Learnable Spectral Activations

2026-10-05

AI Summary

This study addresses the inefficiency of frequency composition and optimization difficulties inherent in implicit neural representations by proposing a learnable spectral activation method. Specifically, this approach replaces fixed nonlinear functions with residual truncated Fourier series to decouple feature selection from spectral shaping. By separating the gradients of linear weights and activation coefficients, it directly optimizes the spectral shape, thereby enhancing the energy concentration of the principal modes within the neural tangent kernel. Integrated with differentiable programming techniques, the proposed method significantly improves both reconstruction quality and optimization efficiency across diverse tasks, including audio, image, and neural radiance field synthesis.

Abstract

Implicit neural representations (INRs) are shaped by the spectral structure induced by their input encodings and activation functions. Existing methods improve fitting primarily by modifying which frequencies are available to the network, through coordinate encodings or periodic nonlinearities. However, frequency access is not the only bottleneck: signals with localized or spatially varying structure require the network to efficiently compose frequencies into multi-harmonic internal responses. We introduce learnable spectral activations (LSA), which replace fixed neuron-level nonlinearities with a residual truncated Fourier series whose harmonic amplitudes are learned during training. LSA does not expand the asymptotic function class. Instead, it changes the factorization of the representation: linear weights select features while activation coefficients control spectral shaping, and the two are updated by separate gradients. Because the activation output is affine in the coefficients given fixed pre-activations, spectral tuning becomes a more direct subproblem compared to architectures where it is entangled with feature selection. Empirically, this factorization concentrates more target-signal energy in the leading eigenmodes of the neural tangent kernel, consistent with improved optimization behavior. Across audio, image, neural radiance field, and neural acoustic field tasks, LSA also improves reconstruction quality.

Problem

• Implicit Neural Representations
• Spectral Structure
• Frequency Composition
• Multi-harmonic Responses
• Signal Reconstruction

Innovation

• Learnable Spectral Activations
• Implicit Neural Representations
• Truncated Fourier Series
• Neural Tangent Kernel
• Spectral Shaping