The article attempts to find an algebraic formula describing the correlation
coefficients between random variables and the principal components representing
them. As a result of the analysis, starting from selected statistics relating
to individual random variables, the equivalents of these statistics relating to
a set of random variables were presented in the language of linear algebra,
using the concepts of vector and matrix. This made it possible, in subsequent
steps, to derive the expected formula. The formula found is identical to the
formula used in Factor Analysis to calculate factor loadings. The discussion
showed that it is possible to apply this formula to optimize the number of
principal components in Principal Component Analysis, as well as to optimize
the number of factors in Factor Analysis.

使用线性代数的向量和矩阵概念，从个体随机变量的统计信息出发，找到了与这些统计信息相关的一组随机变量的等效统计信息，得到了描述随机变量与代表它们的主成分间相关系数的代数公式。该公式与因子分析中用于计算因子载荷的公式相同，可以用于优化主成分分析和因子分析中的主成分和因子数量。

随机变量与其主成分之间的相关性

On the Correlation between Random Variables and their Principal  Components

Similarity measures based purely on word embeddings are comfortably competing
with much more sophisticated deep learning and expert-engineered systems on
unsupervised semantic textual similarity (STS) tasks. In contrast to commonly
used geometric approaches, we treat a single word embedding as e.g. 300
observations from a scalar random variable. Using this paradigm, we first
illustrate that similarities derived from elementary pooling operations and
classic correlation coefficients yield excellent results on standard STS
benchmarks, outperforming many recently proposed methods while being much
faster and trivial to implement. Next, we demonstrate how to avoid pooling
operations altogether and compare sets of word embeddings directly via
correlation operators between reproducing kernel Hilbert spaces. Just like
cosine similarity is used to compare individual word vectors, we introduce a
novel application of the centered kernel alignment (CKA) as a natural
generalisation of squared cosine similarity for sets of word vectors. Likewise,
CKA is very easy to implement and enjoys very strong empirical results.

研究表明，基于单词嵌入的相似性度量方法在无监督语义文本相似性（STS）任务中正在与更复杂的深度学习和专家设计的系统轻松竞争。通过将一个单词嵌入视为标量随机变量的例如 300 个观测值，我们避免了传统的几何方法，使用于基本汇集运算和经典相关系数得到的相似性效果很好，胜过许多最近的方法，同时速度更快且实现容易。此外，研究认为通过重新生成核希尔伯特空间之间的相关算子，可以避免汇集运算并直接比较词嵌入集。正如余弦相似性用于比较单个单词向量一样，我们介绍了中心核对齐（CKA）的一种新颖应用，作为平方余弦相似性的集合的自然推广。同样，CKA 非常易于实现，并享有非常强的实证结果。