论文标题
基于动态系统降低的动力学框架
A dynamical systems based framework for dimension reduction
论文作者
论文摘要
我们提出了一个新的框架，用于学习基于非线性动力学系统的数据的低维表示，我们将其称为动力学尺寸降低（DDR）。在DDR模型中，每个点通过非线性流向较低维的子空间而发展。对子空间的投影给出了低维嵌入的。训练该模型涉及识别非线性流量和子空间。遵循方程发现方法，我们表示使用字典元素的线性组合来定义流的矢量场，其中每个元素都是预先指定的线性/非线性候选函数。最佳运输理论也引入并激发了平均总动能的正则化项。我们证明了由此产生的优化问题是良好的，并建立了DDR方法的几种属性。我们还展示了如何使用基于梯度的优化方法对DDR方法进行训练，在该方法中，使用最佳控制理论的伴随方法计算梯度。在合成和示例数据集上实现了DDR方法，并将其与其他维度减少方法进行了比较，包括PCA，T-SNE和UMAP。
We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems, which we call dynamical dimension reduction (DDR). In the DDR model, each point is evolved via a nonlinear flow towards a lower-dimensional subspace; the projection onto the subspace gives the low-dimensional embedding. Training the model involves identifying the nonlinear flow and the subspace. Following the equation discovery method, we represent the vector field that defines the flow using a linear combination of dictionary elements, where each element is a pre-specified linear/nonlinear candidate function. A regularization term for the average total kinetic energy is also introduced and motivated by optimal transport theory. We prove that the resulting optimization problem is well-posed and establish several properties of the DDR method. We also show how the DDR method can be trained using a gradient-based optimization method, where the gradients are computed using the adjoint method from optimal control theory. The DDR method is implemented and compared on synthetic and example datasets to other dimension reductions methods, including PCA, t-SNE, and Umap.
