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REVIEW 3 major objections 5 minor 2 cited by

A Survey of Dimension Estimation Methods

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read No single dimension estimator works across all datasets; most need careful tuning and many overfit benchmarks.

desk verdict A useful, honest benchmark of dimension estimators whose 'overfitting' claim overstates the evidence: it measures hyperparameter sensitivity, not held-out generalization, but the empirical work is solid and deserves peer review. read the letter →

arxiv 2507.13887 v1 pith:LUSFVIHX submitted 2025-07-18 stat.ML cs.LGmath.DGmath.MGmath.STstat.TH

James A. D. Binnie , Pawe{l} D{l}otko , John Harvey , Jakub Malinowski , Ka Man Yim This is my paper · ORCID
classification stat.MLcs.LGmath.DGmath.MGmath.STstat.TH
keywords intrinsicdimensionestimationbenchmarkcomparisonhyperparametersensitivitycurvatureandnoiserobustnesstangentialestimatorsparametrictopologicaldataanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This survey asks a practical question: when a dataset lives on a low-dimensional structure inside a high-dimensional space, how reliably can existing estimators recover that intrinsic dimension in real conditions? The authors review and categorize the main estimation methods by the geometric information they exploit, then test them on benchmark manifolds. Their central finding is that there is no single estimator or set of hyperparameters that performs well across all settings, and that choosing the best hyperparameters for benchmark datasets frequently amounts to overfitting, so many estimators may not generalize beyond the data they were tuned on.

What carries the argument

The central organizing framework is the categorization of estimators by the geometric information they use: tangential estimators that detect a local affine tangent structure, parametric estimators that rely on dimension-dependent probability distributions of distances or angles, and estimators based on topological or metric invariants such as minimum spanning trees, k-nearest-neighbour graphs, and magnitude. This categorization is what allows the authors to explain why different estimators fail differently on curvature, noise, boundary effects, and high dimensional throttling, where the neighbourhood size k sets a hard ceiling on the dimension an estimator can return.

What would settle it

A concrete falsifier would be to find a single estimator with a fixed hyperparameter setting that performs accurately on the same benchmark manifolds across a wide range of intrinsic dimensions, curvatures, noise levels, and out-of-distribution datasets, contradicting the paper's claim that no universal estimator or hyperparameter set exists.

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Extended reading notes

Core claim

The paper's core claim is stated in its conclusion: "There is no single estimator or set of hyperparameters that can perform well across all settings." This is established empirically by benchmarking a wide cohort of estimators on synthetic manifolds, and by showing that the performance of most estimators depends heavily on the choice of hyperparameters, with the best choice often varying from dataset to dataset. The authors also find that most estimators underestimate intrinsic dimension on high-dimensional datasets, while overestimation can occur on datasets such as SO(n), and that no estimator performs satisfactorily on non-linear manifolds of dimension above six.

Load-bearing premise

The evaluation assumes that the chosen benchmark manifolds and the range of hyperparameters tested represent how dimension estimators will actually be used in practice.

Editorial extensions

If this is right

  • Practitioners should not trust a single dimension estimator with default hyperparameters and should instead use a range of estimators and hyperparameters to assess how sensitive the inferred dimension is to modelling assumptions.
  • Benchmark results obtained by tuning hyperparameters to the test set should not be interpreted as a predictor of real-world performance, because the paper shows that overfitting to benchmark datasets is frequent.
  • On high-dimensional manifolds, tangential estimators face a throttling limit that grows with dimension, and slope-inference estimators can even produce negative estimates, so caution is required.
  • For practitioners with limited data, global estimators are generally more suitable, while local estimators require more samples because each neighbourhood must contain enough points.
  • Future work on automated or adaptive hyperparameter selection is needed, since the paper finds that no simple rule performs well universally.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's conclusion implies that the apparent accuracy of an estimator on a benchmark should be read as a statement about that benchmark and that tuning process, not about the estimator alone.
  • A systematic rule for choosing hyperparameters, for instance based on features of the data before estimation, would be a direct extension of this work and could be tested on the same benchmark suite.
  • The finding that slightly positively curved surfaces can be easier for lPCA than flat ones suggests a testable extension: whether this bias generalizes to other tangential estimators or depends on the specific aggregation and thresholding.
  • The paper's evidence that estimators differ greatly in variance across random samples implies that reporting a single number for intrinsic dimension without uncertainty is misleading; interval estimates could be derived from running estimators across hyperparameters and subsamples.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper surveys intrinsic dimension estimation methods, organizing them into tangential, parametric, and topological/metric families, and extends the scikit-dimension package with new estimators. The empirical core benchmarks thirteen estimators on synthetic manifolds with intrinsic dimensions 1-70 and ambient dimensions 3-96, using four sample sizes, twenty repeated point sets per condition, and multiple hyperparameter choices. The authors report estimator-specific tables, noise and curvature experiments, a throttling analysis, and qualitative assessments. The central claims are that no single estimator or hyperparameter setting works across all settings and that frequent 'overfitting' in hyperparameter identification suggests limited generalization.

Significance. If the results are read as a sensitivity analysis of the tested estimators within the stated hyperparameter grids, the paper is a substantial and useful contribution. The benchmark is unusually careful: twenty repetitions, standard deviations, four sample sizes, and transparent tables; the authors also disclose in Appendix C that the 'best' hyperparameter is selected with knowledge of the true dimension. The code contribution to scikit-dimension is valuable. The main weakness is interpretive: the abstract and Section 3.3 present the oracle-based sensitivity comparison as evidence of 'overfitting' and poor generalization, which is not what the protocol measures.

major comments (3)
  1. [Abstract; Section 3.3; Appendix C] The claim that 'overfitting is frequent' and that 'many estimators may not generalise well beyond the datasets on which they have been tested' is not supported by the experimental protocol. In Appendix C, the 'best' hyperparameter for estimator E on manifold M is selected as the minimizer of |d_hat(E,M,H) - d(M)| and the reported performance uses the same 20 point sets used for selection; the 'med abs' and 'med rel' columns use a hyperparameter chosen by minimizing median absolute or relative error against the true dimensions over the same benchmark manifolds. This measures hyperparameter sensitivity and the tension between per-dataset and global choices, not generalization to new samples or new manifolds. To support the abstract's wording, the authors should add a held-out evaluation: tune H on one collection of samples and evaluate on independent samples from the same manifold, and apply H_best(E,M1) to other benchmark manifolds. Without such a test, the evidence supports only 'no single hyperparameter setting is optimal across benchmarks'.
  2. [Table 3; Appendix C] The qualitative column 'No tailoring of params' in Table 3 is based on the 'med abs' and 'med rel' hyperparameter choices, which are themselves obtained by minimizing the median absolute or relative error against the true dimensions across the benchmark manifolds (Appendix C). This is not an untuned or default setting; it is a globally oracle-tuned fixed parameter. The ticks in Table 3 therefore overstate the 'no tailoring' property. Please either use the scikit-dimension default parameters as the fixed baseline, or rename the column to something like 'insensitivity to a global hyperparameter choice' and adjust the discussion.
  3. [Section 3.3; Table C1] The statement in Section 3.3 that lPCA 'appears to give the correct dimension every time' is contradicted by Table C1, where the best-hyperparameter estimate for M10d Cubic (true dimension 70) is 55.2 at every sample size. The caveat about hyperparameter tuning does not resolve the contradiction, because the quoted value is already the best over the grid. Please correct the claim or restrict it to the datasets on which lPCA actually succeeds.
minor comments (5)
  1. [Abstract; Remark 3.2] The abstract's generalizations should carry the qualifier that the empirical conclusions are relative to the hyperparameter grids listed in Appendix D, as the authors themselves acknowledge in Remark 3.2; as written, 'overfitting is frequent' reads as a statement about all hyperparameter ranges.
  2. [Section 2.3.1] The reference to 'eq. (14)' in the paragraph after the Costa and Hero discussion is a forward reference to Theorem 2.5; please renumber or point to eq. (10), where the Renyi-entropy connection is first visible.
  3. [Section 3.4.2] The WODCap throttling calculation would benefit from a cleaner derivation: the displayed expression for S(d) mixes Gamma-function notation and the resulting approximation for k is hard to parse; in particular, the notation involving Gamma((d+1)/2, 1/2) should presumably be Gamma((d+1)/2).
  4. [Section 4; Algorithm 1; Table C10] There are several typographical errors: 'is is' in Section 4, 'Datas set' in Algorithm 1, 'M20 Norm' in the Table C10 caption, and 'Theorem 2.6' in Figure 6 and Section 3.3 where Remark 2.6 is meant.
  5. [Table 2] The row for M2/M9 ('Affine 3to5 and M9 Affine 3, 20 5, 20') is hard to read; please reformat the table or caption to make the manifold, dimension, and ambient dimension columns unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is an empirical benchmark survey whose conclusions are measurements, not derivations from fitted inputs; the 'overfitting' inference is under-supported but not circular.

full rationale

This is a survey and benchmark study, not a derivation chain. The central empirical claims - that no single estimator or hyperparameter set dominates, and that estimators are sensitive to hyperparameter choice - are measured quantities reported from experiments on standard benchmark manifolds, not quantities derived from fitted inputs. The oracle-tuning protocol in Appendix C selects H_best(E,M) = argmin_H |dhat(E,M,H) - d(M)| and compares it with a fixed H_abs chosen across manifolds; this measures per-dataset hyperparameter sensitivity and the tension between per-dataset and global tuning. It is a measurement protocol, and the paper does not present the resulting best-case performance as a prediction about new data. The abstract's wording that 'overfitting is frequent, indicating that many estimators may not generalise well beyond the datasets on which they have been tested' goes beyond what the in-sample protocol can establish - no held-out evaluation is performed - but that is an evidentiary weakness, not circularity: the inference is not forced by the construction, it is an unverified extrapolation. Remark 3.2 itself acknowledges that 'our empirical study is subject to our particular choices of hyperparameter ranges,' explicitly flagging the scope limitation. Theoretical results (Theorems 2.1, 2.2, 2.5, and eqs. (6)-(9)) are imported from external literature (Kleindessner and von Luxburg, Beardwood-Halton-Hammersley, Steele, Schweinhart, Kozma-Lotker-Stupp, Yukich), not from the authors' own prior work; I find no load-bearing self-citations. The survey's taxonomy (tangential, parametric, topological/metric) is presented as an organizing classification, not as a new unification, and it does not rename a known result. The throttling and error-propagation analyses in Sections 3.4.2 and 2.6 are derived algebraically from the estimators' published definitions. Overall, the paper is self-contained as an empirical study; the 'overfitting' claim would need a held-out test to be fully supported, but no step of the paper's argument reduces by construction to its own inputs.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. Its central claim is empirical and rests on the manifold hypothesis, on classical asymptotic results for Euclidean functionals, and on the representativeness of the chosen benchmark manifolds and hyperparameter grids. The only hand-chosen quantities are the experimental hyperparameter ranges, which the authors explicitly flag as finite and literature-guided.

free parameters (1)
  • Hyperparameter grids and defaults per estimator = Varied per method; ranges listed in Appendix D
    The comparison between best, median-absolute, and median-relative performance depends on these ranges. The authors note in Remark 3.2 that different ranges could change conclusions, so this is a hand-chosen experimental design element rather than a fitted constant in a model.
assumptions (3)
  • domain assumption The finite sample X is drawn i.i.d. from a distribution supported on a compact smooth submanifold M^d of R^D.
    Section 1.4 defines this setup for all estimators. If real data do not lie near such a manifold, the notion of intrinsic dimension and the benchmark conclusions may not transfer.
  • standard math Classical asymptotic results for Euclidean functionals correctly describe MST and kNN graph scaling used by PH0 and KNN estimators.
    Invoked in Theorems 2.2 and 2.5 and the surrounding text in Section 2.3. The paper does not re-derive these and relies on them to justify the slope-inference estimators.
  • domain assumption The standard scikit-dimension benchmark manifolds with known intrinsic dimension provide a representative testbed for practical dimension estimation.
    Section 3.2 describes the datasets and Remark 3.2 acknowledges the finite hyperparameter ranges. The paper's central overfitting claim is measured against this chosen set of manifolds.

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Cite this review

Pith. "Pith review of A Survey of Dimension Estimation Methods." pith.science (2026). https://pith.science/paper/LUSFVIHX

@misc{pith2026250713887,
  author       = {Pith},
  title        = {Pith review of: A Survey of Dimension Estimation Methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LUSFVIHX}},
  note         = {Machine review of arXiv:2507.13887}
}
read the original abstract

It is a standard assumption that datasets in high dimension have an internal structure which means that they in fact lie on, or near, subsets of a lower dimension. In many instances it is important to understand the real dimension of the data, hence the complexity of the dataset at hand. A great variety of dimension estimators have been developed to find the intrinsic dimension of the data but there is little guidance on how to reliably use these estimators. This survey reviews a wide range of dimension estimation methods, categorising them by the geometric information they exploit: tangential estimators which detect a local affine structure; parametric estimators which rely on dimension-dependent probability distributions; and estimators which use topological or metric invariants. The paper evaluates the performance of these methods, as well as investigating varying responses to curvature and noise. Key issues addressed include robustness to hyperparameter selection, sample size requirements, accuracy in high dimensions, precision, and performance on non-linear geometries. In identifying the best hyperparameters for benchmark datasets, overfitting is frequent, indicating that many estimators may not generalise well beyond the datasets on which they have been tested.

Figures

Figures reproduced from arXiv: 2507.13887 by the authors.

Figure 1
Figure 1. Example of a cover and a refinement. There are points on the circle which are contained in three [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. On the left the idea behind the doubling property estimator is represented. The ratio of the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. The best estimates from our benchmark tables for 625, 1250, 2500 and 5000 points for each [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Standard deviation of dimension estimates of M1Sphere (uniform samples on [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: Comparison of dimension estimates of SO(4) and S 6 for estimators with a k-nearest neighbour parameter. The input data consists of 2500 points uniformly sampled on the manifolds, which have intrinsic dimension 6. The k parameter is varied from 4 to 50 in steps of 2. Fo…
Figure 6
Figure 6. Figure 6: Performance of KNN-estimator vs k on benchmark manifolds, for 5000 samples on each manifold. We show the empirical median dimension estimate, and the interquartile range, over 20 sets of random samples. Note the different scaling of the y-axis from dataset to dataset. …
Figure 7
Figure 7. Figure 7: Dimension estimates from lPCA when neighbourhood sizes are varied from 4 to 100, for S 6 ⊂ R 7 . Colours indicate different threshold methods. Solid curves correspond to estimates using knn neighbour￾hoods, dashed curves correspond to using ϵ-neighbourhoods, where for …
Figure 8
Figure 8. Figure 8: Dimension estimates from lPCA when neighbourhood sizes are varied from 4 to 100, for SO(4) ⊂ R 16. Colours indicate different threshold methods. Solid curves correspond to estimates using knn neighbourhoods, dashed curves correspond to using ϵ-neighbourhoods, where for…
Figure 9
Figure 9. Figure 9: We consider 5000 points on M6 Nonlinear and fix the neighbourhood size at 50. Searching a range of values for the hyperparameter α in both alphaFO and alphaRatio, we observe that the methods mirror each other. Both can produce the correct dimension, 6, but only for a n…
Figure 10
Figure 10. Figure 10: Left: We compare the average performance over 20 runs for the best performing hyperparamenters [PITH_FULL_IMAGE:figures/full_fig_p032_10.png]
Figure 11
Figure 11. Figure 11: Demonstration of how curvature affects PCA when points are drawn from a non-linear curve in R 2 . By comparison, using lPCA mitigates the effect of curvature. If the neighbourhoods are chosen too large lPCA will also begin to struggle. Consider, for example, the perfo…
Figure 12
Figure 12. Figure 12: Points drawn from an elliptic paraboloid. The curvature at the origin is positive. [PITH_FULL_IMAGE:figures/full_fig_p034_12.png]
Figure 13
Figure 13. Figure 13: Points drawn from a hyperbolic paraboloid. The curvature at the origin is negative. [PITH_FULL_IMAGE:figures/full_fig_p034_13.png]
Figure 14
Figure 14. Figure 14: Sample from a torus. Red points having negative Gauss curvature, blue points have positive [PITH_FULL_IMAGE:figures/full_fig_p035_14.png]
Figure 15
Figure 15. Figure 15: Both figures show the same trend. As curvature increases from negative to positive, the number [PITH_FULL_IMAGE:figures/full_fig_p035_15.png]
Figure 16
Figure 16. Figure 16: Average pointwise dimension at (0,0,0) for [PITH_FULL_IMAGE:figures/full_fig_p036_16.png]
Figure 17
Figure 17. Figure 17: For MLE there is a trend to underestimate as either κ2 or the neighbourhood size k is increased. The standard deviation stays reasonably constant for changes of κ2, while it decreases with k [PITH_FULL_IMAGE:figures/full_fig_p036_17.png]
Figure 18
Figure 18. Figure 18: For lPCA we recover the same trend in the estimate with changes in κ2 which we saw in [PITH_FULL_IMAGE:figures/full_fig_p036_18.png]
Figure 19
Figure 19. Figure 19: Using a torus embedded in R 3 as an example, we investigate the tendency to overestimate dimension as the local geometry is varied. lPCA and MLE exhibit divergent responses. The x-axis represents the position of points on the torus, with 0 representing the innermost l…

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Reference graph

Works this paper leans on

118 extracted references · 75 canonical work pages · cited by 2 Pith papers

  1. [1]

    A Fractal Dimension for Measures via Persistent Homology

    Henry Adams, Elin Farnell, Manuchehr Aminian, Michael Kirby, Joshua Mirth, Rachel Neville, Chris Peterson, and Clayton Shonkwiler. A Fractal Dimension for Measures via Persistent Homology. In Nils A Baas, Gunnar E Carlsson, Gereon Quick, Markus Szymik, and Marius Thaule, editors, Topo- logical Data Analysis, pages 1–31, Cham, 2020. Springer International ...

    2020

  2. [2]

    Estimating the effective dimension of large biological datasets using Fisher separability analysis

    Luca Albergante, Jonathan Bac, and Andrei Zinovyev. Estimating the effective dimension of large biological datasets using Fisher separability analysis. In 2019 International Joint Conference on Neural Networks (IJCNN) , pages 1–8, Budapest, 2019. IEEE

    2019

  3. [3]

    Houle, Ken ichi Kawarabayashi, and Michael Nett

    Laurent Amsaleg, Oussama Chelly, Teddy Furon, St´ ephane Girard, Michael E. Houle, Ken ichi Kawarabayashi, and Michael Nett. Extreme-value-theoretic estimation of local intrinsic dimension- ality. Data Mining and Knowledge Discovery , 32(6):1768–1805, 11 2018

    2018

  4. [4]

    Intrinsic Dimensionality Estimation within Tight Localities

    Laurent Amsaleg, Oussama Chelly, Michael E Houle, Miloˇ s Radovanovi, and Weeris Treeratanajaru. Intrinsic Dimensionality Estimation within Tight Localities. In Proceedings of the 2019 SIAM Inter- national Conference on Data Mining (SDM) , pages 181–189, Calgary, 2019. SIAM

    2019

  5. [5]

    Metric Space Magnitude and Generalisation in Neural Networks

    Rayna Andreeva, Katharina Limbeck, Bastian Rieck, and Rik Sarkar. Metric Space Magnitude and Generalisation in Neural Networks. In Timothy Doster, Tegan Emerson, Henry Kvinge, Nina Miolane, Mathilde Papillon, Bastian Rieck, and Sophia Sanborn, editors, Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML) , vol...

    2023

  6. [6]

    Mirkes, Alexander N

    Jonathan Bac, Evgeny M. Mirkes, Alexander N. Gorban, Ivan Tyukin, and Andrei Zinovyev. Scikit- Dimension: A Python Package for Intrinsic Dimension Estimation. Entropy, 23(10):1368, 10 2021

    2021

  7. [7]

    Schwartz

    Mukund Balasubramanian and Eric L. Schwartz. The isomap algorithm and topological stability. Science, 295(5552), 2002

    2002

  8. [8]

    Jillian Beardwood, J. H. Halton, and J. M. Hammersley. The shortest path through many points. Mathematical Proceedings of the Cambridge Philosophical Society , 55(4):299–327, 10 1959

    1959

Show all 118 references
  1. [9]

    Er¨ oss, Andr´ as Telcs, and Zolt´ an Somogyv´ ari

    Zsigmond Benk˝ o, Marceli Stippinger, Roberta Rehus, Attila Bencze, D´ aniel Fab´ o, Bogl´ arka Hajnal, Lor´ and G. Er¨ oss, Andr´ as Telcs, and Zolt´ an Somogyv´ ari. Manifold-adaptive dimension estimation revisited. PeerJ Computer Science , 8, 2022

    2022

  2. [10]

    Intrinsic Dimension, Persistent Ho- mology and Generalization in Neural Networks

    Tolga Birdal, Aaron Lou, Leonidas J Guibas, and Umut Simsekli. Intrinsic Dimension, Persistent Ho- mology and Generalization in Neural Networks. In M Ranzato, A Beygelzimer, Y Dauphin, P S Liang, and J Wortman Vaughan, editors, Advances in Neural Information Processing Systems...

    2021

  3. [11]

    Bayesian PCA

    Christopher Bishop. Bayesian PCA. Advances in Neural Information Processing Systems , 11, 1998

    1998

  4. [12]

    Bishop and Yuval Peres

    Christopher J. Bishop and Yuval Peres. Fractals in Probability and Analysis . Cambridge University Press, 2016

    2016

  5. [13]

    Intrinsic Dimension Estimation Using Wasserstein Distance

    Adam Block, Zeyu Jia, Yury Polyanskiy, and Alexander Rakhlin. Intrinsic Dimension Estimation Using Wasserstein Distance. Journal of Machine Learning Research , 23:1–37, 2022

    2022

  6. [14]

    Remarks on Parallel Analysis

    Andreas Buja and Nermin Eyuboglu. Remarks on Parallel Analysis. Multivariate Behavioral Research, 27(4):509–540, 10 1992

    1992

  7. [15]

    Data dimensionality estimation methods: a survey

    Francesco Camastra. Data dimensionality estimation methods: a survey. Pattern Recognition , 36(12):2945–2954, 12 2003

    2003

  8. [16]

    Intrinsic dimension estimation: Advances and open prob- lems

    Francesco Camastra and Antonino Staiano. Intrinsic dimension estimation: Advances and open prob- lems. Information Sciences, 328:26–41, 1 2016. 39

    2016

  9. [17]

    Intrinsic Dimension Estimation of Data: An Approach Based on Grassberger–Procaccia’s Algorithm

    Francesco Camastra and Alessandro Vinciarelli. Intrinsic Dimension Estimation of Data: An Approach Based on Grassberger–Procaccia’s Algorithm. Neural Processing Letters, 14(1):27–34, 8 2001

    2001

  10. [18]

    Campadelli, E

    P. Campadelli, E. Casiraghi, C. Ceruti, and A. Rozza. Intrinsic Dimension Estimation: Relevant Techniques and a Benchmark Framework. Mathematical Problems in Engineering , 2015:1–21, 2015

    2015

  11. [19]

    Abanov, Jeffrey Berger, Cameron J

    Luca Candelori, Alexander G. Abanov, Jeffrey Berger, Cameron J. Hogan, Vahagn Kirakosyan, Kharen Musaelian, Ryan Samson, James E.T. Smith, Dario Villani, Martin T. Wells, and Mengjia Xu. Robust estimation of the intrinsic dimension of data sets with quantum cognition machine l...

    2025

  12. [20]

    DANCo: An intrinsic dimensionality estimator exploiting angle and norm concentration

    Claudio Ceruti, Simone Bassis, Alessandro Rozza, Gabriele Lombardi, Elena Casiraghi, and Paola Campadelli. DANCo: An intrinsic dimensionality estimator exploiting angle and norm concentration. Pattern Recognition, 47(8):2569–2581, 2014

    2014

  13. [21]

    Dimension Detection via Slivers

    Siu-Wing Cheng and Man-Kwun Chiu. Dimension Detection via Slivers. In Proceedings of the Twen- tieth Annual ACM-SIAM Symposium on Discrete Algorithms , pages 1001–1010. Society for Industrial and Applied Mathematics, 2009

    2009

  14. [22]

    Costa and Alfred O

    Jose A. Costa and Alfred O. Hero. Entropic graphs for manifold learning. In Conference Record of the Asilomar Conference on Signals, Systems and Computers , volume 1, 2003

    2003

  15. [23]

    Costa and Alfred O

    Jose A. Costa and Alfred O. Hero. Geodesic entropic graphs for dimension and entropy estimation in manifold learning. IEEE Transactions on Signal Processing , 52(8):2210–2221, 8 2004

    2004

  16. [24]

    Costa and Alfred O

    Jose A. Costa and Alfred O. Hero. Learning intrinsic dimension and intrinsic entropy of high- dimensional datasets. In European Signal Processing Conference, volume 06-10-September-2004, 2015

    2004

  17. [25]

    J. M. Craddock and C. R. Flood. Eigenvectors for representing the 500 mb geopotential surface over the Northern Hemisphere. Quarterly Journal of the Royal Meteorological Society , 95(405):576–593, 7 1969

    1969

  18. [26]

    The generalized ratios intrinsic dimension estimator

    Francesco Denti, Diego Doimo, Alessandro Laio, and Antonietta Mira. The generalized ratios intrinsic dimension estimator. Scientific Reports, 12(1):20005, 11 2022

    2022

  19. [27]

    Beyond the noise: intrinsic dimension estimation with optimal neighbourhood identification

    Antonio Di Noia, Iuri Macocco, Aldo Glielmo, Alessandro Laio, and Antonietta Mira. Beyond the noise: intrinsic dimension estimation with optimal neighbourhood identification. arXiv:2405.15132v2, 5 2024

    2024 arXiv

  20. [28]

    Metric Space Spread, Intrinsic Dimension and the Manifold Hypothesis

    Kevin Dunne. Metric Space Spread, Intrinsic Dimension and the Manifold Hypothesis. arXiv:2308.01382, 8 2023

    2023 arXiv

  21. [29]

    Generalization bounds using data- dependent fractal dimensions

    Benjamin Dupuis, George Deligiannidis, and Umut Simsekli. Generalization bounds using data- dependent fractal dimensions. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engel- hardt, Sivan Sabato, and Jonathan Scarlett, editors, Proceedings of the 40th International...

    2023

  22. [30]

    Eckmann and D

    J.-P. Eckmann and D. Ruelle. Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems. Physica D: Nonlinear Phenomena , 56(2-3):185–187, 5 1992

    1992

  23. [31]

    Intrinsic dimension estimation for locally under- sampled data

    Vittorio Erba, Marco Gherardi, and Pietro Rotondo. Intrinsic dimension estimation for locally under- sampled data. Scientific Reports, 9(1):17133, 11 2019

    2019

  24. [32]

    Estimating the intrinsic dimension of datasets by a minimal neighborhood information

    Elena Facco, Maria d’Errico, Alex Rodriguez, and Alessandro Laio. Estimating the intrinsic dimension of datasets by a minimal neighborhood information. Scientific Reports, 7(1):12140, 9 2017

    2017

  25. [33]

    Alternative Definitions of Dimension

    Kenneth Falconer. Alternative Definitions of Dimension. In Fractal Geometry, chapter 3, pages 39–58. John Wiley & Sons, Ltd, 2003. 40

    2003

  26. [34]

    Intrinsic dimension estimation of data by principal component analysis

    Mingyu Fan, Nannan Gu, Hong Qiao, and Bo Zhang. Intrinsic dimension estimation of data by principal component analysis. arXiv, 2 2010

    2010

  27. [35]

    Manifold-adaptive dimension estimation

    Amir massoud Farahmand, Csaba Szepesv´ ari, and Jean-Yves Audibert. Manifold-adaptive dimension estimation. In Proceedings of the 24th international conference on Machine learning , pages 265–272, New York, NY, USA, 6 2007. ACM

    2007

  28. [36]

    Serge Frontier. ´Etude de la d´ ecroissance des valeurs propres dans une analyse en composantes prin- cipales: Comparaison avec le mod` ele du bˆ aton bris´ e.Journal of Experimental Marine Biology and Ecology, 25(1):67–75, 11 1976

    1976

  29. [37]

    Fukunaga and D.R

    K. Fukunaga and D.R. Olsen. An Algorithm for Finding Intrinsic Dimensionality of Data. IEEE Transactions on Computers, C-20(2):176–183, 2 1971

    1971

  30. [38]

    Supervised nonlinear dimensionality reduction for visualization and classification

    Xin Geng, De Chuan Zhan, and Zhi Hua Zhou. Supervised nonlinear dimensionality reduction for visualization and classification. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cy- bernetics, 35(6):1098–1107, 12 2005

    2005

  31. [39]

    Elements of Dimensionality Re- duction and Manifold Learning

    Benyamin Ghojogh, Mark Crowley, Fakhri Karray, and Ali Ghodsi. Elements of Dimensionality Re- duction and Manifold Learning . Springer International Publishing, Cham, 2023

    2023

  32. [40]

    Persistent magnitude

    Dejan Govc and Richard Hepworth. Persistent magnitude. Journal of Pure and Applied Algebra , 225(3):106517, 3 2021

    2021

  33. [41]

    Accurate Estimation of the Intrinsic Dimension Using Graph Distances: Unraveling the Geometric Complexity of Datasets

    Daniele Granata and Vincenzo Carnevale. Accurate Estimation of the Intrinsic Dimension Using Graph Distances: Unraveling the Geometric Complexity of Datasets. Scientific Reports, 6, 8 2016

    2016

  34. [42]

    Measuring the strangeness of strange attractors

    Peter Grassberger and Itamar Procaccia. Measuring the strangeness of strange attractors. Physica D: Nonlinear Phenomena, 9(1):189–208, 1983

    1983

  35. [43]

    Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula

    Alfred Gray. Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula. Topology, 21(2):201–228, 1982

    1982

  36. [44]

    Intrinsic Dimensionality Estimation of Submanifolds in $Rˆd$

    Hein Matthias and Audibert Jean-Yves. Intrinsic Dimensionality Estimation of Submanifolds in $Rˆd$. In Proceedings of the 22nd International Conference on Machine Learning, pages 289–296, Bonn, 2005. Association for Computing Machinery

    2005

  37. [45]

    Intrinsic dimensionality estimation using Normalizing Flows

    Christian Horvat and Jean-Pascal Pfister. Intrinsic dimensionality estimation using Normalizing Flows. In S Koyejo, S Mohamed, A Agarwal, D Belgrave, K Cho, and A Oh, editors, Advances in Neural Information Processing Systems, volume 35, pages 12225–12236. Curran Associates, I...

    2022

  38. [46]

    Manifold Hypothesis in Data Analysis: Double Geometrically-Probabilistic Approach to Manifold Dimension Estimation

    Alexander Ivanov, Gleb Nosovskiy, Alexey Chekunov, Denis Fedoseev, Vladislav Kibkalo, Mikhail Nikulin, Fedor Popelenskiy, Stepan Komkov, Ivan Mazurenko, and Aleksandr Petiushko. Manifold Hypothesis in Data Analysis: Double Geometrically-Probabilistic Approach to Manifold Dimen...

    2021 arXiv

  39. [47]

    Johnson and Joram Lindenstrauss

    William B. Johnson and Joram Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. In Richard Beals, Anatole Beck, Alexandra Bellow, and Arshag Hajian, editors, Conference on Modern Analysis and Probability, pages 189–206. American Mathematical Society, Provide...

    1984

  40. [48]

    Low bias local intrinsic dimension estimation from expected simplex skewness

    Kerstin Johnsson, Charlotte Soneson, and Magnus Fontes. Low bias local intrinsic dimension estimation from expected simplex skewness. IEEE Transactions on Pattern Analysis and Machine Intelligence , 37(1):196–202, 1 2015

    2015

  41. [49]

    I. T. Jolliffe. Principal Component Analysis. Springer-Verlag, New York, 2002

    2002

  42. [50]

    Manifold Diffusion Geometry: Curvature, Tangent Spaces, and Dimension

    Iolo Jones. Manifold Diffusion Geometry: Curvature, Tangent Spaces, and Dimension. arXiv:2411.04100, 11 2024. 41

    2024

  43. [51]

    Bayesian Estimation Approaches for Local Intrinsic Dimensionality

    Zaher Joukhadar, Hanxun Huang, and Sarah Monazam Erfani. Bayesian Estimation Approaches for Local Intrinsic Dimensionality. In Edgar Ch´ avez, Benjamin Kimia, Jakub Lokoˇ c, Marco Patella, and Jan Sedmidubsky, editors, Similarity Search and Applications , volume 15268 of Lectu...

    2025

  44. [52]

    The Application of Electronic Computers to Factor Analysis

    Henry F Kaiser. The Application of Electronic Computers to Factor Analysis. Educational and Psychological Measurement, 20(1):141–151, 1960

    1960

  45. [53]

    A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models

    Hamidreza Kamkari, Brendan Leigh, Ross Rasa Hosseinzadeh, Jesse C Cresswell, and Gabriel Loaiza- Ganem. A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models. Technical report, 2024

    2024

  46. [54]

    Fractal-Based Methods as a Technique for Estimating the Intrinsic Dimensionality of High-Dimensional Data: A Survey

    Rasa Karbauskaite and Gintautas Dzemyda. Fractal-Based Methods as a Technique for Estimating the Intrinsic Dimensionality of High-Dimensional Data: A Survey. Informatica. 2016;27(2):257-281, 27(2):257–281, 2016

    2016

  47. [55]

    Additive autoencoder for dimension estimation.Neurocomput- ing, 551, 9 2023

    Tommi K¨ arkk¨ ainen and Jan H¨ anninen. Additive autoencoder for dimension estimation.Neurocomput- ing, 551, 9 2023

    2023

  48. [56]

    Is magnitude ’generically continuous’ for finite metric spaces? arXiv:2501.08745, 1 2025

    Hirokazu Katsumasa, Emily Roff, and Masahiko Yoshinaga. Is magnitude ’generically continuous’ for finite metric spaces? arXiv:2501.08745, 1 2025

    2025 arXiv

  49. [57]

    Intrinsic Dimension Estimation Using Packing Numbers

    Bal´ azs K´ egl. Intrinsic Dimension Estimation Using Packing Numbers. In S Becker, S Thrun, and K Obermayer, editors, Advances in Neural Information Processing Systems 15 , pages 697–704. MIT Press, 2002

    2002

  50. [58]

    The central limit theorem for weighted minimal spanning trees on random points

    Harry Kesten and Sungchul Lee. The central limit theorem for weighted minimal spanning trees on random points. The Annals of Applied Probability , 6(2):495–527, 1996

    1996

  51. [59]

    Minimax rates for estimating the dimension of a manifold

    Jisu Kim, Alessandro Rinaldo, and Larry Wasserman. Minimax rates for estimating the dimension of a manifold. Journal of Computational Geometry , 10(1):42–95, 2019

    2019

  52. [60]

    Dimensionality estimation without distances

    Matth¨ aus Kleindessner and Ulrike Luxburg. Dimensionality estimation without distances. In Guy Lebanon and S. V. N. Vishwanathan, editors, Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics , pages 471–479, 2015

    2015

  53. [61]

    The minimal spanning tree and the upper box dimension

    Gady Kozma, Zvi Lotker, and Gideon Stupp. The minimal spanning tree and the upper box dimension. Proceedings of the American Mathematical Society, 134(4):1183–1187, 9 2005

    2005

  54. [62]

    On the connectivity threshold for general uniform metric spaces

    Gady Kozma, Zvi Lotker, and Gideon Stupp. On the connectivity threshold for general uniform metric spaces. Information Processing Letters, 110(10):356–359, 4 2010

    2010

  55. [63]

    Simple correlation dimension estimator and its use to detect causality

    Anna Krakovsk´ a and Martina Chvostekov´ a. Simple correlation dimension estimator and its use to detect causality. Chaos, Solitons and Fractals , 175, 10 2023

    2023

  56. [64]

    J. B. Kruskal. Nonmetric Multidimensional Scaling: A Numerical Method. Psychometrika, 29(2):115– 129, 6 1964

    1964

  57. [65]

    The concentration of measure phenomenon

    Michel Ledoux. The concentration of measure phenomenon. Mathematical surveys and monographs, , 89, 2001

    2001

  58. [66]

    On the asymptotic magnitude of subsets of Euclidean space

    Tom Leinster and Simon Willerton. On the asymptotic magnitude of subsets of Euclidean space. Geometriae Dedicata, 164(1):287–310, 6 2013

    2013

  59. [67]

    Maximum Likelihood Estimation of Intrinsic Dimension

    Elizaveta Levina and Peter J Bickel. Maximum Likelihood Estimation of Intrinsic Dimension. In Advances in Neural Information Processing Systems 17 , 2004

    2004

  60. [68]

    Tangent Space and Dimension Estimation with the Wasserstein Distance

    Uzu Lim, Harald Oberhauser, and Vidit Nanda. Tangent Space and Dimension Estimation with the Wasserstein Distance. SIAM Journal on Applied Algebra and Geometry , 8:650–685, 10 2024. 42

    2024

  61. [69]

    Metric space magnitude for evaluating the diversity of latent representations

    Katharina Limbeck, Rayna Andreeva, Rik Sarkar, and Bastian Rieck. Metric space magnitude for evaluating the diversity of latent representations. In NIPS ’24: Proceedings of the 38th International Conference on Neural Information Processing Systems, pages 123911–123953, Vancouv...

  62. [70]

    Riemannian manifold learning

    Tong Lin and Hongbin Zha. Riemannian manifold learning. IEEE Transactions on Pattern Analysis and Machine Intelligence , 30(5):796–809, 5 2008

    2008

  63. [71]

    Mini- mum Neighbor Distance Estimators of Intrinsic Dimension

    Gabriele Lombardi, Alessandro Rozza, Claudio Ceruti, Elena Casiraghi, and Paola Campadelli. Mini- mum Neighbor Distance Estimators of Intrinsic Dimension. In D Gunopulos, T Hofmann, D Malerba, and M Vazirgiannis, editors, Machine Learning and Knowledge Discovery in Databases. ...

    2011

  64. [72]

    MacKay and Zoubin Ghahramani

    David J.C. MacKay and Zoubin Ghahramani. Comments on ’Maximum Likelihood Estimation of Intrinsic Dimension’ by E. Levina and P. Bickel (2004), 2005

    2004

  65. [73]

    Intrinsic Dimension Estimation for Discrete Metrics

    Iuri Macocco, Aldo Glielmo, Jacopo Grilli, and Alessandro Laio. Intrinsic Dimension Estimation for Discrete Metrics. Physical Review Letters, 130(6), 2 2023

    2023

  66. [74]

    Geometry of sets and measures in Euclidean spaces

    Pertti Mattila. Geometry of sets and measures in Euclidean spaces . Cambridge University Press, Cambridge, 1995

    1995

  67. [75]

    UMAP: Uniform Manifold Approximation and Projection

    Leland McInnes, John Healy, Nathaniel Saul, and Lukas Großberger. UMAP: Uniform Manifold Approximation and Projection. Journal of Open Source Software , 3(29), 2018

    2018

  68. [76]

    Mark W. Meckes. Positive definite metric spaces. Positivity, 17(3):733–757, 9 2013

    2013

  69. [77]

    Mark W. Meckes. Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces. Potential Analysis, 42(2):549–572, 2 2015

    2015

  70. [78]

    Thomas P. Minka. Automatic choice of dimensionality for PCA. In Advances in Neural Information Processing Systems, 2001

    2001

  71. [79]

    Topology (2nd Edition)

    James Raymond Munkres. Topology (2nd Edition). Prentice Hall, Inc, 2000

    2000

  72. [80]

    Oganov and Mario Valle

    Artem R. Oganov and Mario Valle. How to quantify energy landscapes of solids. Journal of Chemical Physics, 130(10), 2009

    2009

  73. [81]

    Alpha magnitude

    Miguel O’Malley, Sara Kalisnik, and Nina Otter. Alpha magnitude. Journal of Pure and Applied Algebra, 227(11):107396, 11 2023

    2023

  74. [82]

    A Novel Approach for Intrinsic Di- mension Estimation

    Kadir ¨Oz¸ coban, Murat Manguo˘ glu, and Emrullah Fatih Yetkin. A Novel Approach for Intrinsic Di- mension Estimation. arXiv:2503.09485v1 [cs.LG] 12 Mar 2025 , 3 2025

    2025 arXiv

  75. [83]

    Papaioannou, Ronen Talmon, Ioannis G

    Panagiotis G. Papaioannou, Ronen Talmon, Ioannis G. Kevrekidis, and Constantinos Siettos. Time- series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics. Chaos, 32(8), 8 2022

    2022

  76. [84]

    Pettis, Thomas A

    Karl W. Pettis, Thomas A. Bailey, Anil K. Jain, and Richard C. Dubes. An Intrinsic Dimensionality Estimator from Near-Neighbor Information. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-1(1):25–37, 1979

    1979

  77. [85]

    Intrinsic dimension estimation based on local adjacency information

    Haiquan Qiu, Youlong Yang, and Benchong Li. Intrinsic dimension estimation based on local adjacency information. Information Sciences, 558:21–33, 5 2021

    2021

  78. [86]

    Underestimation modification for intrinsic dimension estimation

    Haiquan Qiu, Youlong Yang, and Hua Pan. Underestimation modification for intrinsic dimension estimation. Pattern Recognition, 140, 8 2023

    2023

  79. [87]

    Intrinsic dimension estimation method based on correlation dimension and kNN method

    Haiquan Qiu, Youlong Yang, and Saeid Rezakhah. Intrinsic dimension estimation method based on correlation dimension and kNN method. Knowledge-Based Systems, 235, 1 2022. 43

    2022

  80. [88]

    A gen- eral and flexible method for signal extraction from single-cell RNA-seq data

    Davide Risso, Fanny Perraudeau, Svetlana Gribkova, Sandrine Dudoit, and Jean Philippe Vert. A gen- eral and flexible method for signal extraction from single-cell RNA-seq data. Nature Communications, 9(1), 12 2018

    2018

  81. [89]

    Rozza, G

    A. Rozza, G. Lombardi, C. Ceruti, E. Casiraghi, and P. Campadelli. Novel high intrinsic dimensionality estimators. Machine Learning, 89(1-2):37–65, 10 2012

    2012

  82. [90]

    IDEA: Intrinsic Dimension Estimation Algorithm

    Alessandro Rozza, Gabriele Lombardi, Marco Rosa, Elena Casiraghi, and Paola Campadelli. IDEA: Intrinsic Dimension Estimation Algorithm. In Image Analysis and Processing – ICIAP 2011 . Springer, Berlin, Heidelberg., 2011

    2011

  83. [91]

    A Nonlinear Mapping for Data Structure Analysis

    John W Sammon. A Nonlinear Mapping for Data Structure Analysis. IEEE Transactions on Com- puters, C-18(5):401–09, 1969

    1969

  84. [92]

    Estimate manifold dimensionality with LID, 12 2020

    Aaron Schumacher. Estimate manifold dimensionality with LID, 12 2020

    2020

  85. [93]

    Fractal dimension and the persistent homology of random geometric complexes

    Benjamin Schweinhart. Fractal dimension and the persistent homology of random geometric complexes. Advances in Mathematics , 372:107291, 10 2020

    2020

  86. [94]

    Persistent Homology and the Upper Box Dimension

    Benjamin Schweinhart. Persistent Homology and the Upper Box Dimension. Discrete & Computational Geometry, 65(2):331–364, 2021

    2021

  87. [95]

    Dimension Estimation Using Random Connection Models

    Paulo Serra and Michel Mandjes. Dimension Estimation Using Random Connection Models. Journal of Machine Learning Research, 18(138):1–35, 2017

    2017

  88. [96]

    Hausdorff dimension, heavy tails, and generalization in neural networks

    Umut Simsekli, Ozan Sener, George Deligiannidis, and Murat A Erdogdu. Hausdorff dimension, heavy tails, and generalization in neural networks. Advances in Neural Information Processing Systems , 33:5138–5151, 2020

    2020

  89. [97]

    Wasserstein Stability for Persistence Diagrams

    Primoz Skraba and Katharine Turner. Wasserstein Stability for Persistence Diagrams. arXiv:2006.16824, 7 2025

    2006 arXiv

  90. [98]

    Diffusion Models Encode the Intrinsic Dimension of Data Manifolds

    Jan Pawel Stanczuk, Georgios Batzolis, Teo Deveney, and Carola-Bibiane Sch¨ onlieb. Diffusion Models Encode the Intrinsic Dimension of Data Manifolds. In Ruslan Salakhutdinov, Zico Kolter, Katherine Heller, Adrian Weller, Nuria Oliver, Jonathan Scarlett, and Felix Berkenkamp, ...

    2024

  91. [99]

    Michael Steele

    J. Michael Steele. Growth Rates of Euclidean Minimal Spanning Trees with Power Weighted Edges. The Annals of Probability , 16(4), 10 1988

    1988

  92. [100]

    On the Size and Recovery of Submatrices of Ones in a Random Binary Matrix

    Xing Sun and Andrew B Nobel. On the Size and Recovery of Submatrices of Ones in a Random Binary Matrix. Journal of Machine Learning Research , 9(80):2431–2453, 2008

    2008

  93. [101]

    Relative Intrinsic Di- mensionality Is Intrinsic to Learning

    Oliver J Sutton, Qinghua Zhou, Alexander N Gorban, and Ivan Y Tyukin. Relative Intrinsic Di- mensionality Is Intrinsic to Learning. In Lazaros Iliadis, Antonios Papaleonidas, Plamen Angelov, and Chrisina Jayne, editors, Artificial Neural Networks and Machine Learning – ICANN 2...

    2023

  94. [102]

    LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood

    Piotr Tempczyk, Rafa l Michaluk, Lukasz Garncarek, Przemys law Spurek, Jacek Tabor, and Adam Golinski. LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, edito...

    2022

  95. [103]

    A Global Geometric Framework for Non- linear Dimensionality Reduction

    Joshua B Tenenbaum, Vin de Silva, and John C Langford. A Global Geometric Framework for Non- linear Dimensionality Reduction. Science, 290:2319–2323, 2000

    2000

  96. [104]

    G. V. Trunk. Statistical estimation of the intrinsic dimensionality of data collections. Information and Control, 12(5):508–525, 5 1968. 44

    1968

  97. [105]

    Intrinsic dimension estimation for robust detection of ai-generated texts

    Eduard Tulchinskii, Kristian Kuznetsov, Laida Kushnareva, Daniil Cherniavskii, Sergey Nikolenko, Evgeny Burnaev, Serguei Barannikov, and Irina Piontkovskaya. Intrinsic dimension estimation for robust detection of ai-generated texts. Advances in Neural Information Processing Sy...

    2023

  98. [106]

    Ein Fixpunktsatz

    Andrey Tychonoff. Ein Fixpunktsatz. Mathematische Annalen, 11:767–776, 1935

    1935

  99. [107]

    Visualizing data using t-SNE

    Laurens Van Der Maaten and Geoffrey Hinton. Visualizing data using t-SNE. Journal of Machine Learning Research, 9, 2008

    2008

  100. [108]

    Infinite-Dimensional Topology

    Jan van Mill. Infinite-Dimensional Topology. North-Holland, 1st edition, 1988

    1988

  101. [109]

    Verveer and Robert P.W

    Peter J. Verveer and Robert P.W. Duin. An Evaluation of Intrinsic Dimensionality Estimators. IEEE Transactions on Pattern Analysis and Machine Intelligence , 17(1):81–86, 1995

    1995

  102. [110]

    Way, Michael Zietz, Vincent Rubinetti, Daniel S

    Gregory P. Way, Michael Zietz, Vincent Rubinetti, Daniel S. Himmelstein, and Casey S. Greene. Compressing gene expression data using multiple latent space dimensionalities learns complementary biological representations. Genome Biology, 21(1), 5 2020

    2020

  103. [111]

    Heuristic and computer calculations for the magnitude of metric spaces

    Simon Willerton. Heuristic and computer calculations for the magnitude of metric spaces. arXiv:0910.5500, 10 2009

    2009 arXiv

  104. [112]

    Conical dimension as an intrisic dimension estimator and its applications

    Xin Yang, Sebastien Michea, and Hongyuan Zha. Conical dimension as an intrisic dimension estimator and its applications. In Chid Apte, David Skillicorn, Bing Liu, and Srinivasan Parthasarathy, editors, Proceedings of the 2007 SIAM International Conference on Data Mining , page...

    2007

  105. [113]

    Adversarial Estimation of Topological Dimension with Harmonic Score Maps

    Eric Yeats, Cameron Darwin, Frank Liu, and Hai Li. Adversarial Estimation of Topological Dimension with Harmonic Score Maps. arXiv:2312.06869, 12 2023

    2023 arXiv

  106. [114]

    Joseph E. Yukich. Probability Theory of Classical Euclidean Optimization Problems . Springer Berlin, Heidelberg, 1998

    1998

  107. [115]

    Abstract Latent-Space Construction for Analyzing Large Genomic Data Sets

    Wenlan Zang. Abstract Latent-Space Construction for Analyzing Large Genomic Data Sets . PhD thesis, Yale University, 2021. 45 A Comparison of PH and KNN Since PH0 and KNN are derived from the common theory of Euclidean functionals, and are similar in construction, we highlight...

    2021

  108. [117]

    The hyperparameter that minimises the difference between the estimated and intrinsic dimension ˆdbest(E, M) = min H∈HE | ˆd(E, M, H) − d(M )|

  109. [118]

    While a choice of hyperparameter might be optimal for a certain dataset, the same hyperparameter may lead to a poor performance on another

    The performance of the estimator with a fixed choice of hyperparameter; this is either (a) The hyperparameter that minimises the median absolute error across the benchmark manifolds: ˆdabs(E, M) = ˆd(E, M, Habs(E)) Habs(E) = argmin H∈HE median M ∈M | ˆd(E, M, H) − d(M )| (b) T...

  110. [2025]

    Curran Associates Inc

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.