Unifying Nuclear Norm and Bilinear Factorization Approaches for Low-Rank Matrix Decomposition

R. Cabral, F. D. L. Torre, J. Costeira, Alexandre Bernardino
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引用次数: 181

Abstract

Low rank models have been widely used for the representation of shape, appearance or motion in computer vision problems. Traditional approaches to fit low rank models make use of an explicit bilinear factorization. These approaches benefit from fast numerical methods for optimization and easy kernelization. However, they suffer from serious local minima problems depending on the loss function and the amount/type of missing data. Recently, these low-rank models have alternatively been formulated as convex problems using the nuclear norm regularizer, unlike factorization methods, their numerical solvers are slow and it is unclear how to kernelize them or to impose a rank a priori. This paper proposes a unified approach to bilinear factorization and nuclear norm regularization, that inherits the benefits of both. We analyze the conditions under which these approaches are equivalent. Moreover, based on this analysis, we propose a new optimization algorithm and a "rank continuation'' strategy that outperform state-of-the-art approaches for Robust PCA, Structure from Motion and Photometric Stereo with outliers and missing data.
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低秩矩阵分解的统一核范数与双线性分解方法
在计算机视觉问题中,低秩模型被广泛用于形状、外观或运动的表示。传统的低秩模型拟合方法使用显式双线性分解。这些方法得益于快速的数值方法优化和易于核化。然而,由于损失函数和丢失数据的数量/类型,它们存在严重的局部最小问题。最近,这些低秩模型被表述为使用核范数正则化器的凸问题,与因式分解方法不同,它们的数值求解速度很慢,并且不清楚如何对它们进行核化或施加先验秩。本文提出了一种统一的双线性分解和核范数正则化方法,继承了两者的优点。我们分析了这些方法是等价的条件。此外,在此分析的基础上,我们提出了一种新的优化算法和“秩延续”策略,该策略优于具有异常值和缺失数据的鲁棒主成分分析、运动和光度立体结构的最先进方法。
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