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Life sciences · Preprint
arXiv · August 7, 2026
Raises a question worth testing. It does not answer one.
This is a preprint presenting a unified geometric framework for understanding and diagnosing nonlinear dimensionality reduction embeddings, derived from differential and integral geometry. The work is theoretical and has not undergone peer review; experiments validate the framework's mathematical predictions on synthetic and real datasets but lack prospective validation against clinical or established benchmarks.
Theoretical framework derivation with computational validation experiments.
Projection glyphs, map-continuity scores, and transport-based analyses are unified as outcomes of a single geometric object induced by differentiable embeddings. The differential view recovers first-order projection glyphs and second-order curvature as a measure of linear approximation reliability. The integral view detects path-dependence in optimization-based embeddings that pointwise diagnostics cannot distinguish.
Safety was not reported in the material analysed. Check the source before drawing any conclusion about harm.
The source did not state who this applies to in practice.
This is a theoretical framework paper presenting mathematical foundations for embedding diagnostics without empirical validation on clinical or established benchmarks; it raises questions about trustworthiness of embeddings rather than answering them with prospective evidence.
As stated by the source record.
Graded across the dimensions that decide whether you should act, each from what the source actually supports. There is no single score, and where a dimension was not assessed it says so.
What is missing. This record has no reported figures. That is a gap in the analysis, not a judgement about the study.
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.
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