Implicit Neural Representations (INRs) model continuous signals with compact neural networks and have become a standard tool in vision, graphics, and signal processing. A central challenge is accurately capturing fine detail without heavy hand-crafted encodings or brittle training heuristics. Across the literature, periodic activations have emerged as a compelling remedy: from SIREN, which uses a single sinusoid with a fixed global frequency, to more recent architectures employing multiple sinusoids and, in some cases, trainable frequencies and phases. We study this family of sinusoidal activations and develop a principled theoretical and practical framework for trainable sinusoidal activations in INRs. Concretely, we instantiate this framework with Sinusoidal Trainable Activation Functions (STAF), a Fourier-like activation whose amplitudes, frequencies, and phases are learned. Our analysis (i) establishes a Kronecker-equivalence construction that expresses trainable sinusoidal activations with standard sine networks and quantifies expressive growth, (ii) characterizes how the Neural Tangent Kernel (NTK) spectrum changes under trainable sinusoidal parameterization, and (iii) provides an initialization that yields standard normal post-activations without asymptotic central limit theorem (CLT) arguments. Empirically, on images, audio, shapes, inverse problems (super-resolution, denoising) and NeRF, STAF is competitive and often stronger on distortion-oriented reconstruction metrics such as PSNR/SSIM across the evaluated INR tasks, with favorable parameter efficiency under layer-wise sharing. While periodic activations can alleviate practical manifestations of spectral bias, our results indicate they do not eliminate it; instead, trainable sinusoids can improve the observed capacity-optimization trade-off in the evaluated settings.

翻译：隐式神经表示（INRs）利用紧凑型神经网络对连续信号进行建模，已成为视觉、图形和信号处理领域的标准工具。其核心挑战在于无需繁重的手工编码或脆弱的训练启发式方法即可精确捕捉精细细节。文献研究表明，周期激活函数已成为一种引人注目的解决方案：从使用单一固定全局频率正弦函数的SIREN，到采用多个正弦函数甚至可训练频率和相位的最新架构，我们对此类正弦激活函数家族展开研究，并开发了一套用于INRs中可训练正弦激活函数的理论基础与实用性框架。具体而言，我们通过正弦可训练激活函数（STAF）实例化该框架——这是一种类傅里叶激活函数，其振幅、频率和相位均可学习。我们的分析：（i）建立了Kronecker等价构造，将可训练正弦激活函数表达为标准正弦网络并量化表达能力增长；（ii）刻画了神经正切核（NTK）谱在可训练正弦参数化下的变化规律；（iii）提出了一种无需渐近中心极限定理（CLT）论证即可产生标准正态后激活的初始化方法。在图像、音频、形状、逆问题（超分辨率、去噪）和NeRF的实验中，STAF在各类INR任务的失真导向重建指标（如PSNR/SSIM）上具有竞争力且往往表现更优，同时通过层级共享实现了良好的参数效率。尽管周期激活可缓解频谱偏差的实践表现，但我们的结果表明其并未消除该偏差；反之，可训练正弦函数可在评估设置中改善观测到的容量-优化权衡。