In cluster analysis, a common first step is to scale the data aiming to
better partition them into clusters. Even though many different techniques have
throughout many years been introduced to this end, it is probably fair to say
that the workhorse in this preprocessing phase has been to divide the data by
the standard deviation along each dimension. Like division by the standard
deviation, the great majority of scaling techniques can be said to have roots
in some sort of statistical take on the data. Here we explore the use of
multidimensional shapes of data, aiming to obtain scaling factors for use prior
to clustering by some method, like k-means, that makes explicit use of
distances between samples. We borrow from the field of cosmology and related
areas the recently introduced notion of shape complexity, which in the variant
we use is a relatively simple, data-dependent nonlinear function that we show
can be used to help with the determination of appropriate scaling factors.
Focusing on what might be called "midrange" distances, we formulate a
constrained nonlinear programming problem and use it to produce candidate
scaling-factor sets that can be sifted on the basis of further considerations
of the data, say via expert knowledge. We give results on some iconic data
sets, highlighting the strengths and potential weaknesses of the new approach.
These results are generally positive across all the data sets used.

利用多维数据的形状复杂度概念，应用在特定的非线性函数上，制定了一种新的 “中等距离” 下的基于约束的非线性规划问题，从而探索更有效的聚类缩放因子数量的方法。