  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.
