Analyzing multi-way measurements with variations across one mode of the dataset is a challenge in various fields including data mining, neuroscience and chemometrics. For example, measurements may evolve over time or have unaligned time profiles. The PARAFAC2 model has been successfully used to analyze such data by allowing the underlying factor matrices in one mode (i.e., the evolving mode) to change across slices. The traditional approach to fit a PARAFAC2 model is to use an alternating least squares-based algorithm, which handles the constant cross-product constraint of the PARAFAC2 model by implicitly estimating the evolving factor matrices. This approach makes imposing regularization on these factor matrices challenging. There is currently no algorithm to flexibly impose such regularization with general penalty functions and hard constraints. In order to address this challenge and to avoid the implicit estimation, in this paper, we propose an algorithm for fitting PARAFAC2 based on alternating optimization with the alternating direction method of multipliers (AO-ADMM). With numerical experiments on simulated data, we show that the proposed PARAFAC2 AO-ADMM approach allows for flexible constraints, recovers the underlying patterns accurately, and is computationally efficient compared to the state-of-the-art. We also apply our model to two real-world datasets from neuroscience and chemometrics, and show that constraining the evolving mode improves the interpretability of the extracted patterns.
翻译:分析多路测量方法,在数据集的不同模式之间有差异,这是不同领域的挑战,包括数据挖掘、神经科学和化学计量,例如,测量可能随着时间变化而变化,或具有不统一的时间轮廓。PARAFAC2模型已被成功地用于分析这些数据,允许以一种模式(即演进模式)进行基本要素矩阵的跨切片变化。适应PARAFAC2模型的传统方法是使用一种交替的、基于最小方位的算法,这种算法通过隐含地估计演变要素矩阵来处理PARAFAC2模型的经常性交叉产品制约。这种方法使得这些因素矩阵的正规化具有挑战性。目前没有一种算法可以灵活地以一般惩罚功能和硬性限制来实施这种正规化。为了应对这一挑战并避免隐含的估算,在本文件中,我们建议一种算法将PARAFAC2与乘数的交替方向方法(AO-ADMMM)相适应。在模拟数据上进行数字实验后,我们发现拟议的PARAFAC2 AA-ADMMM方法允许灵活地对这些因素矩阵矩阵矩阵进行规范调整。目前采用两种精确的精确的模型,并且测量分析也显示我们不断的精确地采用了模式。