Ensuring Semantics in Weights of Implicit Neural Representations through the Implicit Function Theorem

TL;DR

This paper uses the Implicit Function Theorem to theoretically analyze how neural network weights encode data semantics in Implicit Neural Representations, providing a rigorous understanding of the weight-data relationship.

Contribution

It introduces a novel theoretical framework applying the Implicit Function Theorem to explain semantic encoding in INR weights, bridging a gap in understanding.

Findings

Achieves competitive classification performance on 2D and 3D datasets.

Provides a rigorous theoretical mapping between data and weight space.

Offers a new perspective for future research on neural network weights.

Abstract

Weight Space Learning (WSL), which frames neural network weights as a data modality, is an emerging field with potential for tasks like meta-learning or transfer learning. Particularly, Implicit Neural Representations (INRs) provide a convenient testbed, where each set of weights determines the corresponding individual data sample as a mapping from coordinates to contextual values. So far, a precise theoretical explanation for the mechanism of encoding semantics of data into network weights is still missing. In this work, we deploy the Implicit Function Theorem (IFT) to establish a rigorous mapping between the data space and its latent weight representation space. We analyze a framework that maps instance-specific embeddings to INR weights via a shared hypernetwork, achieving performance competitive with existing baselines on downstream classification tasks across 2D and 3D datasets. These findings offer a theoretical lens for future investigations into network weights.