# Understanding Differentiable Embeddings Through Differential and Integral Geometry

arXiv: https://arxiv.org/abs/2608.06809 · HTML: https://mlsanity.com/paper/2608.06809/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 44 references that are in the map.
- First posted on arXiv 7 Aug 2026.

## Builds on (most cited first, 20 of 20)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [t-viSNE: Interactive Assessment and Interpretation of t-SNE Projections](https://mlsanity.com/paper/2002.06910/) (2020)
- [Measuring and Explaining the Inter-Cluster Reliability of Multidimensional Projections](https://mlsanity.com/paper/2107.07859/) (2021)
- [DimReader: Axis lines that explain non-linear projections](https://mlsanity.com/paper/1710.00992/) (2017)
- [Unveiling High-dimensional Backstage: A Survey for Reliable Visual Analytics with Dimensionality Reduction](https://mlsanity.com/paper/2501.10168/) (2025)
- [Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery](https://mlsanity.com/paper/1305.7255/) (2013)
- [Parallel Transport Unfolding: A Connection-based Manifold Learning Approach](https://mlsanity.com/paper/1806.09039/) (2018)
- [Stop Misusing t-SNE and UMAP for Visual Analytics](https://mlsanity.com/paper/2506.08725/) (2025)
- [FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles](https://mlsanity.com/paper/2206.06513/) (2022)
- [Gauge-invariant representation holonomy](https://mlsanity.com/paper/2601.21653/) (2026)
- [Assessing and improving reliability of neighbor embedding methods: a map-continuity perspective](https://mlsanity.com/paper/2410.16608/) (2024)
- [Conservativeness of untied auto-encoders](https://mlsanity.com/paper/1506.07643/) (2015)
- [Implicit Multidimensional Projection of Local Subspaces](https://mlsanity.com/paper/2009.03259/) (2020)
- [Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction](https://mlsanity.com/paper/2502.20215/) (2025)
- [EmbedOR: Provable Cluster-Preserving Visualizations with Curvature-Based Stochastic Neighbor Embeddings](https://mlsanity.com/paper/2509.03703/) (2025)
- [Into the Void: Mapping the Unseen Gaps in High Dimensional Data](https://mlsanity.com/paper/2501.15273/) (2025)
- [Curvature as a tool for evaluating dimensionality reduction and estimating intrinsic dimension](https://mlsanity.com/paper/2509.13385/) (2025)

## Abstract (arXiv)

How can an analyst decide whether a nonlinear dimensionality reduction embedding can be trusted? Existing diagnostics provide only partial answers: projection glyphs characterize local sensitivity, map-continuity scores measure local conditioning, and transport-based analyses reveal path-dependent inconsistencies. However, these methods appear unrelated and provide no common framework for understanding when they agree or not. We show that they are all derived from a single geometric object induced by every differentiable embedding, whether defined implicitly through optimization or explicitly by a learned mapping. This framework provides two complementary geometric views of an embedding. The differential view explains local behavior: its first-order term recovers projection glyphs, while its second-order curvature quantifies how far their linear approximation remains reliable. The integral view follows the same geometry along high dimensional paths and determines whether an embedding depends only on the current state or also on the path taken to reach it. We further show that map-continuity is a prerequisite for the other analyses. The framework is theoretically complete for diagnostics derived from the embedding geometry, and we prove the integral view irreducible: no amount of local measurement at any number of points, to any order of derivative, reproduces what it detects. Classical rank-based metrics form a complementary class based on finite-scale neighborhood relationships. Experiments on synthetic and real datasets validate theoretical predictions, demonstrate accurate curvature-based trust estimates on single-cell embeddings, and show that the integral analysis distinguishes single-valued embeddings from path-dependent optimization-based embeddings in ways that existing pointwise diagnostics cannot.