UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction

• p. 10 Section 2.2

  Teorem 1. Te functors FinReal

• p. 53 Appendix B

  Teorem 2. Te functors FinReal

• p. 10 Section 2.2

  Appendix B provides a full proof

• p. 10 Sec 2.2

  Appendix B provides a full proof of the theorem.

Judge's note: Checked on p.10 and p.53. The two statements are word-for-word identical. There is no other Theorem 2 in the paper. Lemma 1 keeps its number when restated in Appendix A (text lines 1879–1880). The numbering is inconsistent, but the reader can still trace the proof.

Second check: another model agreed

• p. 25 Pen digits description

  Pen digits [1, 10] is a set of 1797 grayscale images

• p. 58 Reference [1]

  Pen-based recognition of handwrit-

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: that the 1797-sample 8x8 digit images are the UCI optical-recognition digits (as shipped in scikit-learn), and not the pen-based trajectory dataset of [1]. The page shows the description and reference [1] as quoted. The authors should confirm which dataset and citation are intended.

Second check: another model agreed

• p. 6 Definition 1 and |Δn| definition

  {1, . . . , n}, with morphims given by (non-strictly) order-preserving maps.

• p. 9 action of Real

  σ−1(0)

• p. 8 Section 2.2, definition of Real

  is simply inherited from Rn+1

• p. 9 Section 2.2, action of Real on morphisms

  Such a map is clearly non-expansive since 0 ≤a ≤b ≤1 implies that

Only the authors can say what is right here.

Judge's note: I read pp.8–9 on the page images. Page 8 says only that the metric on Real(Δ^n_<a) is "inherited from R^{n+1}". On page 9 the map is x ↦ (log b/log a)·(fibre sums), and the justification given covers only the scalar factor. I checked the counterexample myself. Take σ:[2]→[1] with fibres {0,1} and {2}, a=b (so the factor is 1), and s=−log a. Let x=(s/2,s/2,0) and y=(0,0,s); both lie in Real(Δ^2_<a). Their squared Euclidean distance is s²(1/4+1/4+1)=1.5s². Their images are (s,0) and (0,s), at squared distance 2s². So under the Euclidean metric the map expands distances. Under the l1 metric the claim holds by the triangle inequality, so the step is faulty or under-specified, and the intended Real/Sing construction (taken from Spivak) is not affected. The authors should say which metric is meant. The other two points raised in this cluster are adjudicated in C018 (the [n] indexing) and C017 (the 'morphims' typo).

Second check: another model agreed

• p. 7 para before Definition 3

  Let I be the unit interval (0, 1] ⊆R with topology given by intervals

• p. 7

  unit interval (0, 1] ⊆R w

Judge's note: Checked on the page-7 image. The half-open bracket notation itself is correct, so the 'mismatched brackets' point (reviewer note 3) is not a defect. The real defect is that the stated basis of open sets does not consist of subsets of I=(0,1]. The likely intent is (0,a). The same [0,a) notation is carried into P([0,a)) and ([n],[0,a)).

Second check: another model agreed, code check

• p. 11 Definition 9

  dM(Xj, Xk) −ρ

• p. 8 Definition 6, axiom 1

  x = y implies d(x, y) = 0

Judge's note: Checked on the images of p.8 and p.11. The case split does not exclude j=k. With ρ>0, the diagonal entry for i is −ρ, which is negative, and the other diagonal entries are ∞. Neither fits the extended-pseudo-metric axioms that the definition claims to satisfy. The intended reading is clearly off-diagonal pairs, and the fix (set d_i(x,x)=0) is evident, so the construction is unaffected.

Second check: another model agreed

• p. 17 Section 4.1 inputs

  n, the neighborhood size to use for local metric approximation

• p. 11 Definition 9

  be a dataset in Rn

Judge's note: The p.11 crop shows R^n for the ambient space. The p.17 crop shows n as the neighbourhood size. The text has log2(k) at line 630 (Section 3) and log2(n) at lines 781/809 (Section 4). Each use is clear from context, so this is a notation inconsistency rather than an error.

Second check: another model agreed

• p. 20 P(xi) formula

  {b∈A|d0(b)̸=xi} 1 −µ(b)

Only the authors can say what is right here.

Judge's note: The image confirms the ≠ in the denominator's index set. My own check: with two vertices and equal sums S, P(x1)=P(x2)=1, so the total is 2. The text calls this a 'vertex sampling distribution', so a normaliser over all b∈A seems to be what is meant. The qualitative statement (approximately uniform) is unaffected. Only the authors can confirm the intended form.

Second check: another model agreed

• p. 36 Section 5.4, para 5

  Isomap failed to complete for the Shutle, Fashion-MNIST, MNIST and Google-

• p. 38 Table 3, Flow row, Isomap column

  30654s

• p. 38 Table 3, Flow row

  Flow

• p. 36 Section 5.4

  Isomap failed to complete for the Shutle, Fashion-MNIST, MNIST and Google- News datasets

Only the authors can say what is right here.

Judge's note: In Table 3 the Isomap column is '–' for Shuttle, MNIST, F-MNIST, Flow and Google News. Neither the text nor the caption explains why Flow is missing from the list. The quoted '102s' and '30654s' are the UMAP and Eigenmaps values for Flow, not Isomap values.

Second check: another model agreed

• p. 36 Section 5.4, last para

  Timing were performed for the COIL20 [43], COIL100 [44], Shutle [35],

• p. 38 Table 3

  Pen Digits

• p. 36 Section 5.4

  Timing were performed for the COIL20

• p. 36 Sec 5.4

  Timing were performed for the COIL20 [43], COIL100 [44], Shutle [35], MNIST [32], Fashion-MNIST [63], and GoogleNews [41] datasets.

Only the authors can say what is right here.

Judge's note: Compared the p.36 text with the Table 3 rows on p.38. The table has nine datasets and the list names six.

Second check: another model agreed

• p. 43 Figure 11 caption

  Runtime performance scaling of t-SNE and UMAP on various sized

• p. 42 Section 5.4.3

  We benchmarked all four implementations using subsamples of the Google-

• p. 43 Figure 11 caption

  Runtime performance scaling of t-SNE and UMAP on various sized sub-samples

Judge's note: Checked the p.43 figure, legend and caption, and the p.42 text. The caption is incomplete rather than wrong.

Second check: another model agreed

• p. 2 Introduction, layout paragraph

  where made in developing the algorithm

• p. 2

  the choices that where made in developing the algorithm

Judge's note: The full page image of p.2 shows 'where made'.

Second check: none (judge only)

• p. 2 Introduction, last paragraph

  due to plethora of proposed objec-

Judge's note: The last line of p.2 in the page image reads 'due to plethora of proposed objec-'.

Second check: none (judge only)

• p. 2 Introduction

  Isomap [56]

• p. 36 Section 5.4

  Isomap [55]

• p. 63 References [59], [60]

  [60] Laurens van der Maaten and Geoffrey Hinton.

Judge's note: p.63 shows [59] and [60] as the same work with different capitalisation. [55] (Tenenbaum NIPS 1998) and [56] (Tenenbaum et al., Science 2000) are both Isomap papers, so each citation is defensible on its own. The real defect is the duplicate reference entry; the inconsistent Isomap citation is a minor point of consistency.

Second check: none (judge only)

• p. 63 References [59], [60]

  Visualizing data using

• p. 2 Introduction

  t-SNE [59, 58]

• p. 36 Section 5.4

  The Isomap [55] and Laplacian Eigenmaps [7]

• p. 2 Introduction

  Isomap [56]

Judge's note: I checked the p.63 image: [59] and [60] are the same authors, title, journal, volume, pages and year, so this is a duplicate entry. The p.2 crop shows "t-SNE [59, 58]", and the text layer shows "[60, 58]" at Section 5.1. On the Isomap point, the p.62 image shows two distinct Tenenbaum works: [55] (NIPS 10, 1998) and [56] (Science, 2000). Citing either for Isomap is defensible, so that part is at most an inconsistency of choice and is not confirmed as an error.

Second check: none (judge only)

• p. 3 Introduction

  pracitioners to apply a new field of

• p. 3 Introduction

  weakenesses

• p. 3 Introduction, first paragraph

  pracitioners

• p. 3 Introduction

  allowing pracitioners to apply a new field of mathemtaics

Judge's note: The p.3 page image shows all four misspellings. 'litle' is an extraction artefact: the image reads 'little'.

Second check: none (judge only)

• p. 3 Introduction, top of page

  UMAPs design decisions

Judge's note: The page image of p.3 shows 'UMAPs design decisions'.

Second check: none (judge only)

• p. 4 Section 2, first line

  UMAP, t-SNE and Largeviz located in Appendix C

Judge's note: The crop shows 'Largeviz'. The method is called LargeVis everywhere else in the paper, e.g. p.2 and p.3.

Second check: none (judge only)

• p. 6 Definition 1

  with morphims given by

• p. 14 Section 3.1

  UMAP implemenation

• p. 14 Section 3.1

  our UMAP implemenation

• p. 14

  our UMAP implemenation we prefer

Judge's note: Both misspellings are visible on the page images of p.6 and p.14.

Second check: none (judge only)

• p. 14 Section 3

  an symmetrization function can be found in Section 2.2

Judge's note: The page-14 crop shows "an symmetrization function". The intended word is clearly "and". The preceding sentence repeats itself: "We theoretically justify using the choice of using a k-neighbour graph".

Second check: none (judge only)

• p. 22 Section 4.2

  Laplace-Betrami

Judge's note: The image shows 'Laplace-Betrami' on p.22. The text layer has 'Beltrami operator' at line 717.

Second check: none (judge only)

• p. 25 Section 4.3

  In Figure 2 we provides examples of the same hyperparamter choices

• p. 25 Section 4.3

  have signifcant cluster structure

• p. 23 Section 4.3

  avoiding potential overploting issues

Judge's note: The p.25 image confirms 'we provides' and 'hyperparamter'. The image shows 'significant' (p.25) and 'overplotting' (p.23) spelled correctly, so 'signifcant' and 'overploting' are ligature artefacts of the text layer and are not part of this finding.

Second check: none (judge only)

• p. 35 Section 5.3

  due its large size

• p. 35 Section 5.3

  signifcantly more stable results

• p. 25 Section 4.3, MNIST paragraph

  signifcant cluster structure

• p. 35 Section 5.3

  UMAP has signifcantly more stable results

Judge's note: The images show 'signifcantly' and 'signifcant' as printed. The same pages print 'significantly' and 'significant' correctly elsewhere, so this is not a ligature artefact. 'due its' is also visible on p.35.

Second check: none (judge only)

• p. 28 Flow cytometry dataset description

  CDT4 cells comprised of 1,000,000 samples

• p. 28 Section 5, Flow cytometry

  CDT4 cells

• p. 28 Section 5 dataset list

  measurements of CDT4 cells

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: the standard cell-type name is CD4 (CD4+ T cells), and no cell type 'CDT4' exists. The image shows 'CDT4' as printed, and the paper never defines it.

Second check: none (judge only)

• p. 31

  perfomance in embedding quality

Judge's note: Seen on the p.31 page image.

Second check: none (judge only)

• p. 32 Table 2, MNIST k=1600, PCA

  0.467 (± 0.0233)

Judge's note: Seen on the p.32 image. This is a formatting inconsistency only.

Second check: none (judge only)

• p. 33 Section 5.3

  Pro-

• p. 37 Figure 8 y-axis and caption

  average Procustes distance per point

• p. 37

  Comparison of average Procustes distance per point

Judge's note: On p.37 both the axis label and the caption read 'Procustes'. The p.33 text spells it correctly ('Pro-crustes', hyphenated across the line break), so that location is not an error.

Second check: none (judge only)

• p. 33 Section 5.3

  variety of uses cases

Judge's note: Seen on the p.33 image.

Second check: none (judge only)

• p. 36 Section 5.4

  where subsampled or had their dimension reduced by PCA

• p. 38 Table 3 caption

  where sub-

• p. 38 Table 3 caption

  some of the datasets where sub-sampled

• p. 36 Section 5.4

  some of the datasets where subsampled

Judge's note: Seen on the p.36 and p.38 images.

Second check: none (judge only)

• p. 36 Section 5.4, FIt-SNE paragraph

  MulticoreTNSE is an optimized C++ implementation

• p. 36 Section 5.4

  like MulticoreTNSE is an optimized C++ implementation

Judge's note: Seen on the p.36 image. The same page spells it 'MulticoreTSNE' twice.

Second check: none (judge only)

• p. 45 Section 6

  niques such a Non-Negative Matrix Factorization

Judge's note: Seen on the crop image.

Second check: none (judge only)

• p. 49 Section 6

  region region UMAP would atempt

• p. 49 Section 6

  another region region UMAP would

• p. 56 Appendix C

  strength of the the repulsive contributions

• p. 49

  region region UMAP would

Judge's note: Both repetitions are visible on the images. 'atempt' in the quote is a ligature artefact; the image reads 'attempt'.

Second check: code check

• p. 49 Section 6 Weaknesses, paragraph on knn-graph

  ”reverse-nearest-

Judge's note: The crop of p.49 shows a right double quote both before 'reverse' and after 'neighbors'. This is the usual LaTeX quoting slip and is purely cosmetic.

Second check: none (judge only)

• p. 49 Section 6 Weaknesses, paragraph on metric structure

  mean that

Judge's note: The image of p.49 confirms the wording. The head of the subject is the singular 'This', so the verb should be 'means'.

Second check: none (judge only)

• p. 50 Section 7

  hampered by the lack clear

Judge's note: The crop of p.50 shows the line exactly as quoted. A word is missing.

Second check: none (judge only)

• p. 52 Eq. (3)

  ⟨∥˙c(t), ˙c(t)∥dt

Judge's note: I checked the page 52 image. The radical begins after the opening ⟨. The second line opens ⟨ and then ‖ and closes only with ‖. With g = I/r^2, the integrand √g(ċ,ċ) equals (1/r)√⟨ċ,ċ⟩ = (1/r)‖ċ‖, so the intended chain is clear and the problem is typographic.

Second check: none (judge only)

• p. 53 Appendix B, FinReal(X) colimit

  F([n]).

• p. 53 Appendix B, explicit colimit for FinReal

  y([n],[0,a))→X F([n]).

• p. 53

  y([n],[0,a))→X F

Judge's note: The page 53 image shows F defined as F([n],[0,a)) and the next line using F([n],[0,a)), while the colimit writes only F([n]). The meaning is recoverable from context. The sub-claim about mismatched brackets in '[0,a)' is not a defect, because it is half-open interval notation.

Second check: code check

• p. 54 Appendix C, t-SNE stage 1

  a pair-wise similarity, vij, is calculated

Judge's note: The page 54 image shows 'Quantities with the subscript ij are symmetric' and 'j|i indicates an asymmetric similarity', followed by 'a pair-wise similarity, v_ij,' and then Eq. (5) 'v_{j|i} = exp(...)'. The notation is inconsistent within the paragraph, but the meaning is clear from Eq. (5).

Second check: none (judge only)

• p. 56 Appendix C, after Eq. 12

  the strength of the the repulsive contributions

• p. 56

  weights the strength of the the repulsive contributions

• p. 56

  which weights the strength of the the repulsive contribut

Judge's note: The page 56 crop shows 'of the the repulsive' as printed.

Second check: code check

• p. 63 Reference [64]

  Michigan State Universiy

• p. 58 Ref [10]

  Peter Pretenhofer

• p. 63

  Michigan State Universiy, 2(2):4, 2006.

Judge's note: I checked the p.63 image, and it reads "Universiy", so that part is confirmed. Reviewer note 3 also claimed [10] reads "Pretenhofer". The p.58 image shows "Peter Prettenhofer", spelled correctly, so that part is rejected as a misreading.

Second check: none (judge only)

• p. 59 Reference [18]

  Wei Dong, Charikar Moses, and Kai Li.

• p. 59 Reference [18]

  Wei Dong, Charikar Moses, and Kai Li

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: that the NN-descent co-author is named Moses Charikar (given name Moses). The page 59 crop shows 'Charikar Moses' as quoted, while every other entry is in 'Given Family' order.

Second check: none (judge only)

• p. 60 Ref [21]

  constructing classifier decision boundary maps.

• p. 61 Refs [43], [44]

  Columbia object image library (coil-20.

Judge's note: I checked p.60 in the image. Entry [21] ends after the title, with no venue or year, so the defect is confirmed. This cluster also raised the COIL entries [43]/[44], which I checked on the p.61 image. Those defects are real but are adjudicated under C077, so this verdict covers only [21]. A cosmetic bibliographic defect.

Second check: none (judge only)

• p. 61 References [43], [44]

  object image library (coil-100. Technical report, 1996.

• p. 61 Reference [43]

  image library (coil-20. Technical report, 1996.

• p. 61 Refs [43],[44]

  Columbia object image library (coil-20. Technical report

• p. 61 Ref [44]

  object image library (coil-100.

Judge's note: I checked the p.61 image. Both entries open "(coil-" and never close it, and [44] begins "object image library" with no "Columbia". The [21] and [59]/[60] points raised in reviewer note 3 are handled in C076 and C078. A cosmetic defect in the reference list.

Second check: none (judge only)

• p. 63 Reference [65]

  Lofti A Zadeh

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: that the author's first name is spelled Lotfi. The p.63 image reads "Lofti A Zadeh". This is not a ligature artefact, because the glyph order in the image is L-o-f-t-i.

Second check: none (judge only)