Escaping Spectral Bias without Backpropagation: Fast Implicit Neural Representations with Extreme Learning Machines

TL;DR

This work tackles the bifurcated challenge of spectral bias and slow, backpropagation-reliant training in implicit neural representations (INRs). It introduces ELM-INR, which replaces end-to-end gradient descent with local Extreme Learning Machines solved in closed form and blends their outputs via a partition of unity, enabling fast, stable reconstructions. A Barron-space analysis links global reconstruction error to the maximum local spectral complexity, motivating BEAM, an adaptive, spectrally balanced mesh refinement that equalizes local Barron norms under a capacity constraint. Across images, multispectral imagery, Navier–Stokes simulations, ERA5 climate fields, and MRI, ELM-INR with BEAM achieves substantial quality gains (roughly 2 dB PSNR improvements in benchmarks) with far lower compute, underscoring its practical impact for high-frequency, spectrally rich signals.

Abstract

Training implicit neural representations (INRs) to capture fine-scale details typically relies on iterative backpropagation and is often hindered by spectral bias when the target exhibits highly non-uniform frequency content. We propose ELM-INR, a backpropagation-free INR that decomposes the domain into overlapping subdomains and fits each local problem using an Extreme Learning Machine (ELM) in closed form, replacing iterative optimization with stable linear least-squares solutions. This design yields fast and numerically robust reconstruction by combining local predictors through a partition of unity. To understand where approximation becomes difficult under fixed local capacity, we analyze the method from a spectral Barron norm perspective, which reveals that global reconstruction error is dominated by regions with high spectral complexity. Building on this insight, we introduce BEAM, an adaptive mesh refinement strategy that balances spectral complexity across subdomains to improve reconstruction quality in capacity-constrained regimes.

Figures (23)

• Figure 1: PSNR as a function of wall-clock time on the kodim05 image. Backpropagation-based INR baselines (SIREN, FFN, GaussNet and WIRE) are trained iteratively, while ELM-INR results correspond to closed-form solutions with different subdomain sizes and numbers of hidden nodes.
• Figure 2: Backpropagation-based INR vs. ELM-INR. Standard INRs require iterative gradient-based training, while ELM-INR fits subdomain ELMs via one-shot least squares and blends them with partition-of-unity for a smooth global reconstruction.
• Figure 3: Comparison of Regular Mesh vs. BEAM. (Top) A regular uniform partition results in an unbalanced distribution of spectral Barron norms. (Bottom) BEAM adaptively partitions the domain. By merging simple regions and keeping complex regions small, it achieves a balanced spectral complexity distribution.
• Figure 4: Qualitative comparison on the kodim05 image. All baseline models are trained for the same number of epochs.
• Figure 5: Qualitative comparison on a high-frequency region from a WorldView-3 multispectral image. The image consists of 8 spectral bands with a spatial resolution of $2048 \times 2048$ pixels, while the visualization shows only the RGB channels for clarity.

Theorems & Definitions (7)

• Definition 3.1: Local Extreme Learning Machine
• Lemma 3.2: ELM as Linear Least-Squares
• Definition 3.3: Partition of Unity
• Definition 3.4: Barron Space $\mathcal{B}$ and Function Class $\Gamma$ barron2002universal
• Definition 3.5: Spectral Barron Norm chen2021representationliao2025spectral
• Theorem 3.6: Approximation Error Bound
• Remark 3.7: Error Domination by Spectral Complexity