Abstract Multidimensional scaling (MDS) provides a possibility to present the multidimensional data visually. It is a very popular method of this class. MDS minimizes some stress functions. In this paper, the stress function and multidimensional scaling, in general, have been considered from the geometric point of view. The so-called Geometric MDS has been developed. The new interpretation of the stress allows finding the proper step size, and the descent direction forwards the minimum of the stress function analytically if we consider and move a separate point of the projected space. The exceptional property of the new approach is that we do not need the analytical expression of the stress function. There is no need for numerical evaluation of its derivatives, too. Moreover, we do not need for the linear search that is used for local descent in optimization. Theoretical analysis disclosed that the step direction, determined by Geometric MDS, coincides with the steepest descent direction. The analytically found step size is such that it guarantees the decrease of the stress in this direction. Two realizations of Geometric MDS are proposed and examined. The comparison with SMACOF realization of MDS looks very promising.

中文翻译：

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几何多维缩放：一种新的数据降维方法

摘要 多维标度（MDS）提供了一种可视化呈现多维数据的可能性。这是该类中非常流行的方法。MDS 最小化了一些应力函数。在本文中，应力函数和多维标度，一般是从几何的角度考虑的。已开发出所谓的几何 MDS。对应力的新解释允许找到合适的步长，如果我们考虑并移动投影空间的一个单独的点，下降方向会分析地向前推应力函数的最小值。新方法的特殊属性是我们不需要应力函数的解析表达式。也不需要对其导数进行数值评估。而且，我们不需要在优化中用于局部下降的线性搜索。理论分析表明，Geometric MDS 确定的步长方向与最陡下降方向一致。分析发现的步长是这样的，它保证了在这个方向上的应力减少。提出并检验了 Geometric MDS 的两种实现方式。与 MDS 的 SMACOF 实现的比较看起来非常有希望。