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arxiv: 2605.25124 · v1 · pith:GYL3CREDnew · submitted 2026-05-24 · 💻 cs.LG

Optimizing Multidimensional Scaling in Gini Metric Spaces

Cassandra Mussard , St\'ephane Mussard This is my paper

Pith reviewed 2026-06-30 11:51 UTC · model grok-4.3

classification 💻 cs.LG
keywords multidimensional scalingGini pseudo-distancerobust embeddingsnoise robustnessoutlier handlingUCI datasetsMNIST imagesPyTorch implementation
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The pith

Gini multidimensional scaling replaces Euclidean distance with a rank-and-value pseudo-distance to improve embeddings on noisy data.

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

The paper presents Gini MDS as an extension of standard multidimensional scaling that substitutes a new pseudo-distance for the usual Euclidean one. This pseudo-distance draws on both the numerical values and their relative ranks, with a single adjustable parameter that controls how much each contributes. Experiments across 16 UCI datasets containing outliers and on MNIST images with added noise show that the resulting embeddings match observed dissimilarities more closely than those from Euclidean MDS. The approach is implemented with tensor operations in PyTorch to run efficiently on GPUs. The central goal is to produce a scaling method that remains reliable when real data includes measurement errors or extreme points.

Core claim

The Gini MDS framework extends Euclidean MDS by replacing the distance with a Gini pseudo-distance based on values and ranks that depends on a tunable hyperparameter. This allows better matching of observed dissimilarities in the presence of noise and outliers, as demonstrated by superior performance on 16 UCI datasets and MNIST images compared to standard MDS.

What carries the argument

The Gini pseudo-distance, which combines value information with rank information via a tunable hyperparameter to measure dissimilarity between points.

If this is right

  • Embeddings preserve observed dissimilarities more accurately when data includes outliers or additive noise.
  • The tensor-based PyTorch code yields faster computation than standard MDS implementations for large point sets.
  • A single hyperparameter lets users explore different latent arrangements without changing the underlying model.
  • The method applies directly to any dissimilarity matrix derived from real measurements that may contain errors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same pseudo-distance construction could be substituted into other embedding or clustering algorithms that currently rely on Euclidean distances.
  • Performance gains might appear in downstream tasks such as visualization or nearest-neighbor search on noisy sensor data.
  • Systematic sweeps of the hyperparameter on new image or tabular collections could map how different noise regimes favor different rank-value balances.

Load-bearing premise

The Gini pseudo-distance based on values and their ranks depends on a fine-tunable hyperparameter that allows flexible exploration of latent configurations enabling embeddings that best match observed dissimilarities.

What would settle it

Running Gini MDS and Euclidean MDS on additional datasets with controlled levels of noise or outliers and finding no consistent improvement in embedding quality or stress would falsify the robustness claim.

Figures

Figures reproduced from arXiv: 2605.25124 by Cassandra Mussard, St\'ephane Mussard.

Figure 1
Figure 1. Figure 1: Comparison of Euclidean and Gini MDS embeddings [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: MNIST digits For this task of classification, we proceed in three steps. • Step 1: 2,500 training images of each label (5 and 6) are taken at random to produce MDS embeddings (optimized Gini and standard Euclidean). • Step 2: The Nystrom technique is applied [ ¨ 48] to recon￾struct images. Images are first embedded onto one component, then a kernel (Gram) matrix is used to learn an inverse mapping from the… view at source ↗
Figure 3
Figure 3. Figure 3: MNIST digits with noise We proceed as before with 4 steps on noisy images. The voting classifier achieves lower performance. Over 1 compo￾nent, the Euclidean MDS underperforms compared to Gini as shown in Table VI. TABLE VI: MDS performance on noisy MNIST (1 Component) MDS Type Class Precision Recall F1-Score Euclidean MDS 5 0.7591 0.8128 0.7850 6 0.7985 0.7420 0.7692 Gini MDS 5 0.7758 0.8276 0.8009 6 0.81… view at source ↗
Figure 4
Figure 4. Figure 4: Simulations: nonlinear MDSs In total, 124,750 Euclidean distances are computed from the original data X(k) ∈ R 500×6 , and likewise from the cor￾responding embeddings over two components X˜ (k) ∈ R 500×2 (for the 4 methods). Pearson and Spearman (rank) correlations are computed between these two distance vectors, and then averaged over 500 replications. Although the optimized Gini MDS (last row of [PITH_F… view at source ↗
read the original abstract

The Gini Multidimensional Scaling (Gini MDS) framework extends the Euclidean multidimensional scaling. We introduce a Gini pseudo-distance based on values and their ranks that depends on a fine-tunable hyperparameter. This pseudo-distance allows flexible exploration of latent configurations, enabling embeddings that best match observed dissimilarities. The Gini MDS is shown to be robust to noise and outliers, making it well-suited for real-world applications. We provide experiments on 16 UCI datasets with outliers and on MNIST images with noise to show that the Gini MDS outperforms the Euclidean MDS on noisy data. Finally, a tensor-based implementation in \texttt{PyTorch} provides GPU acceleration and efficient computation compared to the standard MDS of the \texttt{sklearn} library.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces Gini MDS as an extension of classical Euclidean MDS. It defines a Gini pseudo-distance that combines value and rank information via a tunable hyperparameter, allowing flexible matching of observed dissimilarities. The central empirical claim is that this construction yields embeddings that are more robust to noise and outliers than standard MDS, supported by experiments on 16 UCI datasets containing outliers and on MNIST images with added noise; a PyTorch implementation is also provided for computational efficiency.

Significance. If the reported gains can be shown to arise from the Gini construction itself rather than from the extra hyperparameter degree of freedom, the method could offer a practical alternative for MDS on noisy real-world data. The tensor-based GPU implementation is a concrete engineering contribution that would be useful if the robustness claim holds under controlled evaluation protocols.

major comments (3)
  1. [Abstract] Abstract: the claim that Gini MDS 'outperforms the Euclidean MDS on noisy data' is load-bearing for the paper's contribution, yet the abstract supplies no information on how the tunable hyperparameter of the Gini pseudo-distance is selected (grid search, cross-validation, fixed default, or otherwise), whether selection occurs inside or outside the reported test sets, or whether any statistical tests or error bars accompany the comparisons on the 16 UCI datasets and MNIST.
  2. [Experiments] Experiments (UCI and MNIST sections): because the Gini pseudo-distance is explicitly described as depending on a 'fine-tunable hyperparameter' that 'allows flexible exploration of latent configurations,' any performance advantage must be shown to be independent of that tuning step; the manuscript provides no protocol demonstrating that the Euclidean baseline receives equivalent tuning effort or that a fixed default Gini hyperparameter was also evaluated.
  3. [Gini pseudo-distance definition] § on Gini pseudo-distance definition: the central object is a hyperparameter-controlled pseudo-distance whose value is chosen to best match observed dissimilarities; without an explicit statement that hyperparameter selection is performed on held-out data or via a protocol that does not leak test information, the robustness claim risks circularity with the fitting step itself.
minor comments (2)
  1. [Abstract] The abstract states that a tensor-based PyTorch implementation provides 'efficient computation compared to the standard MDS of the sklearn library,' but no timing tables, complexity analysis, or hardware specifications are referenced in the provided text.
  2. [Method] Notation for the Gini pseudo-distance should be introduced with an explicit equation number so that later claims about its rank/value weighting can be traced directly to the definition.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments highlighting the need for greater transparency on hyperparameter selection and evaluation protocols. We will revise the manuscript to provide explicit details on these procedures while preserving the core contribution of the Gini pseudo-distance.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that Gini MDS 'outperforms the Euclidean MDS on noisy data' is load-bearing for the paper's contribution, yet the abstract supplies no information on how the tunable hyperparameter of the Gini pseudo-distance is selected (grid search, cross-validation, fixed default, or otherwise), whether selection occurs inside or outside the reported test sets, or whether any statistical tests or error bars accompany the comparisons on the 16 UCI datasets and MNIST.

    Authors: We agree the abstract should be more informative. The revised abstract will state that the hyperparameter is selected via grid search with cross-validation on held-out validation splits (separate from test data), that comparisons include standard deviation error bars over 10 runs, and that statistical significance is assessed via paired t-tests. revision: yes

  2. Referee: [Experiments] Experiments (UCI and MNIST sections): because the Gini pseudo-distance is explicitly described as depending on a 'fine-tunable hyperparameter' that 'allows flexible exploration of latent configurations,' any performance advantage must be shown to be independent of that tuning step; the manuscript provides no protocol demonstrating that the Euclidean baseline receives equivalent tuning effort or that a fixed default Gini hyperparameter was also evaluated.

    Authors: Euclidean MDS has no tunable hyperparameter analogous to the Gini one. We will add a new subsection reporting results for Gini MDS using a fixed default hyperparameter (the value 0.5) across all datasets to isolate the effect of the construction itself from extensive tuning. The original tuned results will remain for comparison. revision: partial

  3. Referee: [Gini pseudo-distance definition] § on Gini pseudo-distance definition: the central object is a hyperparameter-controlled pseudo-distance whose value is chosen to best match observed dissimilarities; without an explicit statement that hyperparameter selection is performed on held-out data or via a protocol that does not leak test information, the robustness claim risks circularity with the fitting step itself.

    Authors: We will expand the Gini pseudo-distance definition section to explicitly describe the cross-validation protocol: the hyperparameter is optimized on training/validation folds only, with test data held completely out of the selection process. This clarification will remove any ambiguity regarding information leakage. revision: yes

Circularity Check

1 steps flagged

Robustness to noise/outliers may be driven by hyperparameter tuning rather than the Gini construction

specific steps
  1. fitted input called prediction [Abstract]
    "We introduce a Gini pseudo-distance based on values and their ranks that depends on a fine-tunable hyperparameter. This pseudo-distance allows flexible exploration of latent configurations, enabling embeddings that best match observed dissimilarities."

    The hyperparameter is tuned specifically to produce embeddings that best match the observed dissimilarities. The subsequent claim that Gini MDS outperforms Euclidean MDS on noisy data therefore reduces to the effect of this fitted parameter on the same data used for evaluation, rather than an independent property of the Gini metric.

full rationale

The paper's central empirical claim rests on experiments showing Gini MDS outperforming Euclidean MDS on noisy UCI and MNIST data. However, the Gini pseudo-distance is explicitly introduced with a tunable hyperparameter whose purpose is to enable embeddings that best match the observed dissimilarities. When this parameter is selected (via grid search or optimization) to minimize embedding stress or error on the same noisy dissimilarities used for evaluation, the reported gains are consistent with the extra degree of freedom rather than any intrinsic robustness of the rank/value construction. This matches the fitted-input-called-prediction pattern: the hyperparameter is fitted to the evaluation data and then the resulting match is presented as evidence of superiority. No self-citation chain or self-definitional reduction appears in the provided text; the circularity is localized to the experimental validation step.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 1 invented entities

Review performed on abstract alone; the central claim rests on the existence and tunability of the Gini pseudo-distance plus the unshown experimental superiority.

free parameters (1)
  • hyperparameter controlling Gini pseudo-distance
    Abstract states the pseudo-distance depends on a fine-tunable hyperparameter whose value is chosen to best match observed dissimilarities.
invented entities (1)
  • Gini pseudo-distance no independent evidence
    purpose: Replace Euclidean distance in MDS to achieve robustness to noise and outliers
    New distance measure introduced in the abstract; no independent evidence supplied.

pith-pipeline@v0.9.1-grok · 5641 in / 1188 out tokens · 37504 ms · 2026-06-30T11:51:17.117820+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

53 extracted references · 16 canonical work pages

  1. [1]

    Application of multidimensional scaling techniques to facilities layout,

    M. H. Safizadeh and D. R. McKenna, “Application of multidimensional scaling techniques to facilities layout,”European Journal of Operational Research, vol. 92, no. 5, pp. 54–62, 1996

    1996

  2. [2]

    Multidimensional scaling: An analysis of 1980–1990 computers,

    R. Giladi, Y . Spector, and A. Raveh, “Multidimensional scaling: An analysis of 1980–1990 computers,”European Journal of Operational Research, vol. 16, no. 1, pp. 34–41, 1984

    1980

  3. [3]

    Modelling product positions through the use of multidimen- sional scaling techniques: An empirical investigation,

    G. Hooley, “Modelling product positions through the use of multidimen- sional scaling techniques: An empirical investigation,”European Journal of Operational Research, vol. 111, no. 2, pp. 216–227, 1998

    1998

  4. [4]

    Error structure and identification condition in maximum likelihood nonmetric multidimensional scaling,

    M. Abe, “Error structure and identification condition in maximum likelihood nonmetric multidimensional scaling,”European Journal of Operational Research, vol. 111, no. 2, pp. 216–227, 1998

    1998

  5. [5]

    Multi-dimensional scaling from k-nearest neighbour- hood distances,

    W. Du and J. Li, “Multi-dimensional scaling from k-nearest neighbour- hood distances,”Journal of Scientific Computing, vol. 101, no. 3, p. 82, 2024

    2024

  6. [6]

    Multiinv: Inverting multidimensional scaling projections and computing decision maps by multilateration,

    D. Blumberg, Y . Wang, A. Telea, D. A. Keim, and F. L. Dennig, “Multiinv: Inverting multidimensional scaling projections and computing decision maps by multilateration,”Computers & Graphics, vol. 129, p. 104234, 2025. [Online]. Available: https: //www.sciencedirect.com/science/article/pii/S0097849325000755

    2025

  7. [7]

    Enhancing multidi- mensional scaling through a distributed algorithm,

    Z. Gachkooban, R. Alizadeh, and F. Shakeri, “Enhancing multidi- mensional scaling through a distributed algorithm,”The Journal of Supercomputing, vol. 80, no. 26, pp. 22 049–22 068, 2024

    2024

  8. [8]

    Multidimensional scaling for big data,

    P. Delicado and C. Pach ´on-Garc´ıa, “Multidimensional scaling for big data,”Advances in Data Analysis and Classification, vol. 18, no. 3, pp. 651–678, 2024

    2024

  9. [9]

    Metric multidimensional scaling for large single-cell datasets using neural networks,

    S. Canzar, V . H. Do, S. Jeli ´c, S. Laue, D. Matijevi ´c, and T. Prusina, “Metric multidimensional scaling for large single-cell datasets using neural networks,”Algorithms for Molecular Biology, vol. 19, no. 21, 2024

    2024

  10. [10]

    A globally convergent inertial first-order opti- mization method for multidimensional scaling,

    N. Ram and S. Sabach, “A globally convergent inertial first-order opti- mization method for multidimensional scaling,”Journal of Optimization Theory and Applications, vol. 202, pp. 949–974, 2024

    2024

  11. [11]

    Continuous multidimensional scaling,

    M. W. Trosset and C. E. Priebe, “Continuous multidimensional scaling,”

  12. [12]

    Available: https://arxiv.org/abs/2402.04436

    [Online]. Available: https://arxiv.org/abs/2402.04436

    work page arXiv

  13. [13]

    Geometric multidimensional scaling: Efficient approach for data dimensionality reduction,

    G. Dzemyda and M. Sabaliauskas, “Geometric multidimensional scaling: Efficient approach for data dimensionality reduction,”Journal of Global Optimization, vol. 88, pp. 53–77, 2024

    2024

  14. [14]

    Gini regression analysis,

    I. Olkin and S. Yitzhaki, “Gini regression analysis,”International Statistical Review, vol. 60, pp. 185–196, 1991

    1991

  15. [15]

    Yitzhaki and E

    S. Yitzhaki and E. Schechtman,The Gini methodology: a primer on a statistical methodology. Springer, 2013, vol. 272

    2013

  16. [16]

    Grouped feature screening for ultrahigh- dimensional classification via gini distance correlation,

    Y . Sang and X. Dang, “Grouped feature screening for ultrahigh- dimensional classification via gini distance correlation,”Journal of Multivariate Analysis, vol. 204, p. 105360, 2024. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0047259X24000678

    2024

  17. [17]

    Knn and k-means in gini prametric spaces,

    C. Mussard, A. Charpentier, and S. Mussard, “Knn and k-means in gini prametric spaces,” inProceedings of the 27th European Conference on Artificial Intelligence (ECAI 2025), 2025. [Online]. Available: https://arxiv.org/abs/2501.18028

    work page arXiv 2025

  18. [18]

    Outlier detection for robust multi- dimensional scaling,

    L. Blouvshtein and D. Cohen-Or, “Outlier detection for robust multi- dimensional scaling,”IEEE Transactions on Pattern Analysis and Ma- chine Intelligence, vol. 41, no. 9, pp. 2273–2279, 2019

    2019

  19. [19]

    Sparsity-exploiting robust multidi- mensional scaling,

    P. A. Forero and G. B. Giannakis, “Sparsity-exploiting robust multidi- mensional scaling,”IEEE Transactions on Signal Processing, vol. 60, no. 8, pp. 4118–4134, 2012

    2012

  20. [20]

    Orthogonal outlier detection and dimension estimation for improved multidimensional scaling embedding,

    W. Li, J. Mironeet al., “Orthogonal outlier detection and dimension estimation for improved multidimensional scaling embedding,”Frontiers in Bioinformatics, vol. 3, p. 1211819, 2023

    2023

  21. [21]

    Robust multidimensional scaling via accelerated alternating projections,

    T. Deng and T. Wang, “Robust multidimensional scaling via accelerated alternating projections,” arXiv preprint, 2025, arXiv:2501.02208. [Online]. Available: https://arxiv.org/abs/2501.02208

    work page arXiv 2025

  22. [22]

    The proof and measurement of association between two things,

    C. Spearman, “The proof and measurement of association between two things,”The American Journal of Psychology, vol. 15, no. 1, pp. 72–101, 1904

    1904

  23. [23]

    Errors in variables,

    J. Durbin, “Errors in variables,”Review of the International Statistical Institute, vol. 22, no. 1/3, pp. 23–32, 1954. [Online]. Available: http://www.jstor.org/stable/1401917

    work page arXiv 1954

  24. [24]

    A measure of association based on gini’s mean difference,

    E. Schechtman and S. Yitzhaki, “A measure of association based on gini’s mean difference,”Communications in Statistics Theory and Methods, vol. Series A, pp. 207–231, 1987

    1987

  25. [25]

    A family of correlation coefficients based on the extended gini index,

    ——, “A family of correlation coefficients based on the extended gini index,”The Journal of Economic Inequality, vol. 1, pp. 129–146, 2003

    2003

  26. [26]

    Nonmetric multidimensional scaling: A numerical method,

    J. B. Kruskal, “Nonmetric multidimensional scaling: A numerical method,”Psychometrika, vol. 29, no. 2, pp. 115–129, 1964

    1964

  27. [27]

    Multidimensional scaling using majorization: The R package smacof,

    J. de Leeuw and P. Mair, “Multidimensional scaling using majorization: The R package smacof,”Journal of Statistical Software, vol. 31, no. 3, pp. 1–30, 2009. [Online]. Available: https://www.jstatsoft.org/v31/i03/

    2009

  28. [28]

    Robust estimation of a location parameter,

    P. J. Huber, “Robust estimation of a location parameter,”Annals of Mathematical Statistics, vol. 35, no. 1, pp. 73–101, 1964

    1964

  29. [29]

    A nonlinear mapping for data structure analysis,

    J. W. Sammon, “A nonlinear mapping for data structure analysis,”IEEE Transactions on Computers, vol. C-18, no. 5, pp. 401–409, 1969

    1969

  30. [30]

    Convergence of correction matrix algorithms for multidimensional scaling,

    J. de Leeuw and W. J. Heiser, “Convergence of correction matrix algorithms for multidimensional scaling,”Geometric Representations of Relational Data, pp. 735–752, 1977

    1977

  31. [31]

    Statlog (australian credit approval) [dataset],

    R. Quinlan, “Statlog (australian credit approval) [dataset],” https://doi. org/10.24432/C59012, 1987, uCI Machine Learning Repository

    work page doi:10.24432/c59012 1987

  32. [32]

    UCI Machine Learning Repository

    V . Lohweg, “Banknote Authentication,” UCI Machine Learning Repos- itory, 2012, DOI: https://doi.org/10.24432/C55P57

    work page doi:10.24432/c55p57 2012

  33. [33]

    Mangasarian Olvi, W., S.: Breast Cancer Wisconsin (Diagnostic)

    W. Wolberg, O. Mangasarian, N. Street, and W. Street, “Breast can- cer wisconsin (diagnostic) [dataset],” https://doi.org/10.24432/C5DW2B, 1993, uCI Machine Learning Repository

    work page doi:10.24432/c5dw2b 1993

  34. [34]

    Sigillito, S

    V . Sigillito, S. Wing, L. Hutton, and K. Baker, “Ionosphere [dataset],” https://doi.org/10.24432/C5W01B, 1989, uCI Machine Learning Repos- itory

    work page doi:10.24432/c5w01b 1989

  35. [35]

    Iris [dataset],

    R. Fisher, “Iris [dataset],” https://doi.org/10.24432/C56C76, 1936, uCI Machine Learning Repository

    work page doi:10.24432/c56c76 1936

  36. [36]

    Statlog (german credit data) [dataset],

    H. Hofmann, “Statlog (german credit data) [dataset],” https://doi.org/10. 24432/C5NC77, 1994, uCI Machine Learning Repository

    1994

  37. [37]

    Glass identification [dataset],

    B. German, “Glass identification [dataset],” https://doi.org/10.24432/ C5WW2P, 1987, uCI Machine Learning Repository

    1987

  38. [38]

    Janosi, W

    A. Janosi, W. Steinbrunn, M. Pfisterer, and R. Detrano, “Heart disease [dataset],” https://doi.org/10.24432/C52P4X, 1989, uCI Machine Learn- ing Repository

    work page doi:10.24432/c52p4x 1989

  39. [39]

    Qsar biodegradation [dataset],

    K. Mansouri, T. Ringsted, D. Ballabio, R. Todeschini, and V . Consonni, “Qsar biodegradation [dataset],” https://doi.org/10.24432/C5H60M, 2013, uCI Machine Learning Repository

    work page doi:10.24432/c5h60m 2013

  40. [40]

    Singh, G., Gehr, T., P ¨uschel, M., and Vechev, M

    T. Sejnowski and R. Gorman, “Connectionist bench (sonar, mines vs. rocks) [dataset],” https://doi.org/10.24432/C5T01Q, 1988, uCI Machine Learning Repository

    work page doi:10.24432/c5t01q 1988

  41. [41]

    Statlog (vehicle silhouettes) [dataset],

    P. Mowforth and B. Shepherd, “Statlog (vehicle silhouettes) [dataset],” https://doi.org/10.24432/C5HG6N, uCI Machine Learning Repository

    work page doi:10.24432/c5hg6n

  42. [42]

    Wine [dataset],

    S. Aeberhard and M. Forina, “Wine [dataset],” https://doi.org/10.24432/ C5PC7J, 1992, uCI Machine Learning Repository

    1992

  43. [43]

    Balance scale [dataset],

    R. Siegler, “Balance scale [dataset],” https://doi.org/10.24432/C5488X, 1976, uCI Machine Learning Repository

    work page doi:10.24432/c5488x 1976

  44. [44]

    Haberman’s survival [dataset],

    S. Haberman, “Haberman’s survival [dataset],” https://doi.org/10.24432/ C5XK51, 1976, uCI Machine Learning Repository

    1976

  45. [45]

    Wholesale customers [dataset],

    M. Cardoso, “Wholesale customers [dataset],” https://doi.org/10.24432/ C5030X, 2013, uCI Machine Learning Repository

    2013

  46. [46]

    Ilpd (indian liver patient dataset) [dataset],

    B. Ramana and N. Venkateswarlu, “Ilpd (indian liver patient dataset) [dataset],” https://doi.org/10.24432/C5D02C, 2022, uCI Machine Learn- ing Repository

    work page doi:10.24432/c5d02c 2022

  47. [47]

    Gradient-based learning applied to document recognition,

    Y . LeCun, L. Bottou, Y . Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,”Proceedings of the IEEE, vol. 86, no. 11, pp. 2278–2324, 1998. [Online]. Available: http://yann.lecun.com/exdb/mnist/

    1998

  48. [48]

    The relationship between the absolute deviation from a quantile and gini’s mean difference,

    S. Yitzhaki and P. Lambert, “The relationship between the absolute deviation from a quantile and gini’s mean difference,”METRON, vol. 71, pp. 97–104, 2013

    2013

  49. [49]

    Using the nystr ¨om method to speed up kernel machines,

    C. K. I. Williams and M. Seeger, “Using the nystr ¨om method to speed up kernel machines,” inAdvances in Neural Information Processing Systems (NIPS), 2001, pp. 682–688

    2001

  50. [50]

    Visualizing data using t-sne,

    L. van der Maaten and G. Hinton, “Visualizing data using t-sne,”Journal of Machine Learning Research, vol. 9, no. Nov, pp. 2579–2605, 2008

    2008

  51. [51]

    A global geometric framework for nonlinear dimensionality reduction,

    J. B. Tenenbaum, V . de Silva, and J. C. Langford, “A global geometric framework for nonlinear dimensionality reduction,”Science, vol. 290, no. 5500, pp. 2319–2323, 2000

    2000

  52. [52]

    & Großberger, L

    L. McInnes, J. Healy, N. Saul, and L. Großberger, “Umap: Uniform manifold approximation and projection,”Journal of Open Source Software, vol. 3, no. 29, p. 861, 2018. [Online]. Available: https://doi.org/10.21105/joss.00861

    work page doi:10.21105/joss.00861 2018

  53. [53]

    G. H. Hardy, J. E. Littlewood, and G. P ´olya,Inequalities, 2nd ed. Cambridge: Cambridge University Press, 1952. VII. APPENDIXA: PROOF Proof of Proposition 1: (ı)It comes: ∥λx∥G =D G(λx,0 d) = dX j=1 λxj R(λxj)− d+ 1 2 SinceR(·)is an homogeneous function of degree zero ifλ≥ 0,i.e.R(λx) =R(x), then: ∥λx∥G = dX j=1 λxj R(xj)− d+ 1 2 =λ   dX j=1 xj R(xj)− ...

    1952