分析线性动态系统：从建模到编码和学习
Analyzing Linear Dynamical Systems: From Modeling to Coding and Learning
摘要：
用线性动态系统（LDSs）编码时间序列可以产生丰富的模型，其应用范围从动态纹理识别到视频分割等等。在本文中，我们提出了表示LDS与无限维子空间，并推导出一个解析解，以获得稳定的LDS。然后，我们设计了有效的算法来执行稀疏编码和字典学习的无限维子空间的空间。特别地，开发了两种解决方案来稀疏地编码LDS。在第一种方法中，我们将子空间映射到再生核希尔伯特空间（RKHS），并通过核稀疏编码来实现我们的目标。对于第二种解决方案，我们建议将无限维子空间嵌入到对称矩阵空间中，并在诱导空间中相应地制定稀疏编码。对于字典学习，我们通过引入一个新的概念，即双重LDS来编码时间序列。然后，我们利用双重LDS导出用于更新LDS字典的原子的解析形式，即，每个原子本身就是一个LDS。与几种基线和最先进的方法相比，所提出的方法在包括视频分类和触觉识别在内的各种分类任务中具有更高的精度。
英文摘要：
Encoding time-series with Linear Dynamical Systems (LDSs) leads to rich models with applications ranging from dynamical texture recognition to video segmentation to name a few. In this paper, we propose to represent LDSs with infinite-dimensional subspaces and derive an analytic solution to obtain stable LDSs. We then devise efficient algorithms to perform sparse coding and dictionary learning on the space of infinite-dimensional subspaces. In particular, two solutions are developed to sparsely encode an LDS. In the first method, we map the subspaces into a Reproducing Kernel Hilbert Space (RKHS) and achieve our goal through kernel sparse coding. As for the second solution, we propose to embed the infinite-dimensional subspaces into the space of symmetric matrices and formulate the sparse coding accordingly in the induced space. For dictionary learning, we encode time-series by introducing a novel concept, namely the two-fold LDSs. We then make use of the two-fold LDSs to derive an analytical form for updating atoms of an LDS dictionary, i.e., each atom is an LDS itself. Compared to several baselines and state-of-the-art methods, the proposed methods yield higher accuracies in various classification tasks including video classification and tactile recognition.
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