FUTON: Fourier Tensor Network for Implicit Neural Representations

TL;DR

This work introduces FUTON (Fourier Tensor Network), which models signals as generalized Fourier series whose coefficients are parameterized by a low-rank tensor decomposition, and derives an inference algorithm with complexity linear in the spectral resolution and the input dimension.

Abstract

Implicit neural representations (INRs) have emerged as powerful tools for encoding signals, yet dominant MLP-based designs often suffer from slow convergence, overfitting to noise, and poor extrapolation. We introduce FUTON (Fourier Tensor Network), which models signals as generalized Fourier series whose coefficients are parameterized by a low-rank tensor decomposition. FUTON implicitly expresses signals as weighted combinations of orthonormal, separable basis functions, combining complementary inductive biases: Fourier bases capture smoothness and periodicity, while the low-rank parameterization enforces low-dimensional spectral structure. We provide theoretical guarantees through a universal approximation theorem and derive an inference algorithm with complexity linear in the spectral resolution and the input dimension. On image and volume representation, FUTON consistently outperforms state-of-the-art MLP-based INRs while training 2--5$\times$ faster. On inverse problems such as image denoising and super-resolution, FUTON generalizes better and converges faster.

Figures (18)

• Figure 1: FUTON weight tensor visualization. (a) Original image from the Kodak dataset ($C=2$, $D=3$). The FUTON model uses $K=32$ cosine basis functions per dimension, yielding a weight tensor $\bm{\mathcal{W}} \in \mathbb{R}^{32 \times 32 \times 3}$. Visualizations of $\bm{\mathcal{W}}$ as RGB images for (b) full-rank and (c) low-rank ($R=4$). The low-rank representation captures the dominant spectral structure.
• Figure 2: Tensor diagram of FUTON's forward pass. Nodes represent tensors, connected edges represent contracted indices (summed over), and dangling edges correspond to open indices. Refer to \ref{['sec:tensor_diagram_notation']} in the supplementary materials for tensor diagram notation.
• Figure 3: Optimal contraction order in FUTON's forward pass. The computation has linear complexity $\mathcal{O}(CKR + CR + DR)$, proceeding in three stages, where yellow boxes indicate the tensors being contracted at each step: (a) project each basis vector $\bm{\varphi}_c(x_c)$ onto its factor matrix $\bm{U}^{(c)}$ via matrix-vector multiplication \ref{['eq:futon_cp_h']}, incurring $\mathcal{O}(CKR)$ operations; (b) combine the projected features $\bm{h}_c(x_c)$ via Hadamard product \ref{['eq:futon_cp_g']}, incurring $\mathcal{O}(CR)$ operations; and (c) map the resulting vector $\bm{g}(\bm{x})$ to the output via matrix-vector multiplication with $\bm{V}$\ref{['eq:futon_cp']}, incurring $\mathcal{O}(DR)$ operations. This contrasts with the naive approach of directly computing $\bm{\Phi}(\bm{x}) \bullet \bm{\mathcal{W}}$, which requires $\mathcal{O}(K^C D)$ operations and scales exponentially with the input dimension $C$.
• Figure 4: Comparison on image representation. FUTON produces sharper reconstructions with better detail preservation than baselines. The image (kodim17) is from the Kodak dataset.
• Figure 5: Learning speed on signal representation. PSNR and IoU versus training time for image (kodim17) and occupancy volume (Thai Statue) representation tasks.

Theorems & Definitions (28)

• Theorem 1: Separable Basis Orthonormality
• proof
• Theorem 2: GFS Convergence
• proof
• Definition 1: Generalized Dot Product
• Theorem 3: Universal Approximation
• proof
• Definition 2: $L^2$ Space
• Definition 3: $L^2$ Inner Product
• Definition 4: $L^2$ Norm