761
个编辑
更改
→理论
其中<math>\langle \cdot, \cdot \rangle</math>表示RKHS中的内积。这种方法允许我们在不知道显式映射<math>\phi</math>的情况下，直接在核空间中计算相似性。即此时Koopman算子的近似矩阵<math>\mathbf{A}</math>为<ref name=william_2015_kernal>M. O. Williams, C. W. Rowley, and I. G. Kevrekidis, A kernel approach to data-driven Koopman spectral analysis, J. Comput. Dynam., 2 (2015), pp. 247--265. (Cited on pp. 255, 256, 284, 285, 294, 318)</ref><ref name=Gonzalez_2023_kernal>Gonzalez, E., Abudia, M., Jury, M., Kamalapurkar, R. & Rosenfeld, J. A. The kernel perspective on dynamic mode decomposition. Preprint at https://doi.org/10.48550/arXiv.2106.00106 (2023).</ref>：
其中<math>\langle \cdot, \cdot \rangle</math>表示RKHS中的内积。这种方法允许我们在不知道显式映射<math>\phi</math>的情况下，直接在核空间中计算相似性。即此时Koopman算子的近似矩阵<math>\mathbf{A}</math>为<ref name=william_2015_kernal>M. O. Williams, C. W. Rowley, and I. G. Kevrekidis, A kernel approach to data-driven Koopman spectral analysis, J. Comput. Dynam., 2 (2015), pp. 247--265. (Cited on pp. 255, 256, 284, 285, 294, 318)</ref><ref name=Gonzalez_2023_kernal>Gonzalez, E., Abudia, M., Jury, M., Kamalapurkar, R. & Rosenfeld, J. A. The kernel perspective on dynamic mode decomposition. Preprint at https://doi.org/10.48550/arXiv.2106.00106 (2023).</ref>：
<math>\mathbf{K}_{YX} \approx \mathbf{A}\phi \mathbf{K}_{XX}</math>
<math>\mathbf{K}_{YX} \approx \mathbf{A}_\phi \mathbf{K}_{XX}</math>
其中，<math>(\mathbf{K}_{XX})_{ij} = k(\mathbf{x}_i, \mathbf{x}_j), \quad (\mathbf{K}_{YX})_{ij} = k(\mathbf{y}_i, \mathbf{x}_j)</math>
其中，<math>(\mathbf{K}_{XX})_{ij} = k(\mathbf{x}_i, \mathbf{x}_j), \quad (\mathbf{K}_{YX})_{ij} = k(\mathbf{y}_i, \mathbf{x}_j)</math>