Partition Matters in Learning and Learning-to-Learn Implicit Neural Representations

Ke Liu, Feng Liu, Haishuai Wang, Ning Ma, Jiajun Bu, Bo Han

Keywords: implicit neural representations, partition, meta-learning

Abstract: $\textit{Implicit neural representation}$ (INR) aims to learn a $\textit{continuous function}$ (i.e., a neural network) to represent an image, where the input and output of the function are pixel coordinator and RGB/Gray values, respectively. However, images tend to consist of many objects whose colors are not perfectly consistent, resulting in the challenge that image itself is actually a $\textit{discontinuous piecewise function}$ and cannot be well estimated by a continuous function. In this paper, we empirically investigate that if a neural network is enforced to fit a discontinuous piecewise function (e.g., a step function) to reach a fixed small error $\epsilon$, the time costs will increase exponentially. We name this phenomenon as $\textit{exponential-increase}$ hypothesis. Obviously, handling an image with many objects is almost impossible in INR if this hypothesis is true. To address this issue, we first prove that partitioning a complex signal into several sub-regions and utilizing piecewise INRs to fit that signal can significantly reduce the convergence time, even when the exponential-increase hypothesis is true. Based on this fact, we introduce two partition-based INR methods: one for learning INRs, and the other for learning-to-learn INRs. Both methods are designed to partition an image into different sub-regions, and dedicate smaller networks for each part. In addition, we further propose two partition rules based on regular grids and semantic segmentation maps, respectively. Extensive experiments validate the effectiveness of the proposed partitioning methods in terms of learning INR for a single image (ordinary learning framework) and the learning-to-learn framework.

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TL;DR: We use partition techniques to speed up the convergence of learning INRs and learning-to-learn INRs.