摘要:使用对称正定（symmetric positive definite，简称SPD）矩阵将视觉数据建模到黎曼流形（SPD流形），对于模式识别和机器学习中许多任务有较好的效果.其中，将基于稀疏表示的分类算法扩展到SPD流形上样本的分类任务得到了广泛的关注.本文综合考虑了稀疏表示分类算法的特点以及SPD流形的黎曼几何结构，通过核函数将SPD流形嵌入到再生核希尔伯特空间（reproducing kernel Hilbert space，简称RKHS），分别提出了核空间潜在稀疏表示模型和潜在分类方法.但是，原始的视觉数据在核空间中没有明确的表示形式，这给核空间中的潜在字典更新带来了不便.Nyström是一种可以近似表征核特征的方法.因此，我们利用该方法得到训练样本在RKHS中的近似表示，以更新潜在字典和潜在矩阵.最后，通过在5个标准数据集上的分类实验，验证了该方法的有效性.



Abstract:Modeling visual data onto the SPD (symmetric positive definite) manifold using the SPD matrices has been proven to yield high discriminatory power for many visual classification tasks in the domain of pattern recognition and machine learning. Among them, generalising the sparse representation classification algorithm to the SPD manifold-based visual classification tasks has attracted extensive attention. This study first comprehensively reviews the characteristics of the sparse representation classification algorithm and the Riemannian geometrical structure of the SPD manifold. Then, embedding the SPD manifold into the Reproducing Kernel Hilbert Space (RKHS) via a kernel function. Afterwards, the latent sparse representation model and latent classification model in RKHS has been suggested, respectively. However, the original visual data in RKHS is implicitly described, which is impossible to perform the subsequent dictionary learning. To handle this issue, the Nyström method is utilized to obtain the approximate representations of the training samples in RKHS for the sake of updating the latent dictionary and latent matrix. Finally, the classification results obtained on five benchmarking datasets show the effectiveness of the proposed approach.



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