\section{Introduction}
\label{sec:introduction}
It is a common choice to represent data on discretized grids, e.g., to represent an image as a grid of pixels. While this data representation is widely explored, it poorly scales with grid resolution and ignores the often continuous nature of the underlying signal \cite{dupont2022data}.  Recent research demonstrated that NFs provide an interesting, continuous alternative to represent different kinds of data modalities like sound \cite{sitzmann2020implicit}, images \cite{stanley2007compositional}, shapes \cite{mescheder2019occupancy}, videos \cite{chen2022videoinr}, or 3D scenes \cite{mildenhall2021nerf}, by treating data as neural functions that take spatial or temporal positions (e.g., pixel coordinates) as input and output the appropriate measurements (e.g., image intensity values). A detailed review of single-instance and generalizable NFs can be found in \autoref{app:related}.
Single-instance NF training typically involves overfitting a neural network to a single signal. While this training paradigm yields accurate representations for single instances, it is prohibitively expensive when scaled to large datasets. This scalability issue has gained particular importance as researchers increasingly explore using NFs as compressed dataset representations \cite{dupont2022coin++,schurholt2022hyper,ma2024implicit}, where the neural network weights themselves are treated as a data modality. Naively training single-instance NFs additionally leads to highly unordered weight spaces across separately trained networks, which complicates downstream learning on the weights. Although specialized architectures designed to handle the permutation symmetries inherent to the multilayer perceptrons (MLPs) that typically comprise NFs exist \cite{navon2023equivariant,schurholt2024towards}, alternative frameworks that avoid these issues in the first place can greatly simplify downstream learning.
To overcome these challenges, this work introduces a framework, called \textbf{MedFuncta}, that generalizes medical NFs from isolated, single-instance models to dataset-level neural representations. The central idea, borrowed from \emph{Functa} \cite{dupont2022data} and shown in \autoref{fig:network}, is to meta-learn a shared neural representation across the dataset, in which each signal is represented by a unique, signal-specific parameter vector, also referred to as latent, that conditions a shared network. 
\begin{figure}[htbp]
\floatconts
  {fig:network}
  {\caption{\textit{(Left)} The proposed network with shared parameters $\theta$, that is conditioned by a single signal-specific parameter vector $\phi^{(i)}$. \textit{(Right)} The proposed meta-learning strategy that, starting from a random initialization of $\theta$, learns shared network parameters $\theta^{*}$ in a way that we can fit a signal by updating $\phi^{(i)}$ for \emph{few} steps.}}
  {\includegraphics[width=\linewidth]{images/Network.pdf}}
\end{figure}
This structure enables the model to capture and reuse redundancies across different signals, drastically improving computational efficiency and scalability. Unlike prior methods that rely on patch-based representations \cite{dupont2022coin++,bauer2023spatial}, our proposed framework represents each signal, from 1D time series to 3D volumetric data, with a single 1D latent vector. This abstraction enables \emph{consistent downstream processing across diverse data types}, and is especially advantageous in medical applications, where the ability to \emph{unify multiple data modalities under a common representation} is desirable (see \autoref{app:whynopatch}), and where the inherent capability of NFs to \emph{handle irregularly sampled, heterogeneous data} provides further benefits. Our main contributions are threefold: \\
\textbf{(1) Optimization of Learning Dynamics at Scale:} We propose a non-constant, layer-dependent $\omega$-schedule for commonly used SIREN activations, significantly improving both convergence speed and reconstruction quality. We provide theoretical insights into the interplay between a layer's $\omega$-parameter and its \emph{effective learning rate}, connecting these results to recent research on theoretical learning dynamics. \textbf{(2) Scalable Meta-Learning via Context Reduction:} We propose an efficient, context-reduced meta-learning framework to handle high-dimensional medical data. By utilizing sparse supervision during training, we significantly reduce memory consumption and computational overhead while maintaining competitive performance and speeding up the learning process. \textbf{(3) Comprehensive Evaluation and Open Resources} We demonstrate the versatility of MedFuncta across a diverse range of medical datasets and downstream tasks. To accelerate community research, we open-source our implementation, trained network weights, and a comprehensive dataset~-~\textbf{MedNF}~-~containing $> \SI{500}{k}$ latent vectors for multi-instance NFs in medical imaging and machine learning.