Wang Yulong, Tang Yuan Yan, Li Luoqing, Chen Hong, Pan Jianjia
IEEE Trans Pattern Anal Mach Intell. 2019 Jan;41(1):6-19. doi: 10.1109/TPAMI.2017.2780094. Epub 2017 Dec 6.
Representation-based classification (RC) methods such as sparse RC (SRC) have attracted great interest in pattern recognition recently. Despite their empirical success, few theoretical results are reported to justify their effectiveness. In this paper, we establish the theoretical guarantees for a general unified framework termed as atomic representation-based classification (ARC), which includes most RC methods as special cases. We introduce a new condition called atomic classification condition (ACC), which reveals important geometric insights for the theory of ARC. We show that under such condition ARC is provably effective in correctly recognizing any new test sample, even corrupted with noise. Our theoretical analysis significantly broadens the range of conditions under which RC methods succeed for classification in the following two aspects: (1) prior theoretical advances of RC are mainly concerned with the single SRC method while our theory can apply to the general unified ARC framework, including SRC and many other RC methods; and (2) previous works are confined to the analysis of noiseless test data while we provide theoretical guarantees for ARC using both noiseless and noisy test data. Numerical results are provided to validate and complement our theoretical analysis of ARC and its important special cases for both noiseless and noisy test data.
基于表示的分类（RC）方法，如稀疏RC（SRC），近年来在模式识别领域引起了广泛关注。尽管它们在实践中取得了成功，但很少有理论结果能证明其有效性。在本文中，我们为一个称为基于原子表示的分类（ARC）的通用统一框架建立了理论保障，该框架将大多数RC方法作为特殊情况包含在内。我们引入了一个称为原子分类条件（ACC）的新条件，它揭示了ARC理论的重要几何见解。我们表明，在这种条件下，ARC在正确识别任何新的测试样本（即使被噪声干扰）方面被证明是有效的。我们的理论分析在以下两个方面显著拓宽了RC方法成功用于分类的条件范围：（1）RC先前的理论进展主要关注单一的SRC方法，而我们的理论可以应用于通用统一的ARC框架，包括SRC和许多其他RC方法；（2）以前的工作局限于对无噪声测试数据的分析，而我们为ARC提供了使用无噪声和有噪声测试数据的理论保障。提供了数值结果来验证和补充我们对ARC及其无噪声和有噪声测试数据的重要特殊情况的理论分析。
