Bioinformatics and Biostatistics Hub, Institut Pasteur, Paris, France.
The Rotman Research Institute Institution at Baycrest, Toronto, ON, Canada.
PLoS One. 2019 Mar 13;14(3):e0211463. doi: 10.1371/journal.pone.0211463. eCollection 2019.
We propose a new sparsification method for the singular value decomposition-called the constrained singular value decomposition (CSVD)-that can incorporate multiple constraints such as sparsification and orthogonality for the left and right singular vectors. The CSVD can combine different constraints because it implements each constraint as a projection onto a convex set, and because it integrates these constraints as projections onto the intersection of multiple convex sets. We show that, with appropriate sparsification constants, the algorithm is guaranteed to converge to a stable point. We also propose and analyze the convergence of an efficient algorithm for the specific case of the projection onto the balls defined by the norms L1 and L2. We illustrate the CSVD and compare it to the standard singular value decomposition and to a non-orthogonal related sparsification method with: 1) a simulated example, 2) a small set of face images (corresponding to a configuration with a number of variables much larger than the number of observations), and 3) a psychometric application with a large number of observations and a small number of variables. The companion R-package, csvd, that implements the algorithms described in this paper, along with reproducible examples, are available for download from https://github.com/vguillemot/csvd.
我们提出了一种新的奇异值分解稀疏化方法——约束奇异值分解（CSVD），它可以将多个约束（如稀疏化和左右奇异向量的正交性）纳入其中。CSVD 可以结合不同的约束，因为它将每个约束实现为到凸集的投影，并且因为它将这些约束集成到多个凸集的交集的投影中。我们证明，在适当的稀疏化常数下，算法保证收敛到稳定点。我们还提出并分析了针对 L1 和 L2 范数定义的球上投影的特定情况的有效算法的收敛性。我们使用：1）一个模拟示例，2）一组小的人脸图像（对应于变量数远大于观测数的配置），以及 3）一个具有大量观测值和少量变量的心理测量应用程序来展示 CSVD，并将其与标准奇异值分解和一种非正交相关的稀疏化方法进行比较。实现本文中描述的算法的 R 包 csvd 以及可重现的示例可从 https://github.com/vguillemot/csvd 下载。
