The magnitude of a finite metric space is a recently-introduced invariant quantity. Despite beneficial theoretical and practical properties, such as a general utility for outlier detection, and a close connection to Laplace radial basis kernels, magnitude has received little attention by the machine learning community so far. In this work, we investigate the properties of magnitude on individual images, with each image forming its own metric space. We show that the known properties of outlier detection translate to edge detection in images and we give supporting theoretical justifications. In addition, we provide a proof of concept of its utility by using a novel magnitude layer to defend against adversarial attacks. Since naive magnitude calculations may be computationally prohibitive, we introduce an algorithm that leverages the regular structure of images to dramatically reduce the computational cost.
翻译:有限度空间的大小是最近引入的变数量。 尽管理论和实践上具有有益的特性,比如对外部探测的通用工具,以及与Laplace辐射基内核的密切连接,但迄今为止,机器学习界对数量没有多少注意。在这项工作中,我们调查了单个图像的大小特性,每个图像都形成了自己的计量空间。我们显示,已知的异端探测特性转化成图像的边缘探测,我们给出了理论依据。此外,我们还用新的星级层来防御对立攻击,从而证明了其概念的实用性。由于天真的量计算可能令人无法计算,我们引入了一种算法,利用图像的常规结构来大幅降低计算成本。