A Dynamical System-Based Framework for Dimension Reduction
A Dynamical System-Based Framework for Dimension Reduction
复制标题
基于动力系统的降维框架
DOI：
10.1007/s42967-022-00234-w
复制
发表时间：
2023
影响因子：
1.6
通讯作者：
Osting, Braxton
中科院分区：
文献类型：
--
作者：
Yoon, Ryeongkyung;Osting, Braxton
我们提出了一种新的框架来学习基于非线性动态系统的数据的低维表示，我们称之为动态降维（DDR）。在DDR模型中，每个点通过非线性流向低维子空间演化；子空间上的投影给出了低维嵌入。模型的训练包括识别非线性流和子空间。根据公式发现方法，我们使用字典元素的线性组合来表示定义流的向量场，其中每个元素是预先指定的线性/非线性候选函数。引入了平均总动能的正则化项，并利用最优输运理论对其进行了激励。我们证明了所得到的优化问题是适定的，并建立了DDR方法的几个性质。我们还展示了如何使用基于梯度的优化方法来训练DDR方法，其中梯度是使用最优控制理论的伴随方法计算的。在合成数据集和示例数据集上实现了DDR方法，并与其他降维方法（包括PCA、t-SNE和Umap）进行了比较。
We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems, which we call thedynamical 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 the 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 the optimal control theory. The DDR method is implemented and compared on synthetic and example data sets to other dimension reduction methods, including the PCA,t-SNE, and Umap.