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.

中文翻译：

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局部子空间的隐式多维投影

我们提出了一种使用隐函数微分的可视化方法来理解多维投影对局部子空间的影响。在这里，我们将局部子空间理解为数据点的多维局部邻域。现有方法侧重于多维数据点的投影，忽略邻域信息。我们的方法能够分析局部子空间的形状和方向信息，通过对局部结构的感知来更深入地了解数据的全局结构。局部子空间由基向量跨越的多维椭圆拟合。基于表达为隐函数的多维投影的解析微分，提出了一种准确有效的向量变换方法。结果被可视化为字形，并使用我们高效的基于 Web 的可视化工具支持的全套专门设计的交互进行分析。使用各种多维和高维基准数据集证明了我们方法的实用性。我们的隐微分向量变换是通过数值比较来评估的；整体方法通过探索示例和用例进行评估。