Abstract Matrix factorization methods are extensively employed to understand complex data. In this paper, we introduce the cross-product penalized component analysis (X-CAN), a matrix factorization based on the optimization of a loss function that allows a trade-off between variance maximization and structural preservation, with a focus on highlighting differences between groups of observations and/or variables. The approach is based on previous developments, notably (i) the Sparse Principal Component Analysis (SPCA) framework based on the LASSO, (ii) extensions of SPCA to constrain both modes of the factorization, like co-clustering or the Penalized Matrix Decomposition (PMD), and (iii) the Group-wise Principal Component Analysis (GPCA) method. The result is a flexible modeling approach that can be used for data exploration in a large variety of problems. We demonstrate its use with applications from different disciplines.

中文翻译：

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交叉产品惩罚成分分析 (X-CAN)

摘要 矩阵分解方法被广泛用于理解复杂数据。在本文中，我们介绍了交叉乘积惩罚成分分析 (X-CAN)，这是一种基于损失函数优化的矩阵分解，允许在方差最大化和结构保留之间进行权衡，重点是突出两者之间的差异观察和/或变量组。该方法基于先前的发展，特别是 (i) 基于 LASSO 的稀疏主成分分析 (SPCA) 框架，(ii) SPCA 的扩展以约束两种分解模式，如共聚类或惩罚矩阵分解 ( PMD)，以及 (iii) 分组主成分分析 (GPCA) 方法。结果是一种灵活的建模方法，可用于在各种问题中进行数据探索。我们展示了它在不同学科的应用程序中的使用。