# CoPE: Conditional image generation using Polynomial Expansions

arXiv: https://arxiv.org/abs/2104.05077 · HTML: https://mlsanity.com/paper/2104.05077/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 52 of its 94 references that are in the map.
- First posted on arXiv 11 Apr 2021.

## Builds on (most cited first, 30 of 52)

- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Context Encoders: Feature Learning by Inpainting](https://mlsanity.com/paper/1604.07379/) (2016)
- [Semantic Image Synthesis with Spatially-Adaptive Normalization](https://mlsanity.com/paper/1903.07291/) (2019)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [High-Resolution Image Synthesis and Semantic Manipulation with Conditional GANs](https://mlsanity.com/paper/1711.11585/) (2017)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [StarGAN: Unified Generative Adversarial Networks for Multi-Domain Image-to-Image Translation](https://mlsanity.com/paper/1711.09020/) (2017)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Multimodal Unsupervised Image-to-Image Translation](https://mlsanity.com/paper/1804.04732/) (2018)
- [Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks](https://mlsanity.com/paper/1703.10593/) (2017)
- [Holistically-Nested Edge Detection](https://mlsanity.com/paper/1504.06375/) (2015)
- [Generative Adversarial Networks: An Overview](https://mlsanity.com/paper/1710.07035/) (2017)
- [Generative Image Inpainting with Contextual Attention](https://mlsanity.com/paper/1801.07892/) (2018)
- [Conditional Image Synthesis With Auxiliary Classifier GANs](https://mlsanity.com/paper/1610.09585/) (2016)
- [Video-to-Video Synthesis](https://mlsanity.com/paper/1808.06601/) (2018)
- [Generative Visual Manipulation on the Natural Image Manifold](https://mlsanity.com/paper/1609.03552/) (2016)
- [A Learned Representation For Artistic Style](https://mlsanity.com/paper/1610.07629/) (2016)
- [Toward Multimodal Image-to-Image Translation](https://mlsanity.com/paper/1711.11586/) (2017)
- [Diverse Image-to-Image Translation via Disentangled Representations](https://mlsanity.com/paper/1808.00948/) (2018)

## Built on by (most cited first, 12 of 12)

- [Polynomial Neural Fields for Subband Decomposition and Manipulation](https://mlsanity.com/paper/2302.04862/) (2023)
- [Augmenting Deep Classifiers with Polynomial Neural Networks](https://mlsanity.com/paper/2104.07916/) (2021)
- [Multilinear Operator Networks](https://mlsanity.com/paper/2401.17992/) (2024)
- [Hadamard product in deep learning: Introduction, Advances and Challenges](https://mlsanity.com/paper/2504.13112/) (2025)
- [Extrapolation and Spectral Bias of Neural Nets with Hadamard Product: a Polynomial Net Study](https://mlsanity.com/paper/2209.07736/) (2022)
- [Cluster-guided Image Synthesis with Unconditional Models](https://mlsanity.com/paper/2112.12911/) (2021)
- [Adversarial Audio Synthesis with Complex-valued Polynomial Networks](https://mlsanity.com/paper/2206.06811/) (2022)
- [MI-NeRF: Learning a Single Face NeRF from Multiple Identities](https://mlsanity.com/paper/2403.19920/) (2024)
- [MIGS: Multi-Identity Gaussian Splatting via Tensor Decomposition](https://mlsanity.com/paper/2407.07284/) (2024)
- [PolySAE: Modeling Feature Interactions in Sparse Autoencoders via Polynomial Decoding](https://mlsanity.com/paper/2602.01322/) (2026)
- [PNeRV: A Polynomial Neural Representation for Videos](https://mlsanity.com/paper/2406.19299/) (2024)
- [Polynomial-Augmented Neural Networks (PANNs) with Weak Orthogonality Constraints for Enhanced Function and PDE Approximation](https://mlsanity.com/paper/2406.02336/) (2024)

## Abstract (arXiv)

Generative modeling has evolved to a notable field of machine learning. Deep polynomial neural networks (PNNs) have demonstrated impressive results in unsupervised image generation, where the task is to map an input vector (i.e., noise) to a synthesized image. However, the success of PNNs has not been replicated in conditional generation tasks, such as super-resolution. Existing PNNs focus on single-variable polynomial expansions which do not fare well to two-variable inputs, i.e., the noise variable and the conditional variable. In this work, we introduce a general framework, called CoPE, that enables a polynomial expansion of two input variables and captures their auto- and cross-correlations. We exhibit how CoPE can be trivially augmented to accept an arbitrary number of input variables. CoPE is evaluated in five tasks (class-conditional generation, inverse problems, edges-to-image translation, image-to-image translation, attribute-guided generation) involving eight datasets. The thorough evaluation suggests that CoPE can be useful for tackling diverse conditional generation tasks. The source code of CoPE is available at urlhttps://github.com/grigorisg9gr/polynomialnetsforconditionalgeneration.