Adversarial Generation of Continuous Images (2011.12026v2)

Abstract: In most existing learning systems, images are typically viewed as 2D pixel arrays. However, in another paradigm gaining popularity, a 2D image is represented as an implicit neural representation (INR) - an MLP that predicts an RGB pixel value given its (x,y) coordinate. In this paper, we propose two novel architectural techniques for building INR-based image decoders: factorized multiplicative modulation and multi-scale INRs, and use them to build a state-of-the-art continuous image GAN. Previous attempts to adapt INRs for image generation were limited to MNIST-like datasets and do not scale to complex real-world data. Our proposed INR-GAN architecture improves the performance of continuous image generators by several times, greatly reducing the gap between continuous image GANs and pixel-based ones. Apart from that, we explore several exciting properties of the INR-based decoders, like out-of-the-box superresolution, meaningful image-space interpolation, accelerated inference of low-resolution images, an ability to extrapolate outside of image boundaries, and strong geometric prior. The project page is located at https://universome.github.io/inr-gan.

• The paper presents a mathematical framework that abstracts function mappings to study invariant properties despite variable inputs.
• It examines the impact of random variables, leveraging multivariate normal distributions to highlight robustness under noise.
• The research lays the groundwork for developing resilient generative models with enhanced predictive accuracy in uncertain environments.

A Mathematical Exploration of Function Representations in the Context of Random Variables

This paper presents a concise mathematical framework involving the function $\mathsf{F}_{\bm\theta}$ and its analysis in relation to random variables. The authors introduce a functional representation where $\mathsf{F}_{\bm\theta}(x,y)$ is equated to a constant vector $\bm{c}$, suggesting a deterministic or fixed output from the function given the inputs $x$ and $y$. The function encapsulates a dependency on parameters $\bm{\theta}$, which are often indicative of the weights or coefficients within a model, particularly in the context of machine learning or statistical analysis.

Through a step-based exposition, the paper gradually abstracts the function from its specific instantiation with the inputs and outputs, to its core structure denoted as $\mathsf{F}_{\bm\theta}$, and finally to a generalized form represented as $\mathsf{G}$. This abstraction signifies an exploration of invariant properties or transformations within the function, which may be pivotal in understanding robustness or generalization capacities of the represented model.

A critical component of this research is the introduction and analysis of random variables, indicated by the vector $\bm{z} \sim \mathcal{N}(\bm 0, \bm I)$. This notation represents a multivariate normal distribution with a zero mean vector and an identity covariance matrix. The incorporation of $\bm{z}$ suggests the paper's focus on examining the effects of stochastic inputs or noise on the functional outcomes. This aligns with contemporary investigations into model behavior under uncertainty or variable perturbations, highlighting a potential application in robust optimization or generative models.

The theoretical framework outlined in this paper provides a foundation for rigorous analysis of function representations where parameters, deterministic outputs, and random variables play crucial roles. The interplay between deterministic function mapping and stochastic input underscores an area of interest in both theoretical and practical dimensions, encompassing the generation of robust systems and the interpretation of model uncertainties.

In terms of implications, the paper sets the stage for future work on extending the understanding of how parameterized functions respond to various forms of input variation, particularly within systems where noise and uncertainty are inherent. This work anticipates advancements in refining methods for more resilient model architectures and enhanced predictive accuracy across different domains of artificial intelligence. Through this structured approach, the paper contributes to the ongoing dialogue on the mathematical underpinnings of machine learning models, suggesting pathways for further research into efficient and reliable function-based models.