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

TL;DR

This paper addresses how data semantics can be embedded into the weights of Implicit Neural Representations (INRs). It proposes HyperINR, a model in which a hypernetwork maps low-dimensional latent embeddings to INR weights, and leverages the Implicit Function Theorem to formalize the data-to-weight mapping. The authors show that, under a full-rank Jacobian condition and exact reconstruction, a unique local mapping from data to latent weights exists, and they provide empirical evidence on $2$D and $3$D tasks demonstrating data-semantics clustering and strong classification performance. The work also analyzes Hessian conditioning, latent-space continuity, and ablation studies, arguing for a lightweight yet effective approach to weight-space learning with potential broad impact on meta-learning, transfer learning, and model personalization.

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.

Figures (7)

• Figure 1: HyperINR: A hypernetwork generates weights $\bm{w}_j$ from learnable latent vector $\bm{z}_j$, which are then used by the main network $f(\bm{w}_j, \cdot)$ to map coordinates to pixel or Unsigned Distance Function (UDF) values.
• Figure 2: Phase A transforms data samples to latent weight representations using HyperINR, while Phase B utilizes these latent representations for downstream classification tasks. The fire symbol indicates parameters that are updated, while the snowflake symbol indicates parameters that are frozen.
• Figure 3: Distinct clustering of latent embeddings $\bm{z}$ from ShapeNet10 using 3D PCA. Left: Training samples form well-separated clusters. Right: Test samples fall into the neighborhoods of their respective categories under a fixed hypernetwork.
• Figure 4: Distinct clustering of hypernetwork-generated weights for ShapeNet10 using 2D PCA.
• Figure 5: Linear interpolation in the latent space between two ShapeNet chair samples. The results exhibit smooth transitions in geometry, indicating that the latent space is continuous and semantically meaningful.