Principal component analysis (PCA) for binary data, known as logistic PCA, has become a popular alternative to dimensionality reduction of binary data. It is motivated as an extension of ordinary PCA by means of a matrix factorization, akin to the singular value decomposition, that maximizes the Bernoulli log-likelihood. We propose a new formulation of logistic PCA which extends Pearson’s formulation of a low dimensional data representation with minimum error to binary data. Our formulation does not require a matrix factorization, as previous methods do, but instead looks for projections of the natural parameters from the saturated model. Due to this difference, the number of parameters does not grow with the number of observations and the principal component scores on new data can be computed with simple matrix multiplication. We derive explicit solutions for data matrices of special structure and provide a computationally efficient algorithm for solving for the principal component loadings. Through simulation experiments and an analysis of medical diagnoses data, we compare our formulation of logistic PCA to the previous formulation as well as ordinary PCA to demonstrate its benefits.

中文翻译：

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通过自然参数的投影对二进制数据进行降维

二进制数据的主成分分析 (PCA)，称为逻辑 PCA，已成为二进制数据降维的流行替代方法。它是通过矩阵分解作为普通 PCA 的扩展，类似于奇异值分解，最大化伯努利对数似然。我们提出了一种新的逻辑 PCA 公式，它将 Pearson 的低维数据表示公式扩展到二进制数据。我们的公式不像以前的方法那样需要矩阵分解，而是从饱和模型中寻找自然参数的投影。由于这种差异，参数的数量不会随着观察次数的增加而增加，新数据的主成分分数可以通过简单的矩阵乘法来计算。我们为特殊结构的数据矩阵推导出显式解，并提供一种计算效率高的算法来求解主成分载荷。通过模拟实验和对医疗诊断数据的分析，我们将我们的逻辑 PCA 公式与之前的公式以及普通 PCA 进行比较，以证明其优势。