Online Epsilon Net and Piercing Set for Geometric Concepts

VC-dimension and $\varepsilon$-nets are key concepts in Statistical Learning Theory. Intuitively, VC-dimension is a measure of the size of a class of sets. The famous $\varepsilon$-net theorem, a fundamental result in Discrete Geometry, asserts that if the VC-dimension of a set system is bounded, then a small sample exists that intersects all sufficiently large sets. In online learning scenarios where data arrives sequentially, the VC-dimension helps to bound the complexity of the set system, and $\varepsilon$-nets ensure the selection of a small representative set. This sampling framework is crucial in various domains, including spatial data analysis, motion planning in dynamic environments, optimization of sensor networks, and feature extraction in computer vision, among others. Motivated by these applications, we study the online $\varepsilon$-net problem for geometric concepts with bounded VC-dimension. While the offline version of this problem has been extensively studied, surprisingly, there are no known theoretical results for the online version to date. We present the first deterministic online algorithm with an optimal competitive ratio for intervals in $\mathbb{R}$. Next, we give a randomized online algorithm with a near-optimal competitive ratio for axis-aligned boxes in $\mathbb{R}^d$, for $d\le 3$. Furthermore, we introduce a novel technique to analyze similar-sized objects of constant description complexity in $\mathbb{R}^d$, which may be of independent interest. Next, we focus on the continuous version of this problem, where ranges of the set system are geometric concepts in $\mathbb{R}^d$ arriving in an online manner, but the universe is the entire space, and the objective is to choose a small sample that intersects all the ranges.

Power-Efficient Image Storage: Leveraging Super Resolution Generative Adversarial Network for Sustainable Compression and Reduced Carbon Footprint

In recent years, large-scale adoption of cloud storage solutions has revolutionized the way we think about digital data storage. However, the exponential increase in data volume, especially images, has raised environmental concerns regarding power and resource consumption, as well as the rising digital carbon footprint emissions. The aim of this research is to propose a methodology for cloud-based image storage by integrating image compression technology with SuperResolution Generative Adversarial Networks (SRGAN). Rather than storing images in their original format directly on the cloud, our approach involves initially reducing the image size through compression and downsizing techniques before storage. Upon request, these compressed images will be retrieved and processed by SRGAN to generate images. The efficacy of the proposed method is evaluated in terms of PSNR and SSIM metrics. Additionally, a mathematical analysis is given to calculate power consumption and carbon footprint assesment. The proposed data compression technique provides a significant solution to achieve a reasonable trade off between environmental sustainability and industrial efficiency.