Structural Preservation of Data in Implicit Neural Representations via the Implicit Function Theorem

Implicit Neural Representations (INRs) parameterize signals of interest as coordinate-based mappings, where each data sample is uniquely encoded in a specific set of weights. This data-to-weight association provides a convenient platform for studying Weight Space Learning (WSL), an emerging field with potential for tasks like meta-learning or transfer learning. So far, a precise theoretical explanation of how the underlying data structure is preserved in the learned weights is still missing. In this work, we employ the Implicit Function Theorem (IFT) to establish a rigorous correspondence between the data space and its latent weight representation space. We analyze a framework that maps instance-specific latent codes to INR weights via a shared hyper-network. Empirically, we construct a synthetic Fourier-curve testbed to verify data structure recovery. Furthermore, we demonstrate competitive performance against existing approaches on INR weight classification across 2D and 3D datasets using more compact networks. The present work offers a theoretical lens for future investigation into WSL.