归纳矩阵补全中的全局最优性

Mohsen Ghassemi, A. Sarwate, Naveen Goela
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引用次数: 6

摘要

归纳矩阵补全(IMC)是在矩阵补全问题中以未知矩阵的行、列实体的“特征”形式加入边信息的一种模型。作为副信息,特征可以大大减少从给定项重构未知矩阵所需的观察项的数量。IMC问题可以表述为一个低秩矩阵恢复问题,其中观察到的条目被视为一个较小矩阵的测量值,该矩阵模拟了列和行特征之间的相互作用。我们利用这一性质来研究因式IMC问题的优化前景。特别地,我们证明了该问题的目标函数的临界点要么是对应于真解的全局最小值,要么是“可逃避的”鞍点。这一结果表明,任何保证收敛于局部极小值的最小化算法都可以用于求解因式IMC问题。
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Global Optimality in Inductive Matrix Completion
Inductive matrix completion (IMC) is a model for incorporating side information in form of “features” of the row and column entities of an unknown matrix in the matrix completion problem. As side information, features can substantially reduce the number of observed entries required for reconstructing an unknown matrix from its given entries. The IMC problem can be formulated as a low-rank matrix recovery problem where the observed entries are seen as measurements of a smaller matrix that models the interaction between the column and row features. We take advantage of this property to study the optimization landscape of the factorized IMC problem. In particular, we show that the critical points of the objective function of this problem are either global minima that correspond to the true solution or are “escapable” saddle points. This result implies that any minimization algorithm with guaranteed convergence to a local minimum can be used for solving the factorized IMC problem.
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