We propose a visualization method to understand the effect of multidimensional projection on local subspaces, using implicit function differentiation. Here, we understand the local subspace as the multidimensional local neighborhood of data points. Existing methods focus on the projection of multidimensional data points, and the neighborhood information is ignored. Our method is able to analyze the shape and directional information of the local subspace to gain more insights into the global structure of the data through the perception of local structures. Local subspaces are fitted by multidimensional ellipses that are spanned by basis vectors. An accurate and efficient vector transformation method is proposed based on analytical differentiation of multidimensional projections formulated as implicit functions. The results are visualized as glyphs and analyzed using a full set of specifically-designed interactions supported in our efficient web-based visualization tool. The usefulness of our method is demonstrated using various multi- and high-dimensional benchmark datasets. Our implicit differentiation vector transformation is evaluated through numerical comparisons; the overall method is evaluated through exploration examples and use cases.

翻译：我们提出一种基于隐函数微分来理解多维投影对局部子空间影响的可视化方法。此处，我们将局部子空间理解为数据点在多维空间中的局部邻域。现有方法侧重于多维数据点的投影，忽略了邻域信息。我们的方法能够分析局部子空间的形状与方向信息，通过感知局部结构获得对数据全局结构的更深入洞察。局部子空间由基向量张成的多维椭圆拟合，并基于将多维投影表述为隐函数的解析微分，提出一种精确高效的向量变换方法。结果通过字形进行可视化，并在我们基于Web的高效可视化工具中通过一整套专门设计的交互功能进行分析。我们利用多种多维度与高维度的基准数据集验证了该方法的实用性，通过数值比较评估了隐式微分向量变换方法，并通过探索示例与用例评估了整体方法。