黎曼流形的优化核投影空间
Optimized Kernel-based Projection Space of Riemannian Manifolds
摘要：
事实证明，通过对称正定（SPD）矩阵对图像和视频进行编码，并考虑所得空间的黎曼几何，可以提高分类性能。考虑流形几何通常通过将流形嵌入切空间或再生核希尔伯特空间（RKHS）来完成。最近，有研究表明，将这种流形嵌入到随机投影空间（RPS）中，而不是RKHS或切线空间中，会导致更高的分类和聚类性能。然而，基于随机生成的超平面的结构和维度，RPS上的分类性能可能会有很大的不同。此外，微调RPS是数据昂贵的（因为它需要验证数据），耗时和资源需求。在本文中，我们介绍了一种方法来学习一个优化的基于核的投影（具有固定的维度），采用子空间聚类的概念。因此，我们将数据点与每个点的底层子空间的关联进行编码，以生成有意义的超平面。此外，我们采用字典学习和稀疏编码的概念，判别分析，优化基于核的投影空间（OPS）的SPD流形。我们验证我们的算法在几个分类任务。实验结果也表明，该方法优于国家的最先进的方法在这样的流形。
英文摘要：
It is proven that encoding images and videos through Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, can lead to increased classification performance. Taking into account manifold geometry is typically done via embedding the manifolds in tangent spaces, or Reproducing Kernel Hilbert Spaces (RKHS). Recently, it was shown that embedding such manifolds into a Random Projection Spaces (RPS), rather than RKHS or tangent space, leads to higher classification and clustering performance. However, based on structure and dimensionality of the randomly generated hyperplanes, the classification performance over RPS may vary significantly. In addition, fine-tuning RPS is data expensive (as it requires validation-data), time consuming, and resource demanding. In this paper, we introduce an approach to learn an optimized kernel-based projection (with fixed dimensionality), by employing the concept of subspace clustering. As such, we encode the association of data points to the underlying subspace of each point, to generate meaningful hyperplanes. Further, we adopt the concept of dictionary learning and sparse coding, and discriminative analysis, for the optimized kernel-based projection space (OPS) on SPD manifolds. We validate our algorithm on several classification tasks. The experiment results also demonstrate that the proposed method outperforms state-of-the-art methods on such manifolds.
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