Partition Speeds Up Learning Implicit Neural Representations Based on
  Exponential-Increase Hypothesis
Ke Liu Feng Liu Haishuai Wang Ning Ma Jiajun Bu Bo Han
$\textit{Implicit neural representations}$ (INRs) aim 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 coordinates 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 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
to reach a fixed small error, the time costs will increase exponentially with
respect to the boundaries in the spatial domain of the target signal. We name
this phenomenon the $\textit{exponential-increase}$ hypothesis. Under the
$\textit{exponential-increase}$ hypothesis, learning INRs for images with many
objects will converge very slowly. 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 speed up the convergence. Based on
this fact, we introduce a simple partition mechanism to boost the performance
of two INR methods for image reconstruction: one for learning INRs, and the
other for learning-to-learn INRs. In both cases, we 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.