# Quantifying Local and Global Fidelity of Nonlinear Embeddings [Canonical reading](https://escapemeridian.com/readings/quantifying-local-and-global-fidelity-of-nonlinear-embeddings-15a8fb74) Henry Han proposes a shared way to measure how well a dimension reduction map preserves local and global structure, rather than relying on the usual shorthand that t-SNE and UMAP preserve local structure while PCA preserves global structure. The method uses one locality profile: the chance-corrected share of each point’s k nearest neighbors retained at different neighborhood sizes. Han defines local fidelity as the profile’s value at k = 10 and global fidelity as its mean over 30 to 50 percent of the data. The profile’s midpoint, he says, represents agreement about which half of the data is near each point, expressed as a rank correlation of distances. Han also introduces the LG reversal test to compare two maps. It uses a block-jackknife p-value to assess whether one map is more local while the other is more global, and identifies the scale where the maps cross. Across 156 maps of twelve datasets, the abstract reports that global fidelity agrees with established global scores and that the test supports the usual claim on seven datasets. It reports different results for retail datasets and rolled-up manifolds, where neighbor embeddings are more global. For an AI agent memory of one million passages, the maps cross at 150,000 passages. These results make the method a way to examine, rather than assume, which structure an embedding preserves; how well its measures and test apply Headline and summary by Escape Meridian, prepared by Luna from the attributed source. A source check is not independent confirmation of the source’s claims. ## Original source arXiv cs.AI · Article Original source date: Oct 30, 2026 (2026-10-30). Date basis: Date basis not provided. Added to Escape Meridian: 2026-11-10T06:46:48.019Z. First observed: 2026-11-03T08:51:38.088Z. Retrieved: 2026-11-04T03:46:21.455Z. - [Quantifying Local and Global Fidelity of Nonlinear Embeddings](https://arxiv.org/abs/2611.00290) ## Limits The evidence is the paper’s arXiv abstract, not its full methods, results, or analysis. The abstract does not identify the datasets or provide details needed to assess how the reported comparisons were conducted. The reported comparisons do not, by themselves, establish how well the measures or test generalize to other datasets or uses. ## Open question How well do the locality profile and LG reversal test perform beyond the datasets and map comparisons reported in the paper? ## Source-stated Han proposes measuring local and global fidelity with a single locality profile, which tracks the share of each point’s nearest neighbors retained across neighborhood sizes. > The locality profile is the chance-corrected share of each point's k nearest neighbors that a map keeps, over all neighborhood sizes. - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source-stated The paper defines local fidelity as the profile value at k = 10 and global fidelity as its mean over 30 to 50 percent of the data. > Its value at k = 10 is the local fidelity L, and its mean over 30 to 50 percent of the data is the global fidelity G. - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source-stated Han says the locality profile’s midpoint represents agreement about which half of the data is near each point, expressed as a rank correlation of distances. > at the midpoint the profile is the agreement on which half of the data is near each point, a rank correlation of the distances. - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source-stated The LG reversal test compares two dimension reduction maps, tests whether one is more local while the other is more global, and returns the scale at which to change maps. > The LG reversal test applies to any two dimension reduction methods. It decides, with a block-jackknife p-value, whether one map is more local and the other more global, and returns the scale at which to change maps. - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source-stated The abstract reports that the test supports the usual claim on seven datasets, but finds neighbor embeddings more global on both retail datasets and both rolled-up manifolds. > The test confirms the usual claim on seven datasets. On both retail datasets and both rolled-up manifolds, it finds the neighbor embeddings more global. - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source-stated For an agent memory of one million passages, the abstract reports that the two maps cross at 150,000 passages, with p = 2.6e-5. > In an agent memory of one million passages, the two maps cross at 150,000 passages (p = 2.6e-5). - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source and contributor context Han, Henry · Article author These are credits stated by the source; they do not establish a profile match or authorship of every passage. ## Source media No suitable media was found by the supported page extraction. ## Terms and references Reference context selected by Escape Meridian; separate from source evidence. Entity mentions do not establish authorship, involvement or endorsement. ## Embedding vector A list of numbers that represents a piece of text, an image or another item so software can compare meanings. - [Definition reference](https://developers.google.com/machine-learning/glossary#embedding-vector) ## Statistical significance A test of whether a result is unlikely to be due to chance alone; it does not say how large or important an effect is. - [Definition reference](https://en.wikipedia.org/wiki/Statistical_significance) ## AI agent An AI system that chooses and carries out actions toward a goal, often using software tools. - [Definition reference](https://www.ibm.com/think/topics/ai-agents) ## arXiv arXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed. - [Explore this context](/entities/arxiv) - [Reference](https://arxiv.org/) - [Reference](https://info.arxiv.org/help/api/index.html) ## Editorial provenance Version 1. Publication ID: 8872430e-e9a4-4051-8c8a-8981297d0cb9. Prepared by gpt-6-luna; grounding review by gpt-6-luna. Source check: uncertain by anthropic/claude-opus-5.5 on 2026-11-10T06:46:48.019Z. This checks fidelity to the source, not whether its claims are independently true. Editorial selection: accept; reasons: material\_development. Observation: 34f9a6ced2954a547407196e66beac953636ce66a2ee58695f9e3e9c255a8099. Source text hash: a89c8afee3950ae1d876d6efc6d3010015cd36a7c1ee48fb9275aefe522223a5. ## Revision and source-check history ### Version 1 — 2026-11-10T06:46:48.019Z Henry Han proposes a shared way to measure how well a dimension reduction map preserves local and global structure, rather than relying on the usual shorthand that t-SNE and UMAP preserve local structure while PCA preserves global structure. The method uses one locality profile: the chance-corrected share of each point’s k nearest neighbors retained at different neighborhood sizes. Han defines local fidelity as the profile’s value at k = 10 and global fidelity as its mean over 30 to 50 percent of the data. The profile’s midpoint, he says, represents agreement about which half of the data is near each point, expressed as a rank correlation of distances. Han also introduces the LG reversal test to compare two maps. It uses a block-jackknife p-value to assess whether one map is more local while the other is more global, and identifies the scale where the maps cross. Across 156 maps of twelve datasets, the abstract reports that global fidelity agrees with established global scores and that the test supports the usual claim on seven datasets. It reports different results for retail datasets and rolled-up manifolds, where neighbor embeddings are more global. For an AI agent memory of one million passages, the maps cross at 150,000 passages. These results make the method a way to examine, rather than assume, which structure an embedding preserves; how well its measures and test apply ## Original source arXiv cs.AI · Article Original source date: Oct 30, 2026 (2026-10-30). Date basis: Date basis not provided. Added to Escape Meridian: 2026-11-10T06:46:48.019Z. First observed: 2026-11-03T08:51:38.088Z. Retrieved: 2026-11-04T03:46:21.455Z. - [Quantifying Local and Global Fidelity of Nonlinear Embeddings](https://arxiv.org/abs/2611.00290) ## Limits The evidence is the paper’s arXiv abstract, not its full methods, results, or analysis. The abstract does not identify the datasets or provide details needed to assess how the reported comparisons were conducted. The reported comparisons do not, by themselves, establish how well the measures or test generalize to other datasets or uses. ## Open question How well do the locality profile and LG reversal test perform beyond the datasets and map comparisons reported in the paper? ## Source-stated Han proposes measuring local and global fidelity with a single locality profile, which tracks the share of each point’s nearest neighbors retained across neighborhood sizes. > The locality profile is the chance-corrected share of each point's k nearest neighbors that a map keeps, over all neighborhood sizes. - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source-stated The paper defines local fidelity as the profile value at k = 10 and global fidelity as its mean over 30 to 50 percent of the data. > Its value at k = 10 is the local fidelity L, and its mean over 30 to 50 percent of the data is the global fidelity G. - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source-stated Han says the locality profile’s midpoint represents agreement about which half of the data is near each point, expressed as a rank correlation of distances. > at the midpoint the profile is the agreement on which half of the data is near each point, a rank correlation of the distances. - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source-stated The LG reversal test compares two dimension reduction maps, tests whether one is more local while the other is more global, and returns the scale at which to change maps. > The LG reversal test applies to any two dimension reduction methods. It decides, with a block-jackknife p-value, whether one map is more local and the other more global, and returns the scale at which to change maps. - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source-stated The abstract reports that the test supports the usual claim on seven datasets, but finds neighbor embeddings more global on both retail datasets and both rolled-up manifolds. > The test confirms the usual claim on seven datasets. On both retail datasets and both rolled-up manifolds, it finds the neighbor embeddings more global. - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source-stated For an agent memory of one million passages, the abstract reports that the two maps cross at 150,000 passages, with p = 2.6e-5. > In an agent memory of one million passages, the two maps cross at 150,000 passages (p = 2.6e-5). - [Supporting source](https://arxiv.org/abs/2611.00290) ## Source and contributor context Han, Henry · Article author These are credits stated by the source; they do not establish a profile match or authorship of every passage. ## Source media No suitable media was found by the supported page extraction. ## Terms and references Reference context selected by Escape Meridian; separate from source evidence. Entity mentions do not establish authorship, involvement or endorsement. ## Embedding vector A list of numbers that represents a piece of text, an image or another item so software can compare meanings. - [Definition reference](https://developers.google.com/machine-learning/glossary#embedding-vector) ## Statistical significance A test of whether a result is unlikely to be due to chance alone; it does not say how large or important an effect is. - [Definition reference](https://en.wikipedia.org/wiki/Statistical_significance) ## AI agent An AI system that chooses and carries out actions toward a goal, often using software tools. - [Definition reference](https://www.ibm.com/think/topics/ai-agents) ## arXiv arXiv is a research manuscript repository where authors share papers that may not yet have been peer reviewed. - [Explore this context](/entities/arxiv) - [Reference](https://arxiv.org/) - [Reference](https://info.arxiv.org/help/api/index.html) ## Editorial provenance Version 1. Publication ID: 8872430e-e9a4-4051-8c8a-8981297d0cb9. Prepared by gpt-6-luna; grounding review by gpt-6-luna. Source check: uncertain by anthropic/claude-opus-5.5 on 2026-11-10T06:46:48.019Z. This checks fidelity to the source, not whether its claims are independently true. Editorial selection: accept; reasons: material\_development. Observation: 34f9a6ced2954a547407196e66beac953636ce66a2ee58695f9e3e9c255a8099. Source text hash: a89c8afee3950ae1d876d6efc6d3010015cd36a7c1ee48fb9275aefe522223a5. ### uncertain — 2026-11-10T06:46:48.019Z — anthropic/claude-opus-5.5 Publication: 8872430e-e9a4-4051-8c8a-8981297d0cb9 attribution — pass: The paper is correctly attributed to its sole listed author, Henry Han. Its findings are framed as Han's claims or as reports in the abstract, not as independently established facts. > Authors:Henry Han summary — uncertain: The summary is cut off mid-sentence: 'how well its measures and test apply'. That leaves the closing limitation incomplete. Smallest repair: finish the sentence, for example by saying it remains unknown how well the measures and test apply beyond the reported datasets. Several central terms are also unexplained for a general reader: dimension reduction, t-SNE, UMAP, PCA, block-jackknife p-value, neighbor embeddings and rolled-up manifolds. A brief plain-language gloss of dimension reduction, plus term-library tooltips for the rest, would make the reading understandable on its own. The title and question are faithful. date — pass: The originalPublished date of 2026-10-30 matches the v1 arXiv submission date. It is the first submission, not a later revision or the feed date of Nov 3. > \[Submitted on 30 Oct 2026\] support — pass: Every claim matches an exact excerpt from the abstract. The closing framing, that the method lets readers examine rather than assume which structure a map preserves, is a modest reading of the source's stated purpose. The question and limitations correctly note what the abstract does not establish about generalization. No benefits, costs or stakeholders are invented. > On both retail datasets and both rolled-up manifolds, it finds the neighbor embeddings more global. media — pass: The selected media is the verified direct arXiv abstract page for this paper. That is the most direct access to the source. ## Recorded references Explicit citations in public readings and public file attachments to this reading. Shared topics are not citations. Other artifact types need an explicit reference adapter. No public references recorded here yet. [Reference data (JSON)](/readings/15a8fb745dd124cf600b1fe66e804238d51a0503fe96652d0e4828c5e8a643ae/references.json) ## Corrections and challenges [Send a correction or an opposing source](https://escapemeridian.com/corrections/) [JSON record](https://escapemeridian.com/readings/quantifying-local-and-global-fidelity-of-nonlinear-embeddings-15a8fb74.json)