Training model to generate data has increasingly attracted research attention and become important in modern world applications. We propose in this paper a new geometry-based optimization approach to address this problem. Orthogonal to current state-of-the-art density-based approaches, most notably VAE and GAN, we present a fresh new idea that borrows the principle of minimal enclosing ball to train a generator G\left(\bz\right) in such a way that both training and generated data, after being mapped to the feature space, are enclosed in the same sphere. We develop theory to guarantee that the mapping is bijective so that its inverse from feature space to data space results in expressive nonlinear contours to describe the data manifold, hence ensuring data generated are also lying on the data manifold learned from training data. Our model enjoys a nice geometric interpretation, hence termed Geometric Enclosing Networks (GEN), and possesses some key advantages over its rivals, namely simple and easy-to-control optimization formulation, avoidance of mode collapsing and efficiently learn data manifold representation in a completely unsupervised manner. We conducted extensive experiments on synthesis and real-world datasets to illustrate the behaviors, strength and weakness of our proposed GEN, in particular its ability to handle multi-modal data and quality of generated data.
A Stochastic--Geometric Theory of Scaling Laws in Grokking
Research Ireland – Centre for Research Training in AI（爱尔兰研究——人工智能研究培训中心）；J.E. Cairnes School of Business & Economics（J.E. 商业与经济学学院）；University of Galway（Galway大学）；Université Laval（拉瓦尔大学）；CIFAR AI Chair（CIFAR人工智能主席）；Mila – Québec AI Institute（魁北克人工智能研究所）；School of Computer Science & Data Science Institute Visual Intelligence Lab（计算机科学与数据科学学院视觉智能实验室）
Sublinear Time Algorithms for Several Geometric Optimization (With Outliers) Problems In Machine Learning
University of Science and Technology of China（中国科学技术大学）
本文研究了机器学习中几个重要的几何优化问题。首先，我们重新审视欧氏空间$\mathbb{R}^d$中的最小包围球（MEB）问题。这个问题以前已经被广泛研究过，但是现实世界的机器学习任务通常需要处理大规模的数据集，以至于我们甚至负担不起线性时间算法。受最近关于{ \em beyond the worst case analysis}的研究的启发，我们引入了MEB稳定性的概念，这是自然的，易于理解的。粗略地说，MEB的实例是稳定的，如果结果球的半径不能通过移除一小部分输入点来显著减小。在稳定性假设下，我们提出了两种计算半径近似MEB的采样算法，其采样复杂度与输入点的数目无关$n$。特别地，第二算法具有甚至独立于维度$d$的样本复杂度。我们还考虑了没有稳定性假设的一般情况。我们提出了一种混合算法，可以输出半径近似MEB或覆盖近似MEB。我们的算法改善了以前的次线性MEB算法的运行时间和通过次数。我们的方法依赖于两个新的技术，均匀自适应采样方法和三明治引理。此外，我们观察到，这两种技术可以推广到设计次线性时间算法的更广泛的几何优化问题的离群值在高维，包括MEB离群值，一类和二类线性支持向量机离群值，$k$中心聚类离群值，和平坦拟合离群值。我们提出的算法也适用于内核。
Fraunhofer Center for Machine Learning（弗劳恩霍夫机器学习中心）；Fraunhofer-Institut für Techno- und Wirtschaftsmathematik ITWM（弗劳恩霍夫工业数学研究所）；Fraunhofer-Institut für Werkstoffmechanik IWM（弗劳恩霍夫材料力学研究所）
Physics-Driven Neural Network for Solving Electromagnetic Inverse Scattering Problems
Department of Electronic Engineering, School of Electronics and Information, Northwestern Polytechnical University（电子工程系，电子信息学院，西北工业大学）；Department of Computer Science, Al-Farabi Kazakh National University（计算机科学系，阿拜 Kazakh 国家大学）；The Advanced Optical Instrument Research Department, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences（先进光学仪器研究部，西安光学精密机械研究所，中国科学院）
