SIAM Journal on Imaging Sciences, Volume 13, Issue 4, Page 2361-2392, January 2020.
In this paper, we study the image multiview subspace clustering problem via a nonconvex low-rank representation under the framework of tensors. Most of the recent studies of tensor based multiview subspace clustering use the tensor nuclear norm as a convex surrogate of the tensor rank, i.e., the t-SVD based multiview subspace clustering model. However, since the tensor nuclear norm is linearly proportional to the sum of singular values, the tensor rank approximation by using the tensor nuclear norm may become problematic if the ratios of the nonzero singular values are far from 1. In this paper, a nonconvex tensor log-determinant function is proposed as the objective function regularizer, aiming to achieve a better tensor low-rank approximation. Instead of directly solving the minimization problem in its original setting, the corresponding non-convex optimization is conducted in the Fourier domain, which is shown not only to be feasible but also to be quite effective. A corresponding algorithm associated with the augmented Lagrangian multipliers is established and the constructed convergent sequence to the desirable Karush--Kuhn--Tucker critical point solution is mathematically validated in detail. Extensive simulations on eight benchmark image datasets are provided, along with full comparisons with the latest existing approaches. The obtained results demonstrate that our proposed method significantly outperforms those convex approaches currently available in the literature.

中文翻译：

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通过非凸方法将张量秩最小化的图像多视图聚类

SIAM影像科学杂志，第13卷，第4期，第2361-2392页，2020年1月。
本文通过张量框架下的非凸低秩表示研究图像多视图子空间聚类问题。基于张量的多视图子空间聚类的最新研究大多使用张量核范数作为张量秩的凸替代，即基于t-SVD的多视图子空间聚类模型。但是，由于张量核范数与奇异值之和成线性比例，因此如果非零奇异值的比率远非1，则使用张量核范数进行张量秩近似可能会成为问题。提出了对数行列式函数作为目标函数正则化函数，旨在获得更好的张量低秩逼近。与其直接解决原始设置中的最小化问题，不如说是 相应的非凸优化在傅立叶域中进行，这不仅是可行的而且非常有效。建立了与增强拉格朗日乘子相关的相应算法，并在数学上详细验证了所需的Karush-Kuhn-Tucker临界点解的构造收敛序列。提供了对八个基准图像数据集的广泛模拟，以及与最新现有方法的完整比较。获得的结果表明，我们提出的方法明显优于文献中当前可用的那些凸方法。建立了与增强拉格朗日乘子相关的相应算法，并在数学上详细验证了所需的Karush-Kuhn-Tucker临界点解的构造收敛序列。提供了对八个基准图像数据集的广泛模拟，以及与最新现有方法的完整比较。获得的结果表明，我们提出的方法明显优于文献中当前可用的那些凸方法。建立了与增强拉格朗日乘子相关的相应算法，并在数学上详细验证了所需的Karush-Kuhn-Tucker临界点解的构造收敛序列。提供了对八个基准图像数据集的广泛模拟，以及与最新现有方法的完整比较。获得的结果表明，我们提出的方法明显优于文献中当前可用的那些凸方法。