# Partition Speeds Up Learning Implicit Neural Representations Based on Exponential-Increase Hypothesis

arXiv: https://arxiv.org/abs/2310.14184 · HTML: https://mlsanity.com/paper/2310.14184/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 47 references that are in the map.
- First posted on arXiv 22 Oct 2023.

## Builds on (most cited first, 30 of 35)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [On First-Order Meta-Learning Algorithms](https://mlsanity.com/paper/1803.02999/) (2018)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [PIFu: Pixel-Aligned Implicit Function for High-Resolution Clothed Human Digitization](https://mlsanity.com/paper/1905.05172/) (2019)
- [GIRAFFE: Representing Scenes as Compositional Generative Neural Feature Fields](https://mlsanity.com/paper/2011.12100/) (2020)
- [Block-NeRF: Scalable Large Scene Neural View Synthesis](https://mlsanity.com/paper/2202.05263/) (2022)
- [GRAF: Generative Radiance Fields for 3D-Aware Image Synthesis](https://mlsanity.com/paper/2007.02442/) (2020)
- [KiloNeRF: Speeding up Neural Radiance Fields with Thousands of Tiny MLPs](https://mlsanity.com/paper/2103.13744/) (2021)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Mega-NeRF: Scalable Construction of Large-Scale NeRFs for Virtual Fly-Throughs](https://mlsanity.com/paper/2112.10703/) (2021)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [NeRV: Neural Representations for Videos](https://mlsanity.com/paper/2110.13903/) (2021)
- [Learned Initializations for Optimizing Coordinate-Based Neural Representations](https://mlsanity.com/paper/2012.02189/) (2020)
- [MetaSDF: Meta-learning Signed Distance Functions](https://mlsanity.com/paper/2006.09662/) (2020)
- [From data to functa: Your data point is a function and you can treat it like one](https://mlsanity.com/paper/2201.12204/) (2022)
- [SCANimate: Weakly Supervised Learning of Skinned Clothed Avatar Networks](https://mlsanity.com/paper/2104.03313/) (2021)
- [The Convergence Rate of Neural Networks for Learned Functions of Different Frequencies](https://mlsanity.com/paper/1906.00425/) (2019)
- [Adversarial Generation of Continuous Images](https://mlsanity.com/paper/2011.12026/) (2020)
- [i3DMM: Deep Implicit 3D Morphable Model of Human Heads](https://mlsanity.com/paper/2011.14143/) (2020)
- [A Structured Dictionary Perspective on Implicit Neural Representations](https://mlsanity.com/paper/2112.01917/) (2021)
- [DualSDF: Semantic Shape Manipulation using a Two-Level Representation](https://mlsanity.com/paper/2004.02869/) (2020)
- [PatchNets: Patch-Based Generalizable Deep Implicit 3D Shape Representations](https://mlsanity.com/paper/2008.01639/) (2020)
- [Overview frequency principle/spectral bias in deep learning](https://mlsanity.com/paper/2201.07395/) (2022)

## Built on by (most cited first, 13 of 13)

- [Neural Experts: Mixture of Experts for Implicit Neural Representations](https://mlsanity.com/paper/2410.21643/) (2024)
- [In Search of a Data Transformation That Accelerates Neural Field Training](https://mlsanity.com/paper/2311.17094/) (2023)
- [EVOS: Efficient Implicit Neural Training via EVOlutionary Selector](https://mlsanity.com/paper/2412.10153/) (2024)
- [Attention Beats Linear for Fast Implicit Neural Representation Generation](https://mlsanity.com/paper/2407.15355/) (2024)
- [Towards Croppable Implicit Neural Representations](https://mlsanity.com/paper/2409.19472/) (2024)
- [TSINR: Capturing Temporal Continuity via Implicit Neural Representations for Time Series Anomaly Detection](https://mlsanity.com/paper/2411.11641/) (2024)
- [Enhancing Implicit Neural Representations via Symmetric Power Transformation](https://mlsanity.com/paper/2412.09213/) (2024)
- [AC-IND: Sparse CT reconstruction based on attenuation coefficient estimation and implicit neural distribution](https://mlsanity.com/paper/2409.07171/) (2024)
- [MetaNeRV: Meta Neural Representations for Videos with Spatial-Temporal Guidance](https://mlsanity.com/paper/2501.02427/) (2025)
- [ImputeINR: Time Series Imputation via Implicit Neural Representations for Disease Diagnosis with Missing Data](https://mlsanity.com/paper/2505.10856/) (2025)
- [NTK-Guided Implicit Neural Teaching](https://mlsanity.com/paper/2511.15487/) (2025)
- [High-Frequency First: A Two-Stage Approach for Improving Image INR](https://mlsanity.com/paper/2508.15582/) (2025)
- [Neural 4D Evolution under Large Topological Changes from 2D Images](https://mlsanity.com/paper/2411.15018/) (2024)

## Abstract (arXiv)

textitImplicit neural representations (INRs) aim to learn a textitcontinuous 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 textitdiscontinuous 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 textitexponential-increase hypothesis. Under the textitexponential-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.