# Implicit Multidimensional Projection of Local Subspaces

arXiv: https://arxiv.org/abs/2009.03259 · HTML: https://mlsanity.com/paper/2009.03259/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2020.

## Built on by (most cited first, 4 of 4)

- [An Image-based Typology for Visualization](https://mlsanity.com/paper/2403.05594/) (2024)
- [Lens functions for exploring UMAP Projections with Domain Knowledge](https://mlsanity.com/paper/2405.09204/) (2024)
- [Topo-GS: Continuous Volumetric Embedding of High-Dimensional Data via Topological Gaussian Splatting](https://mlsanity.com/paper/2605.17011/) (2026)
- [Understanding Differentiable Embeddings Through Differential and Integral Geometry](https://mlsanity.com/paper/2608.06809/) (2026)

## Abstract (arXiv)

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.