While classical scaling, just like principal component analysis, is parameter-free, most other methods for embedding multivariate data require the selection of one or several parameters. This tuning can be difficult due to the unsupervised nature of the situation. We propose a simple, almost obvious, approach to supervise the choice of tuning parameter(s): minimize a notion of stress. We substantiate this choice by reference to rigidity theory. We extend a result by Aspnes et al. (IEEE Mobile Computing, 2006), showing that general random geometric graphs are trilateration graphs with high probability. And we provide a stability result \`a la Anderson et al. (SIAM Discrete Mathematics, 2010). We illustrate this approach in the context of the MDS-MAP(P) algorithm of Shang and Ruml (IEEE INFOCOM, 2004). As a prototypical patch-stitching method, it requires the choice of patch size, and we use the stress to make that choice data-driven. In this context, we perform a number of experiments to illustrate the validity of using the stress as the basis for tuning parameter selection. In so doing, we uncover a bias-variance tradeoff, which is a phenomenon which may have been overlooked in the multidimensional scaling literature. By turning MDS-MAP(P) into a method for manifold learning, we obtain a local version of Isomap for which the minimization of the stress may also be used for parameter tuning.
翻译:经典缩放, 和主元件分析一样, 经典缩放是没有参数的, 嵌入多变数据的其他方法大多要求选择一个或几个参数。 由于局势不受监督的性质, 调幅可能很难。 我们提议了一个简单、 几乎显而易见的方法来监督调调参数的选择: 尽量减少压力的概念。 我们用僵硬理论来证实这一选择。 我们扩展了Aspnes 等人( IEEE, 移动计算, 2006) 的结果, 显示一般随机几何图形是三变图, 概率很高。 我们做了一些实验, 说明使用压力作为调控基础的 la Anderson etal( SIAM discrete Mathematics, 2010 ) 。 我们用MDS- MAP( P) 算法( Ing and Ruml ( INIEEE INFOCOM, 2004) 来说明这一方法是否正确 。 我们用这种推算方法来调整本地的变数, 也就是变压法的缩缩缩缩缩图。