge in G-NFT | NFT · Time-warped 1D signals, σ = 0.01 | reconstruction error: $2.42$ ($\pm1.19$) $\times 10^{-2}$ -> $2.62$ ($\pm0.26$) $\times 10^{-4}$ | Helped |
| Use known group element identities in g-NFT instead of group-only knowledge in G-NFT | NFT · Time-warped 1D signals, σ = 0.05 | reconstruction error: $5.82$ ($\pm1.15$) $\times 10^{-2}$ -> $1.42$ ($\pm 0.14$) $ \times 10^{-3}$ | Helped |
| Use known group element identities in g-NFT instead of group-only knowledge in G-NFT | NFT · Time-warped 1D signals, σ = 0.1 | reconstruction error: $1.16$ ($\pm 0.22$) $\times 10^{-1}$ -> $2.53$ ($\pm 0.09$) $ \times 10^{-3}$ | Helped |
| Use g-NFT instead of DFT ($N_f = 16$) | Time-warped 1D signal compression · Time-warped 1D signals, noiseless | reconstruction error: $8.10$ -> $3.59$ ($\pm 1.29$) $\times 10^{-5}$ | Helped |
Surprising findings
- Supervised SteerableCNN outperforms all unsupervised baselines in-domain but not on OOD rotated datasets, attributed to overfitting to task-relevant frequencies.
Limitations
- Requires datasets of short tuples/sequences showing transitions, which may be impractical.
- Does not identifiably preserve norms of latent frequency coefficients since scale of encoder is undetermined.
- No proof that the autoencoder loss finds the optimal solution; algorithmic guarantee remains open.
Experimental setup
Datasets: MNIST Fashion-MNIST Kuzushiji-MNIST CIFAR100 ModelNet10-SO3 BRDFs ABO-Material synthetic time-warped signals
Benchmarks: rotated MNIST rotated Fashion-MNIST rotated Kuzushiji-MNIST time-warped signal reconstruction
Metrics: reconstruction MSE linear probe classification accuracy frequency recovery FN/FP
Baselines: AE SimCLR invariant autoencoder (IAE) C_n steerable CNN SO(2) steerable CNN DFT
Builds on: Fourier transform group convolution MSP (unsupervised equivariance learning) steerable CNN
Closest papers by meaning
- Lie Group Decompositions for Equivariant Neural Networks (ICLR 2024 · similarity 0.89)
- Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actions (NeurIPS 2025 · similarity 0.88)
- Approximately Equivariant Neural Processes (NeurIPS 2024 · similarity 0.88)
- Equivariant Neural Tangent Kernels (ICML 2025 · similarity 0.88)
- A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.88)
- Learning symmetries via weight-sharing with doubly stochastic tensors (NeurIPS 2024 · similarity 0.87)
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