CP decomposition (CPD) is prevalent in chemometrics, signal processing, data mining and many more fields. While many algorithms have been proposed to compute the CPD, alternating least squares (ALS) remains one of the most widely used algorithm for computing the decomposition. Recent works have introduced the notion of eigenvalues and singular values of a tensor and explored applications of eigenvectors and singular vectors in areas like signal processing, data analytics and in various other fields. We introduce a new formulation for deriving singular values and vectors of a tensor by considering the critical points of a function different from what is used in the previous work. Computing these critical points in an alternating manner motivates an alternating optimization algorithm which corresponds to alternating least squares algorithm in the matrix case. However, for tensors with order greater than equal to $3$, it minimizes an objective function which is different from the commonly used least squares loss. Alternating optimization of this new objective leads to simple updates to the factor matrices with the same asymptotic computational cost as ALS. We show that a subsweep of this algorithm can achieve a superlinear convergence rate for exact CPD with known rank and verify it experimentally. We then view the algorithm as optimizing a Mahalanobis distance with respect to each factor with ground metric dependent on the other factors. This perspective allows us to generalize our approach to interpolate between updates corresponding to the ALS and the new algorithm to manage the tradeoff between stability and fitness of the decomposition. Our experimental results show that for approximating synthetic and real-world tensors, this algorithm and its variants converge to a better conditioned decomposition with comparable and sometimes better fitness as compared to the ALS algorithm.
翻译:CP 分解( CPD) 概念在色度处理、 信号处理、 数据挖掘和其他多个字段中很普遍。 虽然许多算法都提议计算CPD, 交替最小正方( ALS) 仍然是计算分解最广泛使用的算法之一 。 最近的工作引入了 eigenvalus 概念和单向值概念, 并探索了在信号处理、 数据解析和多个其它领域应用 。 我们引入了一种新的公式, 用于计算 CPD 的单个值和矢量。 我们考虑到与以往工作不同功能的临界点, 从而得出 Exloror 的单个值和矢量。 这些关键点以交替方式进行计算, 促使一种交替的优化算法在矩阵中相当于交替最小正方值的算法。 然而, 对于温度高于 3 的推算器, 将一个目标函数最小向不同于通常使用的最小值损失 。 这个新目标的精度的精度的精度的精度, 和对应的直角值的直径直径直径直方方方方方方方( ), 我们所知道的AMA- salalalalalalalalalalalal) 的算法可以比更接近于每个的亚的亚的直方的直方。