Visualizing Data using t-SNE
2008 · paper · AI and machine learning
Presents t-SNE, a technique that maps high-dimensional data to two or three dimensions, a variation of Stochastic Neighbor Embedding that reduces crowding of points. The authors report it produces significantly better visualizations than other techniques on almost all datasets tested.
Checked: PDF from jmlr.org, fetched 4 Oct 2026. Page numbers are those of that PDF.
3 findings
Substance The momentum update in Section 2 (p.4) and Algorithm 1 (p.9), "Y(t) = Y(t−1) + η δC/δY + α(t)(Y(t−1) − Y(t−2))", adds the gradient, which is a gradient-ascent step; a descent step needs −η δC/δY.
- p. 4 Section 2, momentum update equation
Y (t) = Y (t−1) +η δC
- p. 9 Algorithm 1, update step
set Y (t) = Y (t−1) +η δC
Judge's note: Both images show '+η δC/δY'. Eq. 5 (p.8) gives δC/δyi = 4Σ(pij−qij)(yi−yj)(1+‖yi−yj‖²)^−1, which is the plain derivative of the KL cost (derived in Appendix A). Section 3.4 (p.10) says η is initially set to 100, so η is positive. With these definitions the printed update increases C, which contradicts the stated aim of minimising C by gradient descent. No erratum is among the files.
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Wording Appendix A (p.23) defines dij = ‖yi−yj‖ but writes Eq. 7 as δC/δyi = 2Σj (δC/δdij)(yi−yj), without the factor 1/dij, and gives δC/δdij = 2(pij−qij)(1+dij²)^−1, without the factor dij; the two omissions cancel, so Eq. 5 is unaffected.
- p. 23 Appendix A, Eq. 7
= 2∑ j δC δdi j (yi −y j).
- p. 23 Appendix A, expression for δC/δdij
δC δdi j = 2 pi j qi jZ (1+d2 i j)−2
Only the authors can say what is right here.
Judge's note: The text layer (line 975) and the image confirm dij = ‖yi−yj‖, not its square. My own working: ∂dij/∂yi = (yi−yj)/dij, so Eq. 7 should read 2Σ(δC/δdij)(yi−yj)/dij. Also ∂(1+d²)^−1/∂d = −2d(1+d²)^−2, so the first term of δC/δdij is 2(pij/(qijZ))·dij·(1+dij²)^−2, and the simplified form is 2(pij−qij)·dij·(1+dij²)^−1. The printed forms leave out dij in one place and 1/dij in the other. Multiplying them still gives 4Σ(pij−qij)(1+dij²)^−1(yi−yj), so the final gradient Eq. 5 is correct. The intermediate steps are false as printed, but the stated result stands.
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Wording Appendix B (p.24): "LN = CCᵀ, where C is an upper-triangular matrix ... solving the linear systems Cy = −Bᵀ and CxN = y" — with LN = CCᵀ, the second solve must use Cᵀ, not C.
- p. 24 Appendix B, last paragraph
such that LN =CCT, where C is an upper-triangular matrix
- p. 24 Appendix B, last paragraph
Cy = −BT and CxN = y
Only the authors can say what is right here.
Judge's note: My own working: LN x = b ⇔ C(Cᵀx) = b, so the solves are Cy = b and then Cᵀx = y. Counterexample: take the upper-triangular C = [[1,1],[0,1]], so CCᵀ = [[2,1],[1,1]], and b = (1,0). Solving Cy = b gives y = (1,0); solving Cx = y gives x = (1,0); but LN x = (2,1) ≠ b. The printed procedure therefore does not solve Eq. 8. With an upper-triangular factor, the usual form is LN = CᵀC with solves Cᵀy = b and Cx = y. The printed combination fits neither convention. Eq. 8 itself is unaffected.
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11 minor (typos and the like; most checked by the judge only)
Reference or label Song et al. reference (p.26): "Advances in Neural Information Processing Systems, volume 21 (in press), 2007", while the same list gives NIPS volume 19 the year 2007 (Weinberger et al., p.27).
- p. 26 Reference list, Song et al.
volume 21 (in press), 2007
- p. 27 Reference list, Weinberger et al. 2007
volume 19, 2007
- p. 26 Reference list, Song et al.
Advances in Neural Information Processing Systems, volume 21 (in press), 2007
- p. 27 Reference list, Weinberger et al. 2007
Advances in Neural Information Processing Systems, volume 19, 2007
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that Colored MVU appeared at NIPS 2007, which is volume 20. Within the files, the p.26 and p.27 images confirm that volume 21 and volume 19 are both dated 2007, and volumes 14–15 are dated 2001–2003, so the volume/year pairing for Song et al. is internally inconsistent. The entry is marked 'in press'. Only the authors can confirm which of the volume number or the year is intended.
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Wording σi is called "the variance of the Gaussian" (p.3; also footnotes 2–3), but Eq. 1 uses exp(−‖xi−xj‖²/2σi²), so σi is the standard deviation; the low-dimensional "variance ... to 1/√2" is likewise the standard deviation (variance 1/2); Section 6.1 says "standard deviation of the Gaussian".
- p. 3 Text after Eq. 1
where σi is the variance of the Gaussian that is centered on datapoint xi
- p. 3 Text before q_j|i equation
We set2 the variance of the Gaussian that is employed in the computation of the conditional probabilities q j|i to
- p. 19 Section 6.1, second paragraph
relative to the standard deviation of the Gaussian
Judge's note: Checked Eq. 1 and the q_j|i equation on the p.3 image, footnotes 2–3 (p.3–4) and Section 6.1 (p.19). exp(−‖yi−yj‖²) = exp(−d²/2σ²) requires 2σ² = 1, so σ² = 1/2 and σ = 1/√2 ≈ 0.707. The value 1/√2 fits only the standard-deviation reading, and Section 6.1 uses 'standard deviation' for the same kernel width. The equations themselves are unambiguous, so this is a terminology inconsistency, not a computational error.
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Typo Footnote 1 (p.3): "provided these simiarities can be interpreted as conditional probabilities" — 'simiarities' should be 'similarities'.
- p. 3 Footnote 1
provided these simiarities can be interpreted
- p. 3 Footnote 1
provided these simiarities can be interpreted as conditional probabilities
Judge's note: The misspelling is visible in the footnote on the p.3 page image crop.
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Typo Section 3.4 (p.10) writes the momentum as α^(t) ("α^(t) = 0.5 for t < 250 and α^(t) = 0.8"), whereas Section 2 and Algorithm 1 write α(t).
- p. 10 Section 3.4, final paragraph
was set to α(t) = 0.5 for t < 250
- p. 5 Section 2
α(t) represents the momentum at iteration t
Judge's note: The p.10 image shows α with a superscript (t). The p.4 and p.9 images show α(t). The meaning is unchanged; this is a notation inconsistency only.
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Typo Figure 1(a) and 1(b) (p.8): the colour-bar tick labels are cut off at the right edge (e.g. '1·', '1:', '1(' for the two-digit values), so the upper end of the scale cannot be read.
- p. 8 Figure 1(a),(b) colour bars
(a) Gradient of SNE.
Judge's note: The full p.8 image and crop show truncated two-digit tick labels on the colour bars of panels (a) and (b); panel (c) is fine. The text gives the largest attraction as approximately 19, but this cannot be read off the truncated scale. Presentation only.
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Typo Section 3.4 (p.10): "The number of gradient descent iterations T was set 1000" is missing 'to' ('was set to 1000').
- p. 10 Section 3.4, last paragraph
The number of gradient descent iterations T was set 1000
Judge's note: Confirmed on the p.10 image crop.
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Typo Appendix B (p.24): the expansion of D[xN] contains the cross term "2xNᵀBᵀxM"; the subscript M is never defined, and the term should be 2xNᵀBᵀxL.
- p. 24 Appendix B, expansion of D[x_N]
2xT NBTxM
Judge's note: The p.24 image shows x_M, so this is not an extraction artefact. The text defines only the subscripts L (landmark) and N (non-landmark). Expanding the block quadratic form gives xLᵀBxN + xNᵀBᵀxL = 2xNᵀBᵀxL, because both terms are the same scalar.
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Typo Appendix B (p.24): "the other landmarks points are set to zero" should read 'landmark points'.
- p. 24 Appendix B, first paragraph
the other landmarks points are set to zero
Judge's note: Confirmed on the p.24 image crop.
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Typo Appendix B (p.24): "we may reorder the landmark points such that the landmark points come first" is circular; what is reordered is the set of all points (nodes).
- p. 24 Appendix B
we may reorder the landmark points such that the landmark points come first
Judge's note: Confirmed on the p.24 image. The intended meaning (reorder the points so that the landmarks come first) can be recovered; this is a wording slip.
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Typo The Meytlis and Sirovich (2007) reference gives the journal as "IEEE Transactions of Pattern Analysis and Machine Intelligence"; the paper's other entries for this journal use "on".
- p. 26 Reference: Meytlis and Sirovich 2007
IEEE Transactions of Pattern Analysis and Machine Intelligence
Judge's note: In the crop and on page 26, the Meytlis & Sirovich entry reads 'IEEE Transactions of Pattern Analysis and Machine Intelligence'. The Grady entry (text line 1157) and the Lafon & Lee entry (text line 1176) in the same reference list both read 'IEEE Transactions on Pattern Analysis'. The paper is inconsistent with itself, so this needs no outside knowledge. It is a cosmetic slip in a reference.
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Typo The Weinberger, Sha and Saul (2004) reference misspells Conference: "Proceedings of the 21st International Confernence on Machine Learning".
- p. 27 Reference: Weinberger, Sha and Saul 2004
International Confernence on Machine Learning
- p. 27
21st International Confernence on Machine Learning
Judge's note: Both crops of page 27 show 'Confernence' in italics in the typeset Weinberger, Sha & Saul (2004) entry. It is a misspelling in the published PDF, not an extraction artefact.
Second check: none (judge only)